A Review on Fractional Differential Equations and a Numerical Method to Solve Some Boundary Value Problems

*María I. Troparevsky, Silvia A. Seminara and Marcela A. Fabio*

#### **Abstract**

Fractional differential equations can describe the dynamics of several complex and nonlocal systems with memory. They arise in many scientific and engineering areas such as physics, chemistry, biology, biophysics, economics, control theory, signal and image processing, etc. Particularly, nonlinear systems describing different phenomena can be modeled with fractional derivatives. Chaotic behavior has also been reported in some fractional models. There exist theoretical results related to existence and uniqueness of solutions to initial and boundary value problems with fractional differential equations; for the nonlinear case, there are still few of them. In this work we will present a summary of the different definitions of fractional derivatives and show models where they appear, including simple nonlinear systems with chaos. Existing results on the solvability of classical fractional differential equations and numerical approaches are summarized. Finally, we propose a numerical scheme to approximate the solution to linear fractional initial value problems and boundary value problems.

**Keywords:** fractional derivatives, fractional differential equations, wavelet decomposition, numerical approximation

#### **1. Introduction**

Fractional calculus is the theory of integrals and derivatives of arbitrary real (and even complex) order and was first suggested in works by mathematicians such as Leibniz, L'Hôpital, Abel, Liouville, Riemann, etc. The importance of fractional derivatives for modeling phenomena in different branches of science and engineering is due to their nonlocality nature, an intrinsic property of many complex systems. Unlike the derivative of integer order, fractional derivatives do not take into account only local characteristics of the dynamics but considers the global evolution of the system; for that reason, when dealing with certain phenomena, they provide more accurate models of real-world behavior than standard derivatives.

To illustrate this fact, we will retrieve an example from [1]. Recall the relationship between stress *σ*ð Þ*t* and strain *ε*ð Þ*t* in a material under the influence of external forces: *Nonlinear Systems - Theoretical Aspects and Recent Applications*

$$
\sigma(t) = \eta \frac{d}{dt} \varepsilon(t) \tag{1}
$$

*C* 0*D<sup>α</sup>*

*DOI: http://dx.doi.org/10.5772/intechopen.86273*

was defined as

and

*CF* 0*D<sup>α</sup>*

function.

**5**

*<sup>t</sup>* ½ � *<sup>f</sup>* ðÞ¼ *<sup>t</sup> RL*

*CF* 0*D<sup>α</sup>*

could be changed by any value *a*∈½ Þ �∞*; t* , i.e.,

*<sup>t</sup>* ½ � *f* ðÞ¼ *t*

*CF <sup>a</sup> D<sup>α</sup>*

*C <sup>a</sup> D<sup>α</sup>*

following (convenient) properties:

*Riemann-Lioville sense*:

*<sup>t</sup>* ½ �¼ *k* 0, for any constant *k*.

*ABR* 0*D<sup>α</sup>*

*ABC* 0*D<sup>α</sup>*

replacing the exponential by *<sup>E</sup>α*ð Þ¼ *<sup>z</sup>* ∑∞

0*D<sup>α</sup>*

*<sup>t</sup>* ½ � *f* ðÞ¼ *t*

for *M*ð Þ *α* is a normalizing factor verifying *M*ð Þ¼ 0 *M*ð Þ¼ 1 1.

1 *Γ*ð Þ *n* � *α*

*<sup>t</sup>* ½ � *f* ðÞ¼ *t*

lim*α*!1 *CF* 0*D<sup>α</sup>*

lim*α*!0 *CF* 0*D<sup>α</sup>*

*<sup>t</sup>* ½ � *f* ðÞ¼ *t*

*<sup>t</sup>* ½ � *f* ðÞ¼ *t*

*<sup>t</sup>* ½ � *f* ð Þ�*t* ∑

*A Review on Fractional Differential Equations and a Numerical Method to Solve Some…*

∞ *k*¼0

More recently, the *Caputo-Fabrizio fractional derivative of order α,* with *α* ∈½ Þ 0*;* 1 ,

ð*t* 0 *f* 0 ð Þ*s e* �*α*ð Þ *<sup>t</sup>*�*<sup>s</sup>*

*M*ð Þ *α* 1 � *α*

Let us point out that, both in Eqs. (7) and (9), the lower limit in the integral

ð*t a*

*M*ð Þ *α* 1 � *α*

In [7] the authors prove that the operator defined in Eq. (9) verifies the

ð*t a f* 0 ð Þ*s e* �*α*ð Þ *<sup>t</sup>*�*<sup>s</sup>*

*<sup>t</sup>* ½ � *<sup>f</sup>* ðÞ¼ *<sup>t</sup> df*

Caputo-Fabrizio definition was then generalized by Atangana and Baleanu, who

gave the following definition of the *Atangana-Baleanu fractional derivative in*

*d dt* ð*t* 0

ð*t* 0 *f*0

Other types of fractional derivatives are Grünwald-Letnikov's, Hadamard's, Weyl's, etc. In every definition it is clear that fractional derivative operators are not local, since they need the information of *f* in a whole interval of integration. When defined with 0 as lower limit of integration, as we did, function *f* is usually assumed

Authors choose one definition or the other depending on the real-world phenomena they need to model; the scope of application of each operator is still unknown, and, in relation to some of them, there is neither an agreement about

*M*ð Þ *α* 1 � *α*

and the *Atangana-Baleanu fractional derivative in Caputo sense*

*M*ð Þ *α* 1 � *α*

to be causal (i.e., *f t*ðÞ� 0 for *t* , 0), but this limit can also be changed.

*t k*�*α <sup>Γ</sup>*ð Þ *<sup>k</sup>* � *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> *<sup>f</sup>*

ð Þ *<sup>t</sup>* � *<sup>s</sup> <sup>n</sup>*�1�*<sup>α</sup> dn*

ð Þ*<sup>k</sup>* <sup>0</sup><sup>þ</sup> ð Þ*:* (8)

<sup>1</sup>�*<sup>α</sup> ds,* (9)

*dsn* ½ � *f s*ð Þ *ds* (10)

<sup>1</sup>�*<sup>α</sup> ds:* (11)

*dt :* (12)

� �*ds* (14)

� �*ds,* (15)

*Γ α*ð Þ *<sup>k</sup>*þ<sup>1</sup> , the generalized Mittag-Leffler

*<sup>t</sup>* ½ � *f* ðÞ¼ *t f t*ðÞ� *f*ð Þ 0 *:* (13)

1 � *α*

1 � *α*

*f s*ð Þ*E<sup>α</sup>* � *<sup>α</sup>*ð Þ *<sup>t</sup>* � *<sup>s</sup> <sup>α</sup>*

ð Þ*<sup>s</sup> <sup>E</sup><sup>α</sup>* � *<sup>α</sup>*ð Þ *<sup>t</sup>* � *<sup>s</sup> <sup>α</sup>*

*<sup>k</sup>*¼<sup>0</sup> *<sup>z</sup><sup>k</sup>*

is the Newton's law for a viscous liquid, with *η* the viscosity of the material, and

$$
\sigma(t) = E\varepsilon(t) \tag{2}
$$

is Hooke's law for an elastic solid, with *E* the modulus of elasticity. We can rewrite both Eqs. (1) and (2) as

$$
\sigma(t) = \nu \frac{d^a}{dt^a} \varepsilon(t) \tag{3}
$$

with *α* ¼ 0 for elastic solids and *α* ¼ 1 for a viscous liquid. But, in practice, there exist *viscoelastic* materials that have a behavior intermediate between an elastic solid and a viscous liquid, and it may be convenient to give sense to the operator *<sup>d</sup><sup>α</sup> dt<sup>α</sup>* if 0 , *α* , 1.

There exist various definitions of fractional derivatives. All of them involve integral operators with different regularity properties, and some of them have singular kernels.

Next, we will briefly review the most frequently fractional derivatives cited in the bibliography (see [2] for a more complete review and [1, 3–6] for rigorous theoretical expositions and calculation methods).

The classical Cauchy formula for the *n*-fold iterated integral, with *n* ∈ N, is

$$\, \_0I\_t^n[f](t) = \frac{1}{(n-1)!} \int\_0^t (t-s)^{n-1} f(s)ds. \tag{4}$$

Recalling that gamma function verifies *nΓ*ð Þ¼ *n n*!, an immediate generalization of this formula for a real order *α* is

$$\, \_0I\_t^a[f](t) = \frac{1}{\Gamma(a)} \int\_0^t (t-s)^{a-1} f(s)ds,\tag{5}$$

known as Riemann-Liouville fractional integral operator of order *α* (the term "fractional" is misleading but has a historical origin). From this, the Riemann-Liouville fractional derivative of order *α*, with *n* � 1 , *α* , *n*, is defined as

$${}^{RL}\_{0}D\_{t}^{a}[f](t) = \frac{1}{\Gamma(n-a)} \frac{d^{n}}{dt^{n}} \int\_{0}^{t} \left(t-s\right)^{n-a-1} f(s)ds.\tag{6}$$

while the Caputo fractional derivative of order *α* is defined as

$$\, \_0^C D\_t^a[f](t) = \frac{1}{\Gamma(n-a)} \int\_0^t (t-s)^{n-1-a} \frac{d^n}{ds^n}[f(s)]ds. \tag{7}$$

Both Eqs. (6) and (7) define left inverse operators for the integral operator of Riemann-Liouville of order *α* and are associated to the idea that "deriving *α* times may be thought as integrating *n* � *α* times and deriving *n* times." Of course these definitions aren't equivalent: clearly the domains of the operators *RL* 0*D<sup>α</sup> t* ½ �*:* and *<sup>C</sup>* 0*D<sup>α</sup> <sup>t</sup>* ½ �*:* are different; because of the different hypothesis about *f*, we need to impose to guarantee their existence. Besides that, with the appropriate conditions for *f*,

*A Review on Fractional Differential Equations and a Numerical Method to Solve Some… DOI: http://dx.doi.org/10.5772/intechopen.86273*

$${}^{C}\_{0}D\_{t}^{a}[f](t) = {}^{RL}\_{0}D\_{t}^{a}[f](t) - \sum\_{k=0}^{\infty} \frac{t^{k-a}}{\Gamma(k-a+1)} f^{(k)}(\mathbf{0}^{+}).\tag{8}$$

More recently, the *Caputo-Fabrizio fractional derivative of order α,* with *α* ∈½ Þ 0*;* 1 , was defined as

$${}^{CF}\_{0}D\_{t}^{a}[f](t) = \frac{\mathcal{M}(a)}{\mathbf{1} - a} \int\_{0}^{t} f'(s)e^{-\frac{a(t-s)}{\mathbf{1}-a}}ds,\tag{9}$$

for *M*ð Þ *α* is a normalizing factor verifying *M*ð Þ¼ 0 *M*ð Þ¼ 1 1.

Let us point out that, both in Eqs. (7) and (9), the lower limit in the integral could be changed by any value *a*∈½ Þ �∞*; t* , i.e.,

$$\, \_a^C D\_t^a [f](t) = \frac{1}{\Gamma(n-a)} \int\_a^t \left(t-s\right)^{n-1-a} \frac{d^n}{ds^n} [f(s)] ds \tag{10}$$

and

*σ*ðÞ¼ *t η*

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

*d*

is the Newton's law for a viscous liquid, with *η* the viscosity of the material, and

*dα*

with *α* ¼ 0 for elastic solids and *α* ¼ 1 for a viscous liquid. But, in practice, there exist *viscoelastic* materials that have a behavior intermediate between an elastic solid and a viscous liquid, and it may be convenient to give sense to the operator *<sup>d</sup><sup>α</sup>*

There exist various definitions of fractional derivatives. All of them involve integral operators with different regularity properties, and some of them have

Next, we will briefly review the most frequently fractional derivatives cited in the bibliography (see [2] for a more complete review and [1, 3–6] for rigorous

> ð*t* 0

Recalling that gamma function verifies *nΓ*ð Þ¼ *n n*!, an immediate generalization

ð*t* 0

known as Riemann-Liouville fractional integral operator of order *α* (the term "fractional" is misleading but has a historical origin). From this, the Riemann-Liouville fractional derivative of order *α*, with *n* � 1 , *α* , *n*, is defined as

> *dn dtn* ð*t* 0

ð*t* 0

Both Eqs. (6) and (7) define left inverse operators for the integral operator of Riemann-Liouville of order *α* and are associated to the idea that "deriving *α* times may be thought as integrating *n* � *α* times and deriving *n* times." Of course these definitions aren't equivalent: clearly the domains of the operators *RL*

*<sup>t</sup>* ½ �*:* are different; because of the different hypothesis about *f*, we need

ð Þ *<sup>t</sup>* � *<sup>s</sup> <sup>n</sup>*�<sup>1</sup>

ð Þ *<sup>t</sup>* � *<sup>s</sup> <sup>α</sup>*�<sup>1</sup>

ð Þ *<sup>t</sup>* � *<sup>s</sup> <sup>n</sup>*�*α*�<sup>1</sup>

ð Þ *<sup>t</sup>* � *<sup>s</sup> <sup>n</sup>*�1�*<sup>α</sup> dn*

The classical Cauchy formula for the *n*-fold iterated integral, with *n* ∈ N, is

1 ð Þ *n* � 1 !

> 1 *Γ α*ð Þ

1 *Γ*ð Þ *n* � *α*

while the Caputo fractional derivative of order *α* is defined as

1 *Γ*ð Þ *n* � *α*

to impose to guarantee their existence. Besides that, with the appropriate

is Hooke's law for an elastic solid, with *E* the modulus of elasticity. We can

*σ*ðÞ¼ *t ν*

rewrite both Eqs. (1) and (2) as

theoretical expositions and calculation methods).

0*I n <sup>t</sup>* ½ � *f* ðÞ¼ *t*

> 0*I α <sup>t</sup>* ½ � *f* ðÞ¼ *t*

*<sup>t</sup>* ½ � *f* ðÞ¼ *t*

*<sup>t</sup>* ½ � *f* ðÞ¼ *t*

of this formula for a real order *α* is

*RL* 0*D<sup>α</sup>*

*C* 0*D<sup>α</sup>*

and *<sup>C</sup>* 0*D<sup>α</sup>*

**4**

conditions for *f*,

0 , *α* , 1.

singular kernels.

*dt <sup>ε</sup>*ð Þ*<sup>t</sup>* (1)

*dt<sup>α</sup> <sup>ε</sup>*ð Þ*<sup>t</sup>* (3)

*f s*ð Þ*ds:* (4)

*f s*ð Þ*ds,* (5)

*f s*ð Þ*ds:* (6)

*dsn* ½ � *f s*ð Þ *ds:* (7)

0*D<sup>α</sup> t* ½ �*:*

*dt<sup>α</sup>* if

*σ*ðÞ¼ *t Eε*ð Þ*t* (2)

$$\, \, \_a^{CF} D\_t^a[f](t) = \frac{\mathcal{M}(a)}{\mathbf{1} - a} \int\_a^t f'(s) e^{-\frac{a(t - s)}{\mathbf{1} - a}} ds. \tag{11}$$

In [7] the authors prove that the operator defined in Eq. (9) verifies the following (convenient) properties:

*CF* 0*D<sup>α</sup> <sup>t</sup>* ½ �¼ *k* 0, for any constant *k*.

$$\lim\_{a \to 1} \, ^{\text{CF}}\_0 D\_t^a[f](t) = \frac{d f}{dt} . \tag{12}$$

$$\lim\_{a \to 0} \, ^{CF}\_0 D\_t^a[f](t) = f(t) - f(0). \tag{13}$$

Caputo-Fabrizio definition was then generalized by Atangana and Baleanu, who gave the following definition of the *Atangana-Baleanu fractional derivative in Riemann-Lioville sense*:

$${}^{ABR}\_{0}D\_{t}^{a}[f](t) = \frac{M(a)}{\mathbf{1} - a} \frac{d}{dt} \int\_{0}^{t} f(s)E\_{a}\left(-\frac{a(t-s)^{a}}{\mathbf{1} - a}\right)ds\tag{14}$$

and the *Atangana-Baleanu fractional derivative in Caputo sense*

$${}^{ABC}\_{0}D\_{t}^{a}[f](t) = \frac{\mathcal{M}(a)}{\mathbf{1} - a} \Big|\_{0}^{t} f'(s)E\_{a}\left(-\frac{a(t-s)^{a}}{\mathbf{1} - a}\right)ds,\tag{15}$$

replacing the exponential by *<sup>E</sup>α*ð Þ¼ *<sup>z</sup>* ∑∞ *<sup>k</sup>*¼<sup>0</sup> *<sup>z</sup><sup>k</sup> Γ α*ð Þ *<sup>k</sup>*þ<sup>1</sup> , the generalized Mittag-Leffler function.

Other types of fractional derivatives are Grünwald-Letnikov's, Hadamard's, Weyl's, etc. In every definition it is clear that fractional derivative operators are not local, since they need the information of *f* in a whole interval of integration. When defined with 0 as lower limit of integration, as we did, function *f* is usually assumed to be causal (i.e., *f t*ðÞ� 0 for *t* , 0), but this limit can also be changed.

Authors choose one definition or the other depending on the real-world phenomena they need to model; the scope of application of each operator is still unknown, and, in relation to some of them, there is neither an agreement about

whether it is appropriate or not to call them derivatives (see, for a discussion on this topic, [8, 9]) nor what are the criteria to decide on it ([10, 11]).

Numerical methods have also been proposed to obtain approximate solution to

*A Review on Fractional Differential Equations and a Numerical Method to Solve Some…*

In [6] some numerical approximations to solutions to different fractional differential equations are presented and experimentally verified on various examples, and in [24, 25] complete surveys on numerical methods are offered. But numerous articles appear continuously with new approximation methods: we finish this section commenting briefly some works on numerical methods of quite different

In [26] a local fractional natural homotopy perturbation method is proposed to

A method based on a semi-discrete finite difference approximation in time, and

Galerkin finite element method in space, is proposed in [27] to solve fractional

In [28] a new numerical approximation of Atangana-Baleanu integral, as the summation of the average of the given function and its fractional integral in

Semi-discrete finite element methods are introduced in [29] to solve diffusion equations, and implicit numerical algorithms for the case of spatial and temporal fractional derivatives appeared in [30]. A high-speed numerical scheme for fractional differentiation and fractional integration is proposed in [31]. In [32], a new numerical method to solve partial differential equations involving Caputo derivatives of fractional variable order is obtained in terms of standard (integer order)

In [33], a discrete form is proposed for solving time fractional convectiondiffusion equation. The Laplace transform is used to solve fractional differential

Finally, in Section 4, we will present a numerical method we have developed, based on wavelets, to solve initial and boundary value problems with linear frac-

The purpose of this section is to highlight the role of fractional derivatives

mathematical models of different fields, found in the recent literature. In each of

are used to indistinctly represent any of the fractional derivatives, whose type is

In [35] the authors review the evolution of the general fractional equation:

for *a* . 0*,* 0 , *β* ≤ 2, where *x*∈*S*⊂ R and *t*∈ R . 0 denote the space and time variables. This equation is obtained from the classical D'Alembert wave equation by replacing the second-order time derivative with the Caputo fractional derivative of order *β* ∈ð � 0*;* 2 *:* The authors show that, for 1 , *β* , 2, the behavior of the fundamental solutions turns out to be intermediate between diffusion (for a viscous fluid)

*∂*2 *u*

*∂<sup>β</sup>u <sup>∂</sup>t<sup>β</sup>* <sup>¼</sup> *<sup>a</sup>*

and wave propagation (for an elastic solid), thus justifying the attribute of

*<sup>∂</sup>t<sup>α</sup>* and *<sup>D</sup><sup>α</sup> t*

*<sup>∂</sup>x*<sup>2</sup> (19)

**2. Some mathematical models with fractional derivatives**

for modeling certain real evolution processes. We enumerate several

them, the fractional order of derivation is justified by the nature of the phenomenon that is described. Usually, in the papers, both the symbols *<sup>∂</sup><sup>α</sup>*

found the solution to partial fractional differential equations as a series.

partial differential equations arising in neuronal dynamics.

fractional differential equations.

*DOI: http://dx.doi.org/10.5772/intechopen.86273*

Riemann-Liouville sense, is proposed.

nature.

derivatives.

equations in [34].

clarified in the text.

fractional diffusive waves.

**7**

tional differential equations.

Caputo and Fabrizio ([12]) proposed the following terms to recognize if an integral operator merits to be called a fractional derivative:


$$\text{5. For } n \in \mathbb{N}, \ 0 \le a \le 1: D\_t^a \left[ D\_t^n[f] \right](t) = D\_t^n \left[ D\_t^a[f] \right](t).$$

6.*D<sup>α</sup> <sup>t</sup>* ½ � *f* ð Þ*t* must depend on the past history of *f*.

Many applications of fractional calculus have been reported in areas as diverse as diffusion problems, hydraulics, potential theory, control theory, electrochemistry, viscoelasticity, and nanotechnology, among others (see, e.g., [13, 14], for a profuse listing of application areas). In Section 2 we will briefly exemplify a few of these applications, in quite different fields, and in Section 3, we will even mention some cases of fractional nonlinear systems which exhibit chaotic behavior.

Theoretical results concerning existence and uniqueness of solutions to fractional differential equations have been also developed.

In [15–17] the authors state conditions to guarantee the existence and uniqueness of solution to problems like

$$\begin{cases} \,^C\_0 D\_t^a[f](t) = F(t, f(t))te(0, T), T \le \infty\\ \quad \text{initial or boundary conditions} \end{cases} \tag{16}$$

or

$$\begin{cases} \, ^{RL}\_{0}D\_{t}^{a}[f](t) = F(t, f(t))t \epsilon(0, T), T \le \infty \\ \quad \text{initial or boundary conditions} \end{cases} \tag{17}$$

for 0 , *α* , 2. After rewriting the equation as an integral equation with a kernel whose norm is bounded in a proper Banach space, they use generalizations of the fixed-point theorem. The function *F*, besides being continuous, must satisfy certain conditions that substitute the classical Lipschitz's one.

Similar results are stated in [18] for a coupled system of fractional differential equations involving Riemann-Liouville derivatives.

Analytical calculus of fractional operators is, in general, difficult. In [19–21] a few examples of quite different analytical methods are presented.

In [22, 23], existence and uniqueness, for the solution to a simple case,

$$\int \,^{\rm CF}\_{0} D\_{\rm t} a[f](t) + \mathfrak{H}(t) = \mathfrak{g}(t) f(\mathfrak{O}) = \mathfrak{O} \tag{18}$$

are proved, and explicit formulae are presented when *g* is continuous, causal, and null at the origin. The case of Caputo derivative is also considered. In all cases, the computation of the primitive of the data function is required.

*A Review on Fractional Differential Equations and a Numerical Method to Solve Some… DOI: http://dx.doi.org/10.5772/intechopen.86273*

Numerical methods have also been proposed to obtain approximate solution to fractional differential equations.

In [6] some numerical approximations to solutions to different fractional differential equations are presented and experimentally verified on various examples, and in [24, 25] complete surveys on numerical methods are offered. But numerous articles appear continuously with new approximation methods: we finish this section commenting briefly some works on numerical methods of quite different nature.

In [26] a local fractional natural homotopy perturbation method is proposed to found the solution to partial fractional differential equations as a series.

A method based on a semi-discrete finite difference approximation in time, and Galerkin finite element method in space, is proposed in [27] to solve fractional partial differential equations arising in neuronal dynamics.

In [28] a new numerical approximation of Atangana-Baleanu integral, as the summation of the average of the given function and its fractional integral in Riemann-Liouville sense, is proposed.

Semi-discrete finite element methods are introduced in [29] to solve diffusion equations, and implicit numerical algorithms for the case of spatial and temporal fractional derivatives appeared in [30]. A high-speed numerical scheme for fractional differentiation and fractional integration is proposed in [31]. In [32], a new numerical method to solve partial differential equations involving Caputo derivatives of fractional variable order is obtained in terms of standard (integer order) derivatives.

In [33], a discrete form is proposed for solving time fractional convectiondiffusion equation. The Laplace transform is used to solve fractional differential equations in [34].

Finally, in Section 4, we will present a numerical method we have developed, based on wavelets, to solve initial and boundary value problems with linear fractional differential equations.

#### **2. Some mathematical models with fractional derivatives**

The purpose of this section is to highlight the role of fractional derivatives for modeling certain real evolution processes. We enumerate several mathematical models of different fields, found in the recent literature. In each of them, the fractional order of derivation is justified by the nature of the phenomenon that is described. Usually, in the papers, both the symbols *<sup>∂</sup><sup>α</sup> <sup>∂</sup>t<sup>α</sup>* and *<sup>D</sup><sup>α</sup> t* are used to indistinctly represent any of the fractional derivatives, whose type is clarified in the text.

In [35] the authors review the evolution of the general fractional equation:

$$\frac{\partial^{\beta}u}{\partial t^{\beta}} = a \, \frac{\partial^{2}u}{\partial \mathbf{x}^{2}}\tag{19}$$

for *a* . 0*,* 0 , *β* ≤ 2, where *x*∈*S*⊂ R and *t*∈ R . 0 denote the space and time variables. This equation is obtained from the classical D'Alembert wave equation by replacing the second-order time derivative with the Caputo fractional derivative of order *β* ∈ð � 0*;* 2 *:* The authors show that, for 1 , *β* , 2, the behavior of the fundamental solutions turns out to be intermediate between diffusion (for a viscous fluid) and wave propagation (for an elastic solid), thus justifying the attribute of fractional diffusive waves.

whether it is appropriate or not to call them derivatives (see, for a discussion on this

Caputo and Fabrizio ([12]) proposed the following terms to recognize if an

3. If the order of the derivative is a positive integer, the derivative must be the

*<sup>t</sup> D<sup>α</sup> <sup>t</sup>* ½ � *<sup>f</sup>* � �ð Þ*<sup>t</sup>* .

Many applications of fractional calculus have been reported in areas as diverse as diffusion problems, hydraulics, potential theory, control theory, electrochemistry, viscoelasticity, and nanotechnology, among others (see, e.g., [13, 14], for a profuse listing of application areas). In Section 2 we will briefly exemplify a few of these applications, in quite different fields, and in Section 3, we will even mention some

Theoretical results concerning existence and uniqueness of solutions to frac-

In [15–17] the authors state conditions to guarantee the existence and uniqueness

*<sup>t</sup>* ½ � *f* ðÞ¼ *t F t*ð Þ *; f t*ð Þ *tϵ*ð Þ 0*; T , T* , ∞ *initial or boundary conditions*

*<sup>t</sup>* ½ � *f* ðÞ¼ *t F t*ð Þ *; f t*ð Þ *tϵ*ð Þ 0*; T , T* , ∞ *initial or boundary conditions*

for 0 , *α* , 2. After rewriting the equation as an integral equation with a kernel whose norm is bounded in a proper Banach space, they use generalizations of the fixed-point theorem. The function *F*, besides being continuous, must satisfy certain

Similar results are stated in [18] for a coupled system of fractional differential

Analytical calculus of fractional operators is, in general, difficult. In [19–21] a

are proved, and explicit formulae are presented when *g* is continuous, causal, and null at the origin. The case of Caputo derivative is also considered. In all cases,

<sup>0</sup>*D*t*α*½ � *<sup>f</sup>* ð Þþ*<sup>t</sup>* <sup>β</sup>*f t*ðÞ¼ *g t*ð Þ*f*ð Þ¼ <sup>0</sup> <sup>0</sup> � (18)

In [22, 23], existence and uniqueness, for the solution to a simple case,

(16)

(17)

2. The fractional derivative of an analytic function must be analytic.

*<sup>t</sup>* ½ � *<sup>f</sup>* � �ðÞ¼ *<sup>t</sup> <sup>D</sup><sup>n</sup>*

4.If the order is null, the original function must be recovered.

cases of fractional nonlinear systems which exhibit chaotic behavior.

*<sup>t</sup> Dn*

*<sup>t</sup>* ½ � *f* ð Þ*t* must depend on the past history of *f*.

tional differential equations have been also developed.

*C* 0*D<sup>α</sup>*

*RL* 0*D<sup>α</sup>*

conditions that substitute the classical Lipschitz's one.

few examples of quite different analytical methods are presented.

the computation of the primitive of the data function is required.

equations involving Riemann-Liouville derivatives.

*CF*

(

(

topic, [8, 9]) nor what are the criteria to decide on it ([10, 11]).

integral operator merits to be called a fractional derivative:

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

1. The fractional derivative must be a linear operator.

classical one.

6.*D<sup>α</sup>*

or

**6**

5. For *n* ∈ N*,* 0 , *α* , 1 : *D<sup>α</sup>*

of solution to problems like

In [36] another approach for time fractional wave (Eq. (19)) is proposed. It is solved, for 0 , *β* , 1 and special initial conditions, by the method of separation of variables.

In [37] the authors study the particular linear fractional Klein-Gordon equation:

$$\begin{cases} \frac{\partial^a u}{\partial t^a} - \frac{\partial^2 u}{\partial x^2} + u = 6x^3t + \left(\mathbf{x}^3 - 6\mathbf{x}\right)t^3 & t > 0, \mathbf{x} \in \mathbb{R} \\\\ u(\mathbf{x}, \mathbf{0}) = \mathbf{0} \\\\ u\_t(\mathbf{x}, \mathbf{0}) = \mathbf{0} \end{cases} \tag{20}$$

In [12] the authors use fractional derivatives to model the magnetic hysteresis,

ðÞþ*<sup>t</sup>* <sup>1</sup> � �*M t*ð Þ� *<sup>θ</sup>*ð Þ*<sup>t</sup> M t*ð Þ� *<sup>C</sup>*0*D<sup>α</sup>*

for *λ* . 0*,H t*ð Þ is the magnetic excitation field, *M t*ð Þ is the magnetization vector, *θ*ð Þ*t* is the temperature, *θ<sup>c</sup>* is the Curie temperature below which the hysteresis is observed, and *C*<sup>0</sup> is the tensor with the constitutive properties of the magnetic

derivative with the one that results when the derivative is the Caputo-Fabrizio one. By numerical simulations they obtain examples of the classical hysteresis cycles and conclude that Caputo derivative expresses a stronger memory than the Caputo-

Chaos theory is also an area where fractional derivatives play an important role. In this section we comment on some nonlinear systems modeled with fractional

*<sup>α</sup>*½ � *<sup>x</sup>* ðÞ¼ *<sup>t</sup> <sup>σ</sup> y t*ðÞ� *<sup>α</sup> x t*ð Þ

*<sup>α</sup>*½ � *<sup>z</sup>* ðÞ¼ *<sup>t</sup> x t*ð Þ *y t*ðÞ� *<sup>β</sup> z t*ð Þ

Under certain assumptions on the physical problem, they proved existence of solutions, and, by means of an iterative algorithm, numerical evidence of chaos is shown when 0*:*25 , *α* , 0*:*3 and 0*:*4 , *α* , 0*:*5 for the usual set of parameters

In [42] a three-dimensional fractional-order dynamical system for cancer growth is proposed replacing the standard derivatives in the evolution equations:

> *x*\_ <sup>1</sup>ðÞ¼ *t x*1ðÞ�*t A x*1ð Þ*t x*2ð Þ�*t B x*1ð Þ*t x*3ð Þ*t x*\_ <sup>2</sup>ðÞ¼ *t Cx*2ð Þ*t* ð Þ� 1 � *x*2ð Þ*t D x*1ð Þ*t x*2ð Þ*t*

by the Caputo-Fabrizio and the Atangana-Baleanu (Caputo sense) derivatives. The system parameters are related to the rate of change in the population of the different cells: healthy and tumor ones. The authors prove that the system has a unique solution and show that the system exhibits chaos for a proper choice of the

*<sup>x</sup>*1ðÞþ*<sup>t</sup> <sup>F</sup>* � *G x*1ð Þ*<sup>t</sup> <sup>x</sup>*2ðÞ�*<sup>t</sup> H x*3ð Þ*<sup>t</sup>*

*<sup>α</sup>*½ � *<sup>y</sup>* ðÞ¼ *<sup>t</sup> <sup>ρ</sup> x t*ðÞ� *x t*ð Þ *z t*ðÞ� *y t*ð Þ

In [41] the authors studied a system based on the classical Lorenz one, but described by the Atangana-Baleanu fractional derivative (in the Caputo sense) with

These are just a few examples of the huge variety of problems that can be modeled by means of fractional differential equations. The nonlocality of the associated operators is the key to the success in the description of these

*<sup>t</sup>* ½ � *M* ð Þ*t* (23)

(24)

(25)

*<sup>t</sup>* is the Caputo fractional

a phenomenon where the "memory" of the ferromagnetic material is crucial. They use a nonlinear model for the constitutive law of an isotropic ferromagnetic

*A Review on Fractional Differential Equations and a Numerical Method to Solve Some…*

*<sup>λ</sup>H t*ðÞ¼ *<sup>θ</sup><sup>c</sup> <sup>M</sup>*<sup>2</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.86273*

**3. Some simple systems exhibiting chaos**

derivative, recently published, that exhibit chaos.

*ABC* <sup>0</sup>*Dt*

8 ><

>:

<sup>3</sup> *,* and *β* ¼ 28.

8 >>><

>>>:

parameters values and initial conditions.

*ABC* <sup>0</sup>*Dt*

*ABC* <sup>0</sup>*Dt*

*<sup>x</sup>*\_ <sup>3</sup>ðÞ¼ *<sup>t</sup> <sup>E</sup> <sup>x</sup>*1ð Þ*<sup>t</sup> <sup>x</sup>*3ð Þ*<sup>t</sup>*

material. They compare the resulting behavior when *D<sup>α</sup>*

material:

Fabrizio operator.

phenomena.

0 , *α* , 1:

*<sup>σ</sup>* <sup>¼</sup> <sup>10</sup>*, <sup>ρ</sup>* <sup>¼</sup> <sup>8</sup>

**9**

considering the fractional Caputo derivative with 1 , *α* ≤2. They achieve a numerical solution by variational iteration method and multivariate Padé approximation.

In [38] the following mathematical model, using Fick's law of diffusion, is developed to study the effect of fractional advection diffusion equation (cross flow) for the calcium profile, considering the Caputo fractional derivative (0 , *α* ≤1):

$$\begin{cases} \quad \frac{\partial^a C}{\partial t^a} = D \frac{\partial^2 C}{\partial x^2} & x \ge 0, t \ge 0 \\ \quad C(x, 0) = C\_0 \\ \quad \lim\_{x \to +\infty} C(x, t) = C\_\infty \end{cases} \tag{21}$$

Here, the calcium concentration *C x*ð Þ *; t* varies in time and space, *D* is a diffusion constant, and *C*<sup>∞</sup> is calcium concentration at infinity and is assumed that, at initial state of time and at a long distance, calcium concentration vanishes or becomes zero. The authors note that the physical parameter *α* characterizes the cytosolic calcium ion in astrocytes.

In [39] the authors explain that arteries, like other soft tissues, exhibit viscoelastic behavior and part of the mechanical energy transferred to them is dissipative (viscosity) and the other part is stored in a reversible form (elasticity). They modified the standard model by a fractional-order one and test it in human arterial segments. They conclude that fractional derivatives, in Riemann-Liouville sense, are a good alternative to model arterial viscosity.

The generalized Voigt model consists of a spring in parallel with two *springpots* (a neologism for a model that is between a spring—purely elastic—and a dashpot, purely viscous) of fractional orders *α* and *β*. The governing fractional-order differential equation is

$$
\sigma(t) = E\_0 \varepsilon(t) + \eta\_1 \frac{\partial^a \varepsilon(t)}{\partial t^a} + \eta\_2 \frac{\partial^\beta \varepsilon(t)}{\partial \mathbf{x}^\beta} \tag{22}
$$

where *E*<sup>0</sup> is the elastic constant for a spring and *η*<sup>1</sup> and *η*<sup>2</sup> represent the viscosities of two springpots in parallel with the spring.

In Ref. [40] the authors developed an accurate and efficient numerical method for the fractional-order standard model described by Eq. (22) with *α* ¼ *β*, combining a fast convolution method with the spectral element discretization based on a general Jacobi polynomial basis that can be used to generate 3D polymorphic high-order elements. In that way they model complicated arterial geometries, such as patient-specific aneurysms, and apply it to 3D fluid-structure interaction simulations.

*A Review on Fractional Differential Equations and a Numerical Method to Solve Some… DOI: http://dx.doi.org/10.5772/intechopen.86273*

In [12] the authors use fractional derivatives to model the magnetic hysteresis, a phenomenon where the "memory" of the ferromagnetic material is crucial. They use a nonlinear model for the constitutive law of an isotropic ferromagnetic material:

$$
\lambda H(t) = \theta\_c \left( M^2(t) + \mathbf{1} \right) M(t) - \theta(t) M(t) - C\_0 D\_t^a[M](t) \tag{23}
$$

for *λ* . 0*,H t*ð Þ is the magnetic excitation field, *M t*ð Þ is the magnetization vector, *θ*ð Þ*t* is the temperature, *θ<sup>c</sup>* is the Curie temperature below which the hysteresis is observed, and *C*<sup>0</sup> is the tensor with the constitutive properties of the magnetic material. They compare the resulting behavior when *D<sup>α</sup> <sup>t</sup>* is the Caputo fractional derivative with the one that results when the derivative is the Caputo-Fabrizio one. By numerical simulations they obtain examples of the classical hysteresis cycles and conclude that Caputo derivative expresses a stronger memory than the Caputo-Fabrizio operator.

These are just a few examples of the huge variety of problems that can be modeled by means of fractional differential equations. The nonlocality of the associated operators is the key to the success in the description of these phenomena.

#### **3. Some simple systems exhibiting chaos**

Chaos theory is also an area where fractional derivatives play an important role. In this section we comment on some nonlinear systems modeled with fractional derivative, recently published, that exhibit chaos.

In [41] the authors studied a system based on the classical Lorenz one, but described by the Atangana-Baleanu fractional derivative (in the Caputo sense) with 0 , *α* , 1:

$$\begin{cases} \begin{aligned} \,^{ABC}\_{0}D\_{t}^{a}[\mathbf{x}](t) &= & \sigma \,\boldsymbol{\chi}(t) - a \,\boldsymbol{\varkappa}(t) \\ \,^{ABC}\_{0}D\_{t}^{a}[\mathbf{y}](t) &= & \rho \,\boldsymbol{\varkappa}(t) - \boldsymbol{\varkappa}(t) \,\boldsymbol{\varkappa}(t) - \boldsymbol{\jmath}(t) \\ \,^{ABC}\_{0}D\_{t}^{a}[\mathbf{z}](t) &= & \boldsymbol{\varkappa}(t) \,\boldsymbol{\jmath}(t) - \boldsymbol{\beta} \,\boldsymbol{\varkappa}(t) \end{aligned} \end{cases} \tag{24}$$

Under certain assumptions on the physical problem, they proved existence of solutions, and, by means of an iterative algorithm, numerical evidence of chaos is shown when 0*:*25 , *α* , 0*:*3 and 0*:*4 , *α* , 0*:*5 for the usual set of parameters *<sup>σ</sup>* <sup>¼</sup> <sup>10</sup>*, <sup>ρ</sup>* <sup>¼</sup> <sup>8</sup> <sup>3</sup> *,* and *β* ¼ 28.

In [42] a three-dimensional fractional-order dynamical system for cancer growth is proposed replacing the standard derivatives in the evolution equations:

$$\begin{cases}
\dot{\boldsymbol{x}}\_{1}(t) = & \boldsymbol{\varkappa}\_{1}(t) - A \,\boldsymbol{\varkappa}\_{1}(t) \,\boldsymbol{\varkappa}\_{2}(t) - B \,\boldsymbol{\varkappa}\_{1}(t) \boldsymbol{\varkappa}\_{3}(t) \\
\dot{\boldsymbol{\varkappa}}\_{2}(t) = & \boldsymbol{\mathcal{C}} \boldsymbol{\varkappa}\_{2}(t) (\mathbf{1} - \boldsymbol{\varkappa}\_{2}(t)) - D \,\boldsymbol{\varkappa}\_{1}(t) \,\boldsymbol{\varkappa}\_{2}(t) \\
\dot{\boldsymbol{\varkappa}}\_{3}(t) = & E \frac{\boldsymbol{\varkappa}\_{1}(t) \boldsymbol{\varkappa}\_{3}(t)}{\boldsymbol{\varkappa}\_{1}(t) + F} - G \,\boldsymbol{\varkappa}\_{1}(t) \,\boldsymbol{\varkappa}\_{2}(t) - H \,\boldsymbol{\varkappa}\_{3}(t)
\end{cases} \tag{25}$$

by the Caputo-Fabrizio and the Atangana-Baleanu (Caputo sense) derivatives. The system parameters are related to the rate of change in the population of the different cells: healthy and tumor ones. The authors prove that the system has a unique solution and show that the system exhibits chaos for a proper choice of the parameters values and initial conditions.

In [36] another approach for time fractional wave (Eq. (19)) is proposed. It is solved, for 0 , *β* , 1 and special initial conditions, by the method of separation of

In [37] the authors study the particular linear fractional Klein-Gordon equation:

considering the fractional Caputo derivative with 1 , *α* ≤2. They achieve a

In [38] the following mathematical model, using Fick's law of diffusion, is developed to study the effect of fractional advection diffusion equation (cross flow) for the calcium profile, considering the Caputo fractional derivative

*C*

Here, the calcium concentration *C x*ð Þ *; t* varies in time and space, *D* is a diffusion constant, and *C*<sup>∞</sup> is calcium concentration at infinity and is assumed that, at initial state of time and at a long distance, calcium concentration vanishes or becomes zero. The authors note that the physical parameter *α* characterizes the cytosolic

In [39] the authors explain that arteries, like other soft tissues, exhibit viscoelastic behavior and part of the mechanical energy transferred to them is dissipative (viscosity) and the other part is stored in a reversible form (elasticity). They modified the standard model by a fractional-order one and test it in human arterial segments. They conclude that fractional derivatives, in Riemann-Liouville sense,

The generalized Voigt model consists of a spring in parallel with two *springpots* (a neologism for a model that is between a spring—purely elastic—and a dashpot, purely viscous) of fractional orders *α* and *β*. The governing fractional-order

> *<sup>∂</sup>αε*ð Þ*<sup>t</sup> <sup>∂</sup>t<sup>α</sup>* <sup>þ</sup> *<sup>η</sup>*<sup>2</sup>

*<sup>∂</sup>βε*ð Þ*<sup>t</sup>*

*<sup>∂</sup>x<sup>β</sup>* (22)

*σ*ðÞ¼ *t E*0*ε*ðÞþ*t η*<sup>1</sup>

viscosities of two springpots in parallel with the spring.

where *E*<sup>0</sup> is the elastic constant for a spring and *η*<sup>1</sup> and *η*<sup>2</sup> represent the

In Ref. [40] the authors developed an accurate and efficient numerical method for the fractional-order standard model described by Eq. (22) with *α* ¼ *β*, combining a fast convolution method with the spectral element discretization based on a general Jacobi polynomial basis that can be used to generate 3D polymorphic high-order elements. In that way they model complicated arterial geometries, such as patient-specific aneurysms, and apply it to 3D fluid-structure interaction

*<sup>∂</sup>x*<sup>2</sup> *<sup>x</sup>*≥0*, t*<sup>≥</sup> <sup>0</sup>

numerical solution by variational iteration method and multivariate Padé

*C x*ð Þ¼ *;* 0 *C*<sup>0</sup>

lim*<sup>x</sup>*!þ<sup>∞</sup> *C x*ð Þ¼ *; <sup>t</sup> <sup>C</sup>*<sup>∞</sup>

*<sup>t</sup>* <sup>þ</sup> *<sup>x</sup>*<sup>3</sup> � <sup>6</sup>*<sup>x</sup>* � �*<sup>t</sup>*

<sup>3</sup> *t* . 0*, x*∈ R

(20)

(21)

variables.

approximation.

(0 , *α* ≤1):

calcium ion in astrocytes.

differential equation is

simulations.

**8**

*∂<sup>α</sup>u <sup>∂</sup>t<sup>α</sup>* � *<sup>∂</sup>*<sup>2</sup>

8 >>><

>>>:

*u*

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

*<sup>∂</sup>x*<sup>2</sup> <sup>þ</sup> *<sup>u</sup>* <sup>¼</sup> <sup>6</sup>*x*<sup>3</sup>

*u x*ð Þ¼ *;* 0 0 *ut*ð Þ¼ *x;* 0 0

> *∂αC <sup>∂</sup>t<sup>α</sup>* <sup>¼</sup> *<sup>D</sup> <sup>∂</sup>*<sup>2</sup>

8 >>>><

>>>>:

are a good alternative to model arterial viscosity.

In [43] a fractional Lorenz system is studied considering the *generalized Caputo derivative*, defined, for 0 , *α* , 1*, ρ* . 0, as

$${}^{GC}\_{0}D\_{t}^{a,\rho}[f](t) = \frac{\rho^{a}}{\sigma(\mathbf{1} - a)} \int\_{0}^{t} \frac{f'(s)}{\left(t^{\rho} - s^{\rho}\right)^{a}} \, ds \tag{26}$$

**4.1 An initial value problem**

detailed description).

<sup>3</sup> ([45]).

*Wj* ¼ *span ψjk; j; k*∈ Z

be crucial in the procedure.

in each *Wj* (see [47] for details).

*g t*ðÞ¼ ∑ *Jmax Jmin gj*

approximation of *gj* arises:

basis by means of Eq. (29):

(recall supp∣*ψ*^*jk*∣ ¼ Ω*j*).

The family *ψjk=ψjk* ¼ 2

0 , *β* ≤ *<sup>π</sup>*

*L*2

so that

**11**

Let us consider the initial value problem (IVP):

*f*ð Þ¼ 0 0

<sup>t</sup> ½ � *f* ðÞþ*t σ*<sup>0</sup> *f t*ðÞþ *σ*<sup>1</sup> *f*

*A Review on Fractional Differential Equations and a Numerical Method to Solve Some…*

We look for *f* satisfying Eq. (31), where *g* is a causal and smooth function with *g*ð Þ¼ 0 0. Other situations where the initial condition is not null can also be faced adapting the following scheme. We will consider that the fractional derivative is the Caputo-Fabrizio one; the Caputo case can be solved similarly (see [44] for a

First we choose a wavelet basis with special properties: well localized in both time and frequency domain, smooth, band limited and infinitely oscillating with fast decay. The mother wavelet *ψ* ∈ *S* (the Schwartz space) and its Fourier trans-

n o and *VJ* <sup>¼</sup> <sup>⊕</sup>*<sup>j</sup>* , *JWj*, the wavelet and scale subspaces,

*j* . *J* ∑ *k*∈Z

where the first and second terms are the projections of *g* in *VJ* and *Wj*, respectively. The properties of localization of the wavelets guarantee absolute convergence

Now we choose the levels where the energy of *g* is concentrated, *Jmin, Jmax* ∈Z

for small *ε*, and truncate the component in each level so that the following

Afterwards we obtain the fractional Caputo-Fabrizio derivatives of the wavelet

*M*ð Þ *α* 1 � α ð

*Ωj*

We have also a scale function φ∈*V*<sup>0</sup> so that f g φð Þ *t* � *k ; k*∈ Z is a BON of *V*<sup>0</sup> ([46]). The sets Ω*<sup>j</sup>*�<sup>1</sup>*,* Ω*j,* Ω*<sup>j</sup>*þ<sup>1</sup> have little overlap, and *Wj* is nearly a basis for the set of functions whose Fourier transform has support in Ω*j*. This property of the basis will

*<sup>t</sup>* � *<sup>k</sup>* � �*, j; <sup>k</sup>*∈<sup>Z</sup>

*<sup>g</sup>;* <sup>φ</sup>*Jn* � �*φJn*ðÞþ*<sup>t</sup>* <sup>∑</sup>

ð Þ R associated to a multiresolution analysis (MRA). We denote, by

0

ðÞ¼ *t g t*ð Þ

ð Þ *<sup>π</sup>* � *<sup>β</sup>* <sup>≤</sup>j j *<sup>ω</sup>* <sup>≤</sup> <sup>2</sup>*<sup>j</sup>* ð Þ *<sup>π</sup>* <sup>þ</sup> *<sup>β</sup>* � �, with

is an orthonormal basis (BON) of

*<sup>g</sup>; <sup>ψ</sup>jk* D E*ψjk*ð Þ*<sup>t</sup>* (32)

ð Þ¼ R ⊕*<sup>j</sup>* <sup>∈</sup>Z*Wj* ¼ ⊕*<sup>j</sup>* . *nWj* þ *Vn, n* ∈Z.

*<sup>g</sup>; <sup>ψ</sup>jk* D E*ψjk*ðÞþ*<sup>t</sup> r t*ð Þ*, r*k k<sup>2</sup> <sup>≤</sup>*ε*k k*<sup>g</sup>* <sup>2</sup> (33)

*cjkψjk*ð Þ*<sup>t</sup> , cjk* <sup>¼</sup> *<sup>g</sup>; <sup>ψ</sup>jk* D E*:* (34)

*ψ*^*jk*ð Þ *ω h*ð Þ *ω e*

*iωt*

*dω* (35)

(31)

*CF* 0*D<sup>α</sup>*

(

*DOI: http://dx.doi.org/10.5772/intechopen.86273*

form satisfy supp <sup>∣</sup>*ψ*^*jk*<sup>∣</sup> <sup>¼</sup> <sup>Ω</sup>*<sup>j</sup>* where <sup>Ω</sup>*<sup>j</sup>* <sup>¼</sup> *<sup>ω</sup>* : <sup>2</sup>*<sup>j</sup>*

respectively, and decompose the space *L*<sup>2</sup>

Now we decompose the data *g* as

*g t*ðÞ¼ ∑ *n*∈ N

ð Þþ t *r t*ðÞ¼ ∑

e*gj*

*vjk*ðÞ¼ *<sup>t</sup> CF*

*Jmax Jmin* ∑ *k*∈Z

ðÞ¼ *t* ∑ *k*∈*Kj*

�<sup>∞</sup> *<sup>D</sup><sup>α</sup>*

<sup>t</sup> *<sup>ψ</sup>jk* h iðÞ¼ <sup>t</sup>

*j* <sup>2</sup>*ψ* 2*<sup>j</sup>*

n o

A detailed analysis of the stability of the system is performed. Adomian method is used to find semi-analytical solution to the fractional nonlinear equations. Chaotic behavior and strange attractors are numerically found for some values of *α* and *ρ:*

#### **4. A numerical approximation scheme to solve linear fractional differential equations**

After having exemplified several applications of fractional differential equations to different real-world problems, including chaotic ones, we will show a method that we have developed to obtain numerical solutions to linear fractional initial value and boundary value problems modeled with Caputo or Caputo-Fabrizio derivatives. The idea of the approximation scheme is to transform the derivatives into integral operators acting on the Fourier transform and to perform a wavelet decomposition of the data. The wavelet coefficients of the unknown are then recovered from a linear system of algebraic equations, and the solution is built up from its coefficients. The properties of the chosen wavelet basis guarantee numerical stability and efficiency of the approximation scheme. In the case of singular kernel, this procedure enables us to handle the singularity.

We note that choosing *a* ¼ �∞ in definition of Eqs. (4) or (5) and being *f* ∈ *H*<sup>1</sup> ð Þ �∞*; b* , the Sobolev space of functions with (weak) first derivative in *L*2 ð Þ �∞*; b* , both derivatives can be expressed as a convolution.

For the Caputo-Fabrizio fractional derivative of order 0 , *α* , 1, changing variables in Eq. (9), we have

$$\mathbf{1}\_{-\infty}^{\rm CF} D\_t^a[f](t) = \frac{\mathbf{M}(a)}{\mathbf{1} - a} \Big|\_{0}^{\infty} f'(t - s) e^{-\frac{a}{\mathbf{1} - a}t} ds = \frac{\mathbf{M}(a)}{\mathbf{1} - a} (f' \* k)(t) \tag{27}$$

where *<sup>k</sup>* is a causal function, *k t*ðÞ¼ *<sup>e</sup>*� *<sup>α</sup><sup>t</sup>* <sup>1</sup>�*<sup>α</sup>* for *<sup>t</sup>*≥0, and *k t*ðÞ¼ 0 for *<sup>t</sup>* , 0. Consequently, since ^ *<sup>k</sup>*ð Þ¼ *<sup>ω</sup>* <sup>1</sup>�*<sup>α</sup> <sup>α</sup>*þ*iω*ð Þ <sup>1</sup>�*<sup>α</sup>* ,

$$\hat{L}\_{-\infty}^{CF} D\_t^{\alpha}[f](t) = \frac{M(\alpha)}{2\pi(1-\alpha)} \int\_{\mathbb{R}} \hat{f}'(\alpha)\hat{k}(\alpha)\mathcal{e}^{i\alpha t} d\alpha. \tag{28}$$

Using the properties of the Fourier transform, we can rewrite the last equality:

$$\mathbf{1}\_{-\infty}^{\rm CF} \mathbf{D}\_{\mathbf{t}}^{a}[f](t) = \frac{\mathbf{M}(a)}{\mathbf{1} - a} \left[ \hat{f}(a) h(a) e^{i\alpha t} d\alpha \right] \tag{29}$$

where *<sup>h</sup>*ð Þ¼ *<sup>ω</sup> <sup>i</sup><sup>ω</sup>* 2*π* ^ *k*ð Þ *ω* . Meanwhile, in the Caputo case, we have

$$\hat{\Gamma}^{\rm CF}\_{-\infty} D\_t^a[f](t) = \frac{1}{\Gamma(1-a)} \int\_{-\infty}^t \frac{f'(s)}{\left(t-s\right)^a} ds = \frac{1}{2\pi \Gamma(1-a)} \int\_{\mathbb{R}} \hat{f}(\omega) \hat{k}(\omega) e^{i\omega t} d\omega \tag{30}$$

where *k t*ðÞ¼ <sup>1</sup> *<sup>t</sup><sup>α</sup>* and ^ *<sup>k</sup>*ð Þ¼ *<sup>ω</sup> <sup>Γ</sup>*ð Þ <sup>1</sup>�*<sup>α</sup>* ð Þ *<sup>i</sup><sup>ω</sup>* <sup>1</sup>�*<sup>α</sup>* . *A Review on Fractional Differential Equations and a Numerical Method to Solve Some… DOI: http://dx.doi.org/10.5772/intechopen.86273*

#### **4.1 An initial value problem**

In [43] a fractional Lorenz system is studied considering the *generalized Caputo*

ð*t* 0

*f* 0 ð Þ*s*

*<sup>t</sup><sup>ρ</sup>* � *<sup>s</sup><sup>ρ</sup>* ð Þ*<sup>α</sup> ds* (26)

*σ*ð Þ 1 � *α*

A detailed analysis of the stability of the system is performed. Adomian method is used to find semi-analytical solution to the fractional nonlinear equations. Chaotic behavior and strange attractors are numerically found for some values of *α* and *ρ:*

After having exemplified several applications of fractional differential equations to different real-world problems, including chaotic ones, we will show a method that we have developed to obtain numerical solutions to linear fractional initial value and boundary value problems modeled with Caputo or Caputo-Fabrizio derivatives. The idea of the approximation scheme is to transform the derivatives into integral operators acting on the Fourier transform and to perform a wavelet decomposition of the data. The wavelet coefficients of the unknown are then recovered from a linear system of algebraic equations, and the solution is built up from its coefficients. The properties of the chosen wavelet basis guarantee numerical stability and efficiency of the approximation scheme. In the case of singular

We note that choosing *a* ¼ �∞ in definition of Eqs. (4) or (5) and being

ð Þ �∞*; b* , the Sobolev space of functions with (weak) first derivative in

� *<sup>α</sup>* <sup>1</sup>�*α<sup>s</sup>*

ð

^ *f* 0 ð Þ *<sup>ω</sup>* ^ *k*ð Þ *ω e iωt*

^*f*ð Þ *<sup>ω</sup> <sup>h</sup>*ð Þ *<sup>ω</sup> <sup>e</sup>*

2*πΓ*ð Þ 1 � *α*

*iωt*

ð *R* ^*f*ð Þ *<sup>ω</sup>* ^ *k*ð Þ *ω e iωt*

*R*

Using the properties of the Fourier transform, we can rewrite the last equality:

ð

*R*

*M*ð Þ *α* 1 � *α*

ð Þ *<sup>t</sup>* � *<sup>s</sup> <sup>α</sup> ds* <sup>¼</sup> <sup>1</sup>

*ds* <sup>¼</sup> *<sup>M</sup>*ð Þ *<sup>α</sup>* <sup>1</sup> � *<sup>α</sup> <sup>f</sup>* 0

<sup>∗</sup> *<sup>k</sup>* � �ð Þ*<sup>t</sup>* (27)

*dω:* (28)

*dω* (29)

*dω* (30)

For the Caputo-Fabrizio fractional derivative of order 0 , *α* , 1, changing

where *<sup>k</sup>* is a causal function, *k t*ðÞ¼ *<sup>e</sup>*� *<sup>α</sup><sup>t</sup>* <sup>1</sup>�*<sup>α</sup>* for *<sup>t</sup>*≥0, and *k t*ðÞ¼ 0 for *<sup>t</sup>* , 0.

*M*ð Þ *α* 2*π*ð Þ 1 � *α*

*<sup>t</sup>* ½ � *<sup>f</sup>* ðÞ¼ *<sup>t</sup> <sup>ρ</sup><sup>α</sup>*

**4. A numerical approximation scheme to solve linear fractional**

kernel, this procedure enables us to handle the singularity.

ð Þ �∞*; b* , both derivatives can be expressed as a convolution.

*M*ð Þ *α* 1 � *α*

*<sup>k</sup>*ð Þ¼ *<sup>ω</sup>* <sup>1</sup>�*<sup>α</sup>*

*<sup>t</sup>* ½ � *f* ðÞ¼ *t*

<sup>t</sup> ½ � *f* ðÞ¼ *t*

*f*0 ð Þ*s*

ð<sup>∞</sup> 0 *f* 0 ð Þ *t* � s *e*

*<sup>α</sup>*þ*iω*ð Þ <sup>1</sup>�*<sup>α</sup>* ,

*derivative*, defined, for 0 , *α* , 1*, ρ* . 0, as

**differential equations**

variables in Eq. (9), we have

Consequently, since ^

where *<sup>h</sup>*ð Þ¼ *<sup>ω</sup> <sup>i</sup><sup>ω</sup>*

where *k t*ðÞ¼ <sup>1</sup>

*<sup>t</sup>* ½ � *f* ðÞ¼ *t*

*CF* �<sup>∞</sup> *<sup>D</sup><sup>α</sup>*

**10**

2*π* ^ *k*ð Þ *ω* . Meanwhile, in the Caputo case, we have

> 1 *Γ*ð Þ 1 � *α*

*<sup>t</sup><sup>α</sup>* and ^

ð*t*

�∞

*<sup>k</sup>*ð Þ¼ *<sup>ω</sup> <sup>Γ</sup>*ð Þ <sup>1</sup>�*<sup>α</sup>* ð Þ *<sup>i</sup><sup>ω</sup>* <sup>1</sup>�*<sup>α</sup>* .

*CF* �<sup>∞</sup> *<sup>D</sup><sup>α</sup>*

*<sup>t</sup>* ½ � *f* ðÞ¼ *t*

*CF* �<sup>∞</sup> *<sup>D</sup><sup>α</sup>*

> *CF* �<sup>∞</sup> *<sup>D</sup><sup>α</sup>*

*f* ∈ *H*<sup>1</sup>

*L*2

*GC* <sup>0</sup> *<sup>D</sup><sup>α</sup>, <sup>ρ</sup>*

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

Let us consider the initial value problem (IVP):

$$\begin{cases} \, ^{CF}\_{0}D\_{t}^{a}[f](t) + \sigma\_{0}f(t) + \sigma\_{1}f'(t) = \mathbf{g}(t) \\ f(\mathbf{0}) = \mathbf{0} \end{cases} \tag{31}$$

We look for *f* satisfying Eq. (31), where *g* is a causal and smooth function with *g*ð Þ¼ 0 0. Other situations where the initial condition is not null can also be faced adapting the following scheme. We will consider that the fractional derivative is the Caputo-Fabrizio one; the Caputo case can be solved similarly (see [44] for a detailed description).

First we choose a wavelet basis with special properties: well localized in both time and frequency domain, smooth, band limited and infinitely oscillating with fast decay. The mother wavelet *ψ* ∈ *S* (the Schwartz space) and its Fourier transform satisfy supp <sup>∣</sup>*ψ*^*jk*<sup>∣</sup> <sup>¼</sup> <sup>Ω</sup>*<sup>j</sup>* where <sup>Ω</sup>*<sup>j</sup>* <sup>¼</sup> *<sup>ω</sup>* : <sup>2</sup>*<sup>j</sup>* ð Þ *<sup>π</sup>* � *<sup>β</sup>* <sup>≤</sup>j j *<sup>ω</sup>* <sup>≤</sup> <sup>2</sup>*<sup>j</sup>* ð Þ *<sup>π</sup>* <sup>þ</sup> *<sup>β</sup>* � �, with 0 , *β* ≤ *<sup>π</sup>* <sup>3</sup> ([45]).

$$\begin{aligned} \text{The family } \left\{ \boldsymbol{\nu}\_{jk} / \boldsymbol{\nu}\_{jk} = 2^{\underset{\cdot}{\cdot}} \boldsymbol{\nu} (2^{\underset{\cdot}{\cdot}} - k), \boldsymbol{j}, k \in \mathbb{Z} \right\} \text{ is an orthonormal basis (BON) of }\\ L^2(\mathbb{R}) \text{ associated to a multiresolution analysis (MRA). We denote, by } \end{aligned}$$

ð Þ R associated to a multiresolution analysis (MRA). We denote, by *Wj* ¼ *span ψjk; j; k*∈ Z n o and *VJ* <sup>¼</sup> <sup>⊕</sup>*<sup>j</sup>* , *JWj*, the wavelet and scale subspaces, respectively, and decompose the space *L*<sup>2</sup> ð Þ¼ R ⊕*<sup>j</sup>* <sup>∈</sup>Z*Wj* ¼ ⊕*<sup>j</sup>* . *nWj* þ *Vn, n* ∈Z.

We have also a scale function φ∈*V*<sup>0</sup> so that f g φð Þ *t* � *k ; k*∈ Z is a BON of *V*<sup>0</sup> ([46]). The sets Ω*<sup>j</sup>*�<sup>1</sup>*,* Ω*j,* Ω*<sup>j</sup>*þ<sup>1</sup> have little overlap, and *Wj* is nearly a basis for the set of functions whose Fourier transform has support in Ω*j*. This property of the basis will be crucial in the procedure.

Now we decompose the data *g* as

$$\log(\mathbf{t}) = \sum\_{n \in \mathbb{N}} \left< \mathbf{g}, \mathbf{q}\_{\restriction n} \right> \boldsymbol{\uprho}\_{\restriction n}(\mathbf{t}) + \sum\_{j \geq 1} \sum\_{k \in \mathbb{Z}} \left< \mathbf{g}, \boldsymbol{\uprho}\_{jk} \right> \boldsymbol{\uprho}\_{jk}(\mathbf{t}) \tag{32}$$

where the first and second terms are the projections of *g* in *VJ* and *Wj*, respectively.

The properties of localization of the wavelets guarantee absolute convergence in each *Wj* (see [47] for details).

Now we choose the levels where the energy of *g* is concentrated, *Jmin, Jmax* ∈Z so that

$$\log(\mathbf{t}) = \sum\_{f\_{\min}}^{f\_{\max}} \mathbf{g}\_{j}(\mathbf{t}) + r(\mathbf{t}) = \sum\_{f\_{\min}}^{f\_{\max}} \sum\_{k \in \mathbb{Z}} \left< \mathbf{g}, \boldsymbol{\upmu}\_{jk} \right> \boldsymbol{\upmu}\_{jk}(\mathbf{t}) + r(\mathbf{t}),\\ \left\| r \right\|\_{2} \le \epsilon \left\| \mathbf{g} \right\|\_{2} \tag{33}$$

for small *ε*, and truncate the component in each level so that the following approximation of *gj* arises:

$$\widetilde{\mathbf{g}}\_j(\mathbf{t}) = \sum\_{k \in K\_j} c\_{jk} \boldsymbol{\mu}\_{jk}(\mathbf{t}),\\ c\_{jk} = \left< \mathbf{g}, \boldsymbol{\mu}\_{jk} \right>. \tag{34}$$

Afterwards we obtain the fractional Caputo-Fabrizio derivatives of the wavelet basis by means of Eq. (29):

$$\boldsymbol{\nu}\_{jk}(\mathbf{t}) = \ \_ {-\infty}^{CF} D\_{\mathbf{t}}^{a} \left[ \boldsymbol{\nu}\_{jk} \right](\mathbf{t}) = \frac{M(a)}{\mathbf{1} - \mathbf{a}} \int\_{\tilde{\Omega}\_{\tilde{\mathbf{t}}}} \hat{\boldsymbol{\nu}}\_{jk}(\boldsymbol{\alpha}) \boldsymbol{h}(\boldsymbol{\alpha}) \boldsymbol{\epsilon}^{i \boldsymbol{a} \boldsymbol{a}} d\boldsymbol{\alpha} \tag{35}$$

(recall supp∣*ψ*^*jk*∣ ¼ Ω*j*).

Let us consider for a moment that in Eq. (31) we have *CF* �<sup>∞</sup> *<sup>D</sup><sup>α</sup> <sup>t</sup>* , i.e., *a* ¼ �∞. Note that, since *supp* ∣*vjk*∣ ⊂ Ω*j*, *vjk* ∈ *Wj*�<sup>1</sup>⋃*Wj*⋃*Wj*þ1, but, from the properties of the chosen basis, we can consider *vjk* ∈*Wj*. This fact enables us to work on each level *j* separately (details can be found in [44, 48]).

Then, since the unknown *f* can be expressed as *f t*ðÞ¼ ∑<sup>j</sup> <sup>∈</sup><sup>Z</sup> *fj* ð Þt *,* where *fj* ðÞ¼ <sup>t</sup> <sup>∑</sup>*<sup>k</sup>*∈<sup>Z</sup> *bjkψjk*ð Þ<sup>t</sup> and *bjk* <sup>¼</sup> *<sup>f</sup>; <sup>ψ</sup>jk* D E*,* we have

$$f(t) \approx \sum\_{J\_{\min}}^{J\_{\max}} f\_j(\mathbf{t}) \approx \sum\_{J\_{\min}}^{J\_{\max}} \sum\_{k \in K\_j} b\_{jk} \boldsymbol{\nu}\_{jk}(\mathbf{t}).\tag{36}$$

causal function defined as *g t*ðÞ¼ *v t*ð Þsin 2ð Þ *πt* cos 0ð Þ *:*5*πt ,* where *v* is a smooth window in the interval 0½ � *;* 4 . Wavelet analysis indicates that the energy of the data *g* is concentrated in the subspaces *W*<sup>0</sup> and *W*1; thus, we consider levels �1≤*j*≤2 for

*A Review on Fractional Differential Equations and a Numerical Method to Solve Some…*

formula for the "exact" solution to Eq. (31) (see [22]). The approximate solution e*f* to the IVP is plotted (in green) in **Figure 1**, together with the exact solution (in

*k*∈ *Kj*

Once more the matrix of the resulting linear system is diagonal dominant, and

Now we consider IVP described by Eq. (31) for *σ*<sup>0</sup> ¼ 0*:*9*, σ*<sup>1</sup> ¼ 0*:*3*, α* ¼ 0*:*5, and

<sup>2</sup>*v t*ð Þð Þ 2 sin 2ð Þ� *:*5*π<sup>t</sup>* <sup>0</sup>*:*5 cos 6ð Þ *<sup>π</sup><sup>t</sup>* , with *<sup>v</sup>* as a smooth window over interval [0, 7]. Since *σ*<sup>1</sup> 6¼ 0 and we have no formula to test the performance of the approximation, for this example we will set *f,* calculate *g* from Eq. (31), and then apply the

> *j*¼�1 e*fj*

This scheme can be adapted to solve boundary value problems. We show the

The following example shows the performance of the method for *σ*<sup>1</sup> ¼ 0*:*3.

<sup>R</sup>*iω*^*<sup>f</sup>* ð Þ *<sup>ω</sup> <sup>e</sup>iω<sup>t</sup>*

*bjk <sup>ψ</sup>jk; <sup>ψ</sup>jm* D E <sup>þ</sup> *<sup>σ</sup>*<sup>1</sup> <sup>∑</sup>

<sup>1</sup>�*<sup>α</sup>*, *<sup>g</sup>* <sup>∈</sup>*C*ð Þ <sup>0</sup>*;* <sup>∞</sup> , and *<sup>g</sup>*ð Þ¼ <sup>0</sup> 0, there exists a

*dω*, we obtain similar equations for the

*k*0 ∈ *Kj*

. The plots of the *f* (blue) and e*f*

*cjk*<sup>0</sup> *<sup>ψ</sup>jk*<sup>0</sup> *; <sup>ψ</sup>jm* D E

(40)

*bjk <sup>i</sup>ωψjk; <sup>ψ</sup>jm* D E <sup>¼</sup> <sup>∑</sup>

the reconstruction.

If *σ*<sup>1</sup> 6¼ 0, since *f*

coefficients on each level:

*bjk vjk; <sup>ψ</sup>jm* D E <sup>þ</sup> *<sup>σ</sup>*<sup>0</sup> <sup>∑</sup>

proposed method to recover *f*.

(green) appear in **Figure 2**.

Choosing �1≤*<sup>j</sup>* <sup>≤</sup>2, it results in <sup>e</sup>*<sup>f</sup>* <sup>¼</sup> <sup>∑</sup><sup>2</sup>

*The approximation and the exact solution for the Example 1.*

procedure finding the solution to the fractional heat equation.

blue).

∑ *k*∈ *Kj*

*4.1.2 Example 2*

*f t*ðÞ¼ *t*

**Figure 1.**

**13**

For this case, being *<sup>σ</sup>*<sup>1</sup> <sup>¼</sup> 0, *<sup>σ</sup>*<sup>0</sup> 6¼ � <sup>1</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.86273*

0 ðÞ¼ *<sup>t</sup>* <sup>1</sup> 2*π* Ð

*k*∈ *Kj*

the system can be solved efficiently (see [49] for details).

Finally, we replace this last expression in Eq. (31), where for simplicity we will first consider *σ*<sup>1</sup> ¼ 0 and look for the wavelet coefficients *bjk* that satisfy, for each *j*∈ *Jmin; Jmax* ½ �:

$$\sum\_{k \in K\_j} b\_{jk} \upsilon\_{jk}(\mathbf{t}) + \sigma\_0 \sum\_{k \in K\_j} b\_{jk} \upsilon\_{jk}(\mathbf{t}) = \sum\_{k' \in K\_j} c\_{jk'} \upsilon\_{jk'}(\mathbf{t}) \tag{37}$$

or

$$\sum\_{k \in K\_j} b\_{jk} \left< \boldsymbol{\nu}\_{jk}, \boldsymbol{\nu}\_{jm} \right> + \sigma\_0 \sum\_{k \in K\_j} b\_{jk} \left< \boldsymbol{\nu}\_{jk}, \boldsymbol{\nu}\_{jm} \right> = \sum\_{k' \in K\_j} c\_{jk'} \left< \boldsymbol{\nu}\_{jk'}, \boldsymbol{\nu}\_{jm} \right> \tag{38}$$

that, in matrix form, results in

$$
\mathcal{M}^{\dot{j}} \mathbf{b}^{\dot{j}} = \boldsymbol{\sigma}^{\dot{i}} \tag{39}
$$

$$\begin{aligned} \text{where } & \mathbf{b}^{j} = \left( b\_{jk} \right)\_{k \in K\_{j}}, \sigma^{j} = \left( c\_{jk'} \right)\_{k' \in K\_{j}}, \mathcal{M}^{j} \in \mathbb{R}^{K\_{j} \times K\_{j}} \text{and } \left( \mathcal{M}^{j} \right)\_{kl} = \left\langle v\_{kl}, \mu\_{kl} \right\rangle + \varepsilon \\ \sigma\_{0} \left\langle \boldsymbol{\nu}\_{kj}, \boldsymbol{\nu}\_{kl} \right\rangle. \end{aligned}$$

From the properties of the wavelet basis, and those of *vjk*, it results that **M***<sup>j</sup>* is a diagonal dominant matrix and, consequently, the vector of coefficients *b<sup>j</sup>* can be computed in a stable and accurate way. The solution *f* can be obtained from Eq. (36). Moreover, it can be shown that *f*ð Þ¼ 0 0. To correct the effect of having considered *CF* �<sup>∞</sup> *<sup>D</sup><sup>α</sup> <sup>t</sup>* instead of *CF* 0*D<sup>α</sup> <sup>t</sup>* , we set e*f* ¼ *f:*χ½ � <sup>0</sup>*;<sup>T</sup>* , where χ½ � <sup>0</sup>*;<sup>T</sup>* is the characteristic function of the interval 0½ � *; T* . Finally, e*f* is an approximate solution to Eq. (31).

The error introduced in the approximation can be controlled and reduced: a more accurate truncated projection of the data into the wavelet subspaces can be considered, and the elements of the matrix can be computed with good precision since they can be expressed as integrals over compact subsets; finally, the matrix of the resulting linear system is a diagonal dominant matrix, and the solution can be computed accurately. In summary, the good properties of the basis and the operator guarantee that the resulting approximation scheme is efficient and numerically stable and no additional conditions need to be imposed.

#### *4.1.1 Example 1*

We illustrate the performance of the proposed approximation scheme by solving the IVP described by Eq. (31) for *σ*<sup>0</sup> ¼ 0*:*9*, σ*<sup>1</sup> ¼ 0*,* and *α* ¼ 0*:*5 and *g* a

#### *A Review on Fractional Differential Equations and a Numerical Method to Solve Some… DOI: http://dx.doi.org/10.5772/intechopen.86273*

causal function defined as *g t*ðÞ¼ *v t*ð Þsin 2ð Þ *πt* cos 0ð Þ *:*5*πt ,* where *v* is a smooth window in the interval 0½ � *;* 4 . Wavelet analysis indicates that the energy of the data *g* is concentrated in the subspaces *W*<sup>0</sup> and *W*1; thus, we consider levels �1≤*j*≤2 for the reconstruction.

For this case, being *<sup>σ</sup>*<sup>1</sup> <sup>¼</sup> 0, *<sup>σ</sup>*<sup>0</sup> 6¼ � <sup>1</sup> <sup>1</sup>�*<sup>α</sup>*, *<sup>g</sup>* <sup>∈</sup>*C*ð Þ <sup>0</sup>*;* <sup>∞</sup> , and *<sup>g</sup>*ð Þ¼ <sup>0</sup> 0, there exists a formula for the "exact" solution to Eq. (31) (see [22]). The approximate solution e*f* to the IVP is plotted (in green) in **Figure 1**, together with the exact solution (in blue).

If *σ*<sup>1</sup> 6¼ 0, since *f* 0 ðÞ¼ *<sup>t</sup>* <sup>1</sup> 2*π* Ð <sup>R</sup>*iω*^*<sup>f</sup>* ð Þ *<sup>ω</sup> <sup>e</sup>iω<sup>t</sup> dω*, we obtain similar equations for the coefficients on each level:

$$\sum\_{k \in K\_j} b\_{jk} \left< \boldsymbol{\nu}\_{jk}, \boldsymbol{\nu}\_{jm} \right> + \sigma\_0 \sum\_{k \in K\_j} b\_{jk} \left< \boldsymbol{\nu}\_{jk}, \boldsymbol{\nu}\_{jm} \right> + \sigma\_1 \sum\_{k \in K\_j} b\_{jk} \left< i\alpha \boldsymbol{\nu}\_{jk}, \boldsymbol{\nu}\_{jm} \right> = \sum\_{k' \in K\_j} c\_{jk'} \left< \boldsymbol{\nu}\_{jk'}, \boldsymbol{\nu}\_{jm} \right> \tag{40}$$

Once more the matrix of the resulting linear system is diagonal dominant, and the system can be solved efficiently (see [49] for details).

The following example shows the performance of the method for *σ*<sup>1</sup> ¼ 0*:*3.

#### *4.1.2 Example 2*

Let us consider for a moment that in Eq. (31) we have *CF*

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

Then, since the unknown *f* can be expressed as *f t*ðÞ¼ ∑<sup>j</sup> <sup>∈</sup><sup>Z</sup> *fj*

*f t*ð Þ≈ ∑ *Jmax Jmin fj* ð Þt ≈ ∑ *Jmax Jmin* ∑ *k*∈*Kj*

*bjkvjk*ð Þþ t *σ*<sup>0</sup> ∑

þ *σ*<sup>0</sup> ∑ *k*∈*Kj*

, *<sup>c</sup><sup>j</sup>* <sup>¼</sup> *cjk*<sup>0</sup> � �

*<sup>t</sup>* instead of *CF*

*k*∈*Kj*

*bjk ψjk; ψjm* D E

> **M***<sup>j</sup> <sup>b</sup><sup>j</sup>* <sup>¼</sup> *<sup>c</sup>*

> > , **M***<sup>j</sup>*

From the properties of the wavelet basis, and those of *vjk*, it results that **M***<sup>j</sup>* is a diagonal dominant matrix and, consequently, the vector of coefficients *b<sup>j</sup>* can be computed in a stable and accurate way. The solution *f* can be obtained from Eq. (36). Moreover, it can be shown that *f*ð Þ¼ 0 0. To correct the effect of

0*D<sup>α</sup>*

The error introduced in the approximation can be controlled and reduced: a more accurate truncated projection of the data into the wavelet subspaces can be considered, and the elements of the matrix can be computed with good precision since they can be expressed as integrals over compact subsets; finally, the matrix of the resulting linear system is a diagonal dominant matrix, and the solution can be computed accurately. In summary, the good properties of

scheme is efficient and numerically stable and no additional conditions need to

We illustrate the performance of the proposed approximation scheme by solving the IVP described by Eq. (31) for *σ*<sup>0</sup> ¼ 0*:*9*, σ*<sup>1</sup> ¼ 0*,* and *α* ¼ 0*:*5 and *g* a

characteristic function of the interval 0½ � *; T* . Finally, e*f* is an approximate

the basis and the operator guarantee that the resulting approximation

∈ R*Kj*�*Kj*

*k*<sup>0</sup> ∈*Kj*

D E

separately (details can be found in [44, 48]).

ðÞ¼ t ∑*<sup>k</sup>*∈<sup>Z</sup> *bjkψjk*ð Þt and *bjk* ¼ *f; ψjk*

∑ *k*∈*Kj*

*bjk vjk; ψjm* D E

that, in matrix form, results in

� � *k*∈*Kj*

�<sup>∞</sup> *<sup>D</sup><sup>α</sup>*

*fj*

*j*∈ *Jmin; Jmax* ½ �:

or

∑ *k*∈*Kj*

where *<sup>b</sup><sup>j</sup>* <sup>¼</sup> *bjk*

having considered *CF*

solution to Eq. (31).

be imposed.

**12**

*4.1.1 Example 1*

.

*σ*<sup>0</sup> *ψkj,ψkl* D E

that, since *supp* ∣*vjk*∣ ⊂ Ω*j*, *vjk* ∈ *Wj*�<sup>1</sup>⋃*Wj*⋃*Wj*þ1, but, from the properties of the chosen basis, we can consider *vjk* ∈*Wj*. This fact enables us to work on each level *j*

*,* we have

Finally, we replace this last expression in Eq. (31), where for simplicity we will first consider *σ*<sup>1</sup> ¼ 0 and look for the wavelet coefficients *bjk* that satisfy, for each

*bjkψjk*ðÞ¼ *t* ∑

*k*<sup>0</sup> ∈*Kj*

*cjk*<sup>0</sup> *ψjk*<sup>0</sup> *; ψjm;* D E

*<sup>j</sup>* (39)

*kl* ¼ *vkl; ψkl* h iþ

and **M***<sup>j</sup>* � �

*<sup>t</sup>* , we set e*f* ¼ *f:*χ½ � <sup>0</sup>*;<sup>T</sup>* , where χ½ � <sup>0</sup>*;<sup>T</sup>* is the

¼ ∑ *k*0 ∈*Kj* �<sup>∞</sup> *<sup>D</sup><sup>α</sup>*

*<sup>t</sup>* , i.e., *a* ¼ �∞. Note

ð Þt *,* where

*cjk*0*ψjk*0ð Þ*t* (37)

(38)

*bjkψjk*ð Þt *:* (36)

Now we consider IVP described by Eq. (31) for *σ*<sup>0</sup> ¼ 0*:*9*, σ*<sup>1</sup> ¼ 0*:*3*, α* ¼ 0*:*5, and *f t*ðÞ¼ *t* <sup>2</sup>*v t*ð Þð Þ 2 sin 2ð Þ� *:*5*π<sup>t</sup>* <sup>0</sup>*:*5 cos 6ð Þ *<sup>π</sup><sup>t</sup>* , with *<sup>v</sup>* as a smooth window over interval [0, 7]. Since *σ*<sup>1</sup> 6¼ 0 and we have no formula to test the performance of the approximation, for this example we will set *f,* calculate *g* from Eq. (31), and then apply the proposed method to recover *f*.

Choosing �1≤*<sup>j</sup>* <sup>≤</sup>2, it results in <sup>e</sup>*<sup>f</sup>* <sup>¼</sup> <sup>∑</sup><sup>2</sup> *j*¼�1 e*fj* . The plots of the *f* (blue) and e*f* (green) appear in **Figure 2**.

This scheme can be adapted to solve boundary value problems. We show the procedure finding the solution to the fractional heat equation.

**Figure 1.** *The approximation and the exact solution for the Example 1.*

with *Bk*ðÞ¼ *t* 2

is null.

**Figure 3.**

**Figure 4.**

**15**

*smooth window in* ½ � 0*;* 3 *.*

Ð 1 0

*DOI: http://dx.doi.org/10.5772/intechopen.86273*

*g x*ð Þ *; t* sin ð Þ *kπx dx* and the Fourier coefficients of *g x*ð Þ *; t* for each

*t*∈ð Þ 0*; T* . The uniqueness of solution is guaranteed because *u x*ð Þ¼ *;* 0 0, so *Bk*ð Þ 0

In this case we only need to solve Eq. (43) for *<sup>k</sup>* <sup>¼</sup> <sup>2</sup>*,* with *<sup>B</sup>*<sup>2</sup> <sup>¼</sup> *v t*ð Þ*e*�*t=*<sup>2</sup> sin 5ð Þ *<sup>π</sup><sup>t</sup> :*

We show the approximate solution to Eq. (41) for *α* ¼ 0*:*5*, T* ¼ 3*,* and

*A Review on Fractional Differential Equations and a Numerical Method to Solve Some…*

Wavelet analysis indicates that the 95% of the energy of *B*<sup>2</sup> is concentrated in subspaces *W*1*,W*2, and *W*3, and we obtained the following condition numbers for

*g x*ð Þ¼ *; <sup>t</sup> v t*ð Þ*e*�*t=*<sup>2</sup> sin 5ð Þ *<sup>π</sup><sup>t</sup>* sin 2ð Þ *<sup>π</sup><sup>x</sup>* , with *<sup>v</sup>* a smooth window in 0½ � *;* <sup>3</sup> .

*The approximate and the exact solutions to Eq. (41) with <sup>α</sup>* <sup>¼</sup> <sup>0</sup>*:*8*, T* <sup>¼</sup> <sup>3</sup>*, and <sup>B</sup>*<sup>3</sup> <sup>¼</sup> *v t*ð Þ*e*�*<sup>t</sup>*

*The approximate solutions to Eq. (39) with <sup>α</sup>* <sup>¼</sup> <sup>0</sup>*:*8*, T* <sup>¼</sup> <sup>3</sup>*,gx*ð Þ¼ *; <sup>t</sup> v t*ð Þ*e*�*t=*<sup>2</sup> sin 5ð Þ *<sup>π</sup><sup>t</sup>* sin 2ð Þ *<sup>π</sup><sup>x</sup> , and v <sup>a</sup>*

<sup>2</sup> sin 5ð Þ *πt .*

**Figure 2.** *f and* e*f of Example 2.*

#### **4.2 Boundary value problem**

We show how to adapt the scheme used for initial value problem for solving boundary value problems with fractional partial differential equations in an example.

We will consider a fractional heat problem where we have replaced the classical time derivative by the Caputo-Fabrizio fractional derivative of order *α*:

$$\begin{cases} \, \, \_0^{\text{CF}} D\_t^a [u](\mathbf{x}, t) - u\_{\text{xx}}(\mathbf{x}, t) = \mathbf{g}(\mathbf{x}, t), & \mathbf{x} \in [0, 1], t \in (0, T) \\\ u(\mathbf{x}, \mathbf{0}) = \mathbf{0} & \mathbf{x} \in [0, 1] \\\ u(\mathbf{0}, t) = u(\mathbf{1}, t) = \mathbf{0} & t \in (0, T) \end{cases} \tag{41}$$

This equation models the evolution of temperatures in a bar of length 1, constituted by a heterogeneous material which has "memory," due to the fluctuations introduced by elements at different dimension scales ([7]).

The smooth and causal function *g* represents an external source. We look for smooth solutions *u* ∈*C*<sup>2</sup> ð Þ� 0*;* 1 ð Þ 0*; T* by separating variables and pose

$$u(\mathbf{x}, t) = \sum\_{k \in \mathbb{Z}} u\_k(t) \sin \left(k \pi \mathbf{x} \right) \tag{42}$$

where *uk*ð Þ*t* is the Fourier coefficients of *u x*ð Þ *; t* for each *t*∈ð Þ 0*; T* .

For the temporal part of the function, after replacing in Eq. (36), we obtain an initial value problem like that described by Eq. (31) for each coefficient *uk*ð Þ*t* :

$$\begin{cases} D\_0^a[u\_k](t) + (k\pi)^2 u\_k(t) = B\_k(t), t \in (0, T) \\ u\_k(0) = 0 \end{cases} \tag{43}$$

*A Review on Fractional Differential Equations and a Numerical Method to Solve Some… DOI: http://dx.doi.org/10.5772/intechopen.86273*

with *Bk*ðÞ¼ *t* 2 Ð 1 0 *g x*ð Þ *; t* sin ð Þ *kπx dx* and the Fourier coefficients of *g x*ð Þ *; t* for each

*t*∈ð Þ 0*; T* . The uniqueness of solution is guaranteed because *u x*ð Þ¼ *;* 0 0, so *Bk*ð Þ 0 is null.

We show the approximate solution to Eq. (41) for *α* ¼ 0*:*5*, T* ¼ 3*,* and *g x*ð Þ¼ *; <sup>t</sup> v t*ð Þ*e*�*t=*<sup>2</sup> sin 5ð Þ *<sup>π</sup><sup>t</sup>* sin 2ð Þ *<sup>π</sup><sup>x</sup>* , with *<sup>v</sup>* a smooth window in 0½ � *;* <sup>3</sup> .

In this case we only need to solve Eq. (43) for *<sup>k</sup>* <sup>¼</sup> <sup>2</sup>*,* with *<sup>B</sup>*<sup>2</sup> <sup>¼</sup> *v t*ð Þ*e*�*t=*<sup>2</sup> sin 5ð Þ *<sup>π</sup><sup>t</sup> :* Wavelet analysis indicates that the 95% of the energy of *B*<sup>2</sup> is concentrated in subspaces *W*1*,W*2, and *W*3, and we obtained the following condition numbers for

**Figure 3.** *The approximate and the exact solutions to Eq. (41) with <sup>α</sup>* <sup>¼</sup> <sup>0</sup>*:*8*, T* <sup>¼</sup> <sup>3</sup>*, and <sup>B</sup>*<sup>3</sup> <sup>¼</sup> *v t*ð Þ*e*�*<sup>t</sup>* <sup>2</sup> sin 5ð Þ *πt .*

#### **Figure 4.**

*The approximate solutions to Eq. (39) with <sup>α</sup>* <sup>¼</sup> <sup>0</sup>*:*8*, T* <sup>¼</sup> <sup>3</sup>*,gx*ð Þ¼ *; <sup>t</sup> v t*ð Þ*e*�*t=*<sup>2</sup> sin 5ð Þ *<sup>π</sup><sup>t</sup>* sin 2ð Þ *<sup>π</sup><sup>x</sup> , and v <sup>a</sup> smooth window in* ½ � 0*;* 3 *.*

**4.2 Boundary value problem**

*CF* 0*D<sup>α</sup>*

8 ><

>:

smooth solutions *u* ∈*C*<sup>2</sup>

example.

**14**

**Figure 2.**

*f and* e*f of Example 2.*

We show how to adapt the scheme used for initial value problem for solving boundary value problems with fractional partial differential equations in an

time derivative by the Caputo-Fabrizio fractional derivative of order *α*:

introduced by elements at different dimension scales ([7]).

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

*Dα*

(

*uk*ð Þ¼ 0 0

*u x*ð Þ¼ *; t* ∑

<sup>0</sup>½ � *uk* ð Þþ*<sup>t</sup>* ð Þ *<sup>k</sup><sup>π</sup>* <sup>2</sup>

where *uk*ð Þ*t* is the Fourier coefficients of *u x*ð Þ *; t* for each *t*∈ð Þ 0*; T* .

We will consider a fractional heat problem where we have replaced the classical

<sup>t</sup> ½ � *u* ð Þ� *x; t uxx*ð Þ¼ *x; t g x*ð Þ *; t , x*∈½ � 0*;* 1 *, t*∈ð Þ 0*; T u x*ð Þ¼ *;* 0 0 *x*∈½ � 0*;* 1 *u*ð Þ¼ 0*; t u*ð Þ¼ 1*; t* 0 *t* ∈ð Þ 0*; T*

This equation models the evolution of temperatures in a bar of length 1, constituted by a heterogeneous material which has "memory," due to the fluctuations

The smooth and causal function *g* represents an external source. We look for

For the temporal part of the function, after replacing in Eq. (36), we obtain an

*k*∈Z

initial value problem like that described by Eq. (31) for each coefficient *uk*ð Þ*t* :

ð Þ� 0*;* 1 ð Þ 0*; T* by separating variables and pose

*uk*ðÞ¼ *t Bk*ð Þ*t , t*∈ð Þ 0*; T*

*uk*ð Þ*t* sin ð Þ *kπx* (42)

(41)

(43)

the band matrices of Eq. (39): *cond*<sup>∞</sup> **<sup>M</sup>**<sup>0</sup> � � <sup>¼</sup> <sup>1</sup>*:*1153, *cond*<sup>∞</sup> **<sup>M</sup>**<sup>1</sup> � � <sup>¼</sup> <sup>1</sup>*:*0663, *cond*<sup>∞</sup> **<sup>M</sup>**<sup>2</sup> � � <sup>¼</sup> <sup>1</sup>*:*0132, *cond*<sup>∞</sup> **<sup>M</sup>**<sup>3</sup> � � <sup>¼</sup> <sup>1</sup>*:*0098, and *cond*<sup>∞</sup> **<sup>M</sup>**<sup>4</sup> � � <sup>¼</sup> <sup>1</sup>*:*0076. We consider levels 0≤*j* ≤ 4 for the reconstruction, and the mean square error in this case is *<sup>u</sup>*<sup>2</sup> � <sup>e</sup>*u*2k*L*<sup>2</sup> <sup>¼</sup> <sup>3</sup>*:*5020 10�<sup>4</sup> � � � .

*α* ¼ 1, as expected (for *α* ¼ 1, Eq. (41) describes classical heat problem, for a bar

*A Review on Fractional Differential Equations and a Numerical Method to Solve Some…*

In this chapter we have presented a summary of some recent works showing the relevance and the intense research work in the area of fractional calculus and its applications. We have focused on nonlinear models describing different phenomena where fractional differentiation plays an important role. In the last section we have presented an approximation scheme that we have developed to solve linear initial and boundary value problems based on wavelet decomposition, and the performance of the method is illustrated by examples. Possible extensions and

This work was partially supported by University of Buenos Aires through

\*, Silvia A. Seminara<sup>1</sup> and Marcela A. Fabio<sup>2</sup>

1 Faculty of Engineering, University of Buenos Aires, Buenos Aires, Argentina

2 School of Science and Technology, University of San Martín, Buenos Aires,

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: mariainestro@gmail.com

provided the original work is properly cited.

The authors declare no conflicts of interest regarding this chapter.

made of a homogeneous material).

*DOI: http://dx.doi.org/10.5772/intechopen.86273*

adaptation to nonlinear equations are still under study.

UBACyT (2018–2021) 20020170100350BA.

**5. Conclusions**

**Acknowledgements**

**Conflict of interest**

**Author details**

Argentina

**17**

María I. Troparevsky<sup>1</sup>

**Figure 3** shows the approximate and the exact solution to Eq. (43). In **Figure 4** we draw the approximate solution *u* of the heat problem described by Eq. (41), and in **Figure 5** the difference between the true solution *u x*ð Þ *; t* and its approximation is plotted. The mean square error obtained in this case is 4*:*0016 10�4*:*

Finally, in **Figure 6**, we show the approximate solutions *u*<sup>2</sup> for different orders of derivation *α*, *α* ! 1, exhibiting the tendency to the solution of Eq. (43) with

#### **Figure 5.**

*The difference between the true solution u x*ð Þ *; t to Eq. (39) and its approximation by means of the wavelet scheme.*

**Figure 6.**

*The approximate solutions to Eq. (41) with α* ! 1*.*

*A Review on Fractional Differential Equations and a Numerical Method to Solve Some… DOI: http://dx.doi.org/10.5772/intechopen.86273*

*α* ¼ 1, as expected (for *α* ¼ 1, Eq. (41) describes classical heat problem, for a bar made of a homogeneous material).

### **5. Conclusions**

the band matrices of Eq. (39): *cond*<sup>∞</sup> **<sup>M</sup>**<sup>0</sup> � � <sup>¼</sup> <sup>1</sup>*:*1153, *cond*<sup>∞</sup> **<sup>M</sup>**<sup>1</sup> � � <sup>¼</sup> <sup>1</sup>*:*0663, *cond*<sup>∞</sup> **<sup>M</sup>**<sup>2</sup> � � <sup>¼</sup> <sup>1</sup>*:*0132, *cond*<sup>∞</sup> **<sup>M</sup>**<sup>3</sup> � � <sup>¼</sup> <sup>1</sup>*:*0098, and *cond*<sup>∞</sup> **<sup>M</sup>**<sup>4</sup> � � <sup>¼</sup> <sup>1</sup>*:*0076. We consider levels 0≤*j* ≤ 4 for the reconstruction, and the mean square error in this

plotted. The mean square error obtained in this case is 4*:*0016 10�4*:*

**Figure 3** shows the approximate and the exact solution to Eq. (43). In **Figure 4** we draw the approximate solution *u* of the heat problem described by Eq. (41), and in **Figure 5** the difference between the true solution *u x*ð Þ *; t* and its approximation is

Finally, in **Figure 6**, we show the approximate solutions *u*<sup>2</sup> for different orders of derivation *α*, *α* ! 1, exhibiting the tendency to the solution of Eq. (43) with

*The difference between the true solution u x*ð Þ *; t to Eq. (39) and its approximation by means of the wavelet*

case is *<sup>u</sup>*<sup>2</sup> � <sup>e</sup>*u*2k*L*<sup>2</sup> <sup>¼</sup> <sup>3</sup>*:*5020 10�<sup>4</sup>

� .

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

� �

**Figure 5.**

**Figure 6.**

**16**

*The approximate solutions to Eq. (41) with α* ! 1*.*

*scheme.*

In this chapter we have presented a summary of some recent works showing the relevance and the intense research work in the area of fractional calculus and its applications. We have focused on nonlinear models describing different phenomena where fractional differentiation plays an important role. In the last section we have presented an approximation scheme that we have developed to solve linear initial and boundary value problems based on wavelet decomposition, and the performance of the method is illustrated by examples. Possible extensions and adaptation to nonlinear equations are still under study.

#### **Acknowledgements**

This work was partially supported by University of Buenos Aires through UBACyT (2018–2021) 20020170100350BA.

### **Conflict of interest**

The authors declare no conflicts of interest regarding this chapter.

### **Author details**

María I. Troparevsky<sup>1</sup> \*, Silvia A. Seminara<sup>1</sup> and Marcela A. Fabio<sup>2</sup>

1 Faculty of Engineering, University of Buenos Aires, Buenos Aires, Argentina

2 School of Science and Technology, University of San Martín, Buenos Aires, Argentina

\*Address all correspondence to: mariainestro@gmail.com

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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[2] Capelas de Oliveira E, Tenreiro Machado J. A review of definitions for fractional derivatives and integral. Mathematical Problems in Engineering. 2014;**2014**:238459. DOI: 10.1155/2014/ 238459. 6p

[3] Oldham K, Spanier J. The Fractional Calculus. New York-London: Academic Press Inc.; 1974. 322p. ISBN: 9780080956206

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[5] Samko S, Kilbas A, Marichev O. Fractional Integrals and Derivatives. Theory and Applications. Amsterdam: Gordon and Breach Science Publishers; 1993. 1016p. ISBN: 2881248640

[6] Podlubny I. Fractional Differential Equations. San Diego: Academic Press; 1998. 340p. ISBN: 9780125588409

[7] Caputo M, Fabrizio M. A new definition of fractional derivative without singular kernel. Progress in Fractional Differentiation and Applications. 2015;**1**(2):73-85. DOI: 10.12785/pfda/010201

[8] Ortigueira M, Tenreiro Machado J. A critical analysis of the Caputo-Fabrizio operator. Communications in Nonlinear Science and Numerical Simulation. 2017;**59**:608-611. DOI: 10.1016/j. cnsns.2017.12.001

[9] Giusti A. A comment on some new definitions of fractional derivative. Nonlinear Dynamics. 2018;**93**(3):

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2018. pp. 35-55. DOI: 10.5772/ intechopen.75523. Ch 2

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[20] Sabatier J, Agrawal O, Tenreiro Machado J, editors. Advances in Fractional Calculus. The Netherlands: Springer; 2007. 549p. ISBN: 978-1-

[21] Odzijewicz T, Malinowska A, Torres D. Fractional calculus of variations in terms of a generalized fractional integral with applications to physics. Abstract and Applied Analysis. 2012;**2012**:871912.

DOI: 10.1155/2012/871912. 24p

[22] Al Salti N, Karimov E, Kerbal S. Boundary value problems for fractional heat equation involving Caputo-Fabrizio derivative. New Trends in Mathematical Science. 2017;**4**(4):79-89. DOI: 10.20852/ntmsci.2016422308

[23] Losada J, Nieto J. Properties of a new fractional derivative without singular kernel. Progress in Fractional Differentiation and Applications. 2015; **1**(2):87-92. DOI: 10.12785/pfda/010202

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[11] Sales Teodoro G, Oliveira D, Capelas de Oliveira E. Sobre derivadas fracionárias. Revista Brasileira de Ensino de Física. 2018;**40**(2):e2307. DOI: 10.1590/1806–9126-RBEF-2017-0213

[12] Caputo M, Fabrizio M. On the notion of fractional derivative and applications to the hysteresis phenomena. Meccanica. 2017;**52**(1): 3043-3052. DOI: 10.1007/s11012-017- 0652-y

[13] Mainardi F. Fractional calculus. In: Carpinteri A, Mainardi F, editors. Fractals and Fractional Calculus in Continuum Mechanics. International Centre for Mechanical Sciences (Courses and Lectures). Vol. 378. Vienna: Springer Verlag; 1997. DOI: 10.1007/978-3-7091-2664-6\_7

[14] Tenreiro Machado J, Silva M, Barbosa R, Jesus I, Reis C, Marcos M, et al. Some applications of fractional calculus in engineering. Mathematical Problems in Engineering. 2010;**2010**: 639801. DOI: 10.1155/2010/639801. 34p

[15] Baleanu D, Rezapour A, Mohammadi H. Some existence results on nonlinear fractional differential equations. Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences. 2013;**371**(1990): 20120144. DOI: 10.1098/rsta.2012.0144

[16] Baleanu D, Agarwal R, Mohammadi H, Rezapour S. Some existence results

*A Review on Fractional Differential Equations and a Numerical Method to Solve Some… DOI: http://dx.doi.org/10.5772/intechopen.86273*

for a nonlinear fractional differential equation on partially ordered Banach spaces. Boundary Value Problems. 2013; **2013**:112. DOI: 10.1186/1687-2770- 2013-112

**References**

978-3-642-14574-2

238459. 6p

9780080956206

0471588849

[1] Diethelm K. The Analysis of Fractional Differential Equations. Berlin: Springer Verlag; 2010. 264p. ISBN: 9783642145735. DOI: 10.1007/

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

1757-1763. DOI: 10.1007/s11071-018-

[11] Sales Teodoro G, Oliveira D, Capelas

fracionárias. Revista Brasileira de Ensino de Física. 2018;**40**(2):e2307. DOI: 10.1590/1806–9126-RBEF-2017-0213

[10] Sales Teodoro G, Capelas de Oliveira E. Derivadas fracionárias: criterios para classificação. Revista Eletrônica Paulista de Matemática. 2017;

**10**:10-19. DOI: 10.21167/ cqdvol10ermacic201723169664

de Oliveira E. Sobre derivadas

[12] Caputo M, Fabrizio M. On the notion of fractional derivative and applications to the hysteresis phenomena. Meccanica. 2017;**52**(1): 3043-3052. DOI: 10.1007/s11012-017-

[13] Mainardi F. Fractional calculus. In: Carpinteri A, Mainardi F, editors. Fractals and Fractional Calculus in Continuum Mechanics. International Centre for Mechanical Sciences (Courses and Lectures). Vol. 378. Vienna: Springer Verlag; 1997. DOI: 10.1007/978-3-7091-2664-6\_7

[14] Tenreiro Machado J, Silva M, Barbosa R, Jesus I, Reis C, Marcos M, et al. Some applications of fractional calculus in engineering. Mathematical Problems in Engineering. 2010;**2010**: 639801. DOI: 10.1155/2010/639801. 34p

[15] Baleanu D, Rezapour A,

Mohammadi H. Some existence results on nonlinear fractional differential equations. Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences. 2013;**371**(1990): 20120144. DOI: 10.1098/rsta.2012.0144

[16] Baleanu D, Agarwal R, Mohammadi H, Rezapour S. Some existence results

4289-8

gsteco1019

0652-y

[2] Capelas de Oliveira E, Tenreiro Machado J. A review of definitions for fractional derivatives and integral. Mathematical Problems in Engineering. 2014;**2014**:238459. DOI: 10.1155/2014/

[3] Oldham K, Spanier J. The Fractional Calculus. New York-London: Academic

[4] Miller K, Ross B. An Introduction to the Fractional Calculus and Fractional Differential Equations. New York: John Wiley & Sons, Inc; 1993. 382p. ISBN:

[5] Samko S, Kilbas A, Marichev O. Fractional Integrals and Derivatives. Theory and Applications. Amsterdam: Gordon and Breach Science Publishers;

1993. 1016p. ISBN: 2881248640

[7] Caputo M, Fabrizio M. A new definition of fractional derivative without singular kernel. Progress in Fractional Differentiation and Applications. 2015;**1**(2):73-85. DOI:

10.12785/pfda/010201

cnsns.2017.12.001

**18**

[6] Podlubny I. Fractional Differential Equations. San Diego: Academic Press; 1998. 340p. ISBN: 9780125588409

[8] Ortigueira M, Tenreiro Machado J. A critical analysis of the Caputo-Fabrizio operator. Communications in Nonlinear Science and Numerical Simulation. 2017;**59**:608-611. DOI: 10.1016/j.

[9] Giusti A. A comment on some new definitions of fractional derivative. Nonlinear Dynamics. 2018;**93**(3):

Press Inc.; 1974. 322p. ISBN:

[17] Abbas S. Existence of solutions to fractional order ordinary and delay differential equations and applications. Electronic Journal of Differential Equations. 2011;**2011**(9):1-11. ISSN: 1072-6691

[18] Shah K, Li Y. Existence theory of differential equations of arbitrary. In: Differential Equations—Theory and Current Research. London: IntechOpen; 2018. pp. 35-55. DOI: 10.5772/ intechopen.75523. Ch 2

[19] Maitama S, Abdullahi I. New analytical method for solving linear and nonlinear fractional partial differential equations. Progress in Fractional Differentiation and Applications. 2016; **2**(4):247-256. DOI: 10.18576/pfda/ 020402

[20] Sabatier J, Agrawal O, Tenreiro Machado J, editors. Advances in Fractional Calculus. The Netherlands: Springer; 2007. 549p. ISBN: 978-1- 4020-6041-0

[21] Odzijewicz T, Malinowska A, Torres D. Fractional calculus of variations in terms of a generalized fractional integral with applications to physics. Abstract and Applied Analysis. 2012;**2012**:871912. DOI: 10.1155/2012/871912. 24p

[22] Al Salti N, Karimov E, Kerbal S. Boundary value problems for fractional heat equation involving Caputo-Fabrizio derivative. New Trends in Mathematical Science. 2017;**4**(4):79-89. DOI: 10.20852/ntmsci.2016422308

[23] Losada J, Nieto J. Properties of a new fractional derivative without singular kernel. Progress in Fractional Differentiation and Applications. 2015; **1**(2):87-92. DOI: 10.12785/pfda/010202 [24] Baleanu D, Diethelm K, Scalas E, Trujillo J. Fractional Calculus: Models and Numerical Methods. Singapore: World Scientific Publishing; 2012. 400p. ISBN: 978-981-4355-20-9

[25] Landy A. Fractional differential equations and numerical methods [Thesis]. University of Chester; 2009

[26] Maitama S. Local fractional natural Homotopy perturbation method for solving partial differential equations with local fractional derivative. Progress in Fractional Differentiation and Applications. 2018;**4**(3):219-228. DOI: 10.18576/pfda/040306

[27] Zhuang P, Liu F, Turner I, Anh V. Galerkin finite element method and error analysis for the fractional cable equation. Numerical Algorithms. 2016; **72**(2):447-466. DOI: 10.1007/ s11075-015-0055-x

[28] Djida J, Area I, Atangana A. New Numerical Scheme of Atangana-Baleanu Fractional Integral: An Application to Groundwater Flow within Leaky Aquifer. 2016. arXiv: 1610.08681

[29] Sun H, Chen W, Sze K. A semidiscrete finite element method for a class of time-fractional diffusion equations. Philosophical Transactions of the Royal Society A - Mathematical Physical and Engineering Sciences. 2013;**371**(1990):20120268. DOI: 10.1098/rsta.2012.0268

[30] Yu Q, Liu F, Turner I, Burrage K. Stability and convergence of an implicit numerical method for the space and time fractional Bloch–Torrey equation. Philosophical Transactions of the Royal Society A - Mathematical Physical and Engineering Sciences. 2013;**371**(1990): 20120150. DOI: 10.1098/rsta.2012.0150

[31] Fukunaga M, Shimizu N. A highspeed algorithm for computation of fractional differentiation and fractional integration. Philosophical Transactions

of the Royal Society A - Mathematical Physical and Engineering Sciences. 2013;**371**:20120152. DOI: 10.1098/ rsta.2012.0152

[32] Tavares D, Almeida R, Torres D. Caputo derivatives of fractional variable order: numerical approximations. Communications in Nonlinear Science and Numerical Simulation. 2016;**35**:69-87. DOI: 10.1016/j. cnsns.2015.10.027

[33] Zhang J, Zhang X, Yang B. An approximation scheme for the time fractional convection–diffusion equation. Applied Mathematics and Computation. 2018;**335**:305-312. DOI: 10.1016/j.amc.2018.04.019

[34] Lin S, Lu C. Laplace transform for solving some families of fractional differential equations and its applications. Adv. Difference Equ. 2013; **2013**:137. DOI: 10.1186/1687-1847- 2013-137. 9p

[35] Mainardi F, Paradisi P. Fractional diffusive waves. Journal of Computational Acoustics. 2001;**9**(4): 1417-1436. DOI: 10.1016/S0218-396X (01)00082-6

[36] Parsian H. Time fractional wave equation: Caputo sense. Advanced Studies in Theoretical Physics. 2012; **6**(2):95-100

[37] Turut V, Güzel N. On solving partial differential equations of fractional oder by using the variational iteration method and multivariate Padé approximations. European Journal of Pure and Applied Mathematics. 2013; **6**(2):147-171. ISSN: 1307-5543

[38] Agarwal R, Sonal J. Mathematical modeling and analysis of dynamics of cytosolic calcium ion in astrocytes using fractional calculus. Journal of Fractional Calculus and Applications. 2018;**9**(2): 1-12. ISSN: 2090-5858

[39] Craiem D, Rojo F, Atienza J, Guinea G, Armentano R. Fractional calculus applied to model arterial viscoelasticity. Latin American Applied Research. 2008; **38**(2):141-145. ISNN: 0327–0793

[47] Fabio M, Serrano E. Infinitely oscillating wavelets and an efficient implementation algorithm based on the FFT. Revista de Matemática: Teoría y Aplicaciones. 2015;**22**(1):61-69. ISSN: 1409–2433 (Print), 2215–3373 (Online)

*DOI: http://dx.doi.org/10.5772/intechopen.86273*

*A Review on Fractional Differential Equations and a Numerical Method to Solve Some…*

[48] Fabio M, Troparevsky M. Numerical solution to initial value problems for fractional differential equations. Progress in Fractional Differentiation and Applications. 2019;

**5**(3):1-12. DOI: 10.18576/pfda/ INitialVP18NSP29nov. [in press]

[49] Troparevsky M, Fabio M.

3522

**21**

Approximate solutions to initial value problems with combined derivatives [in Spanish]. Mecánica Computacional. 2018;**XXXVI**(11):449-459. ISSN: 2591–

[40] Yu Y, Perdikaris P, Karniadakis G. Fractional modeling of viscoelasticity in 3D cerebral arteries and aneurysms. Journal of Computational Physics. 2016; **323**:219-242. DOI: 10.1016/j. jcp.2016.06.038

[41] Atangana A, Koca I. Chaos in a simple nonlinear system with Atangana–Baleanu derivatives with fractional order. Chaos, Solitons & Fractals. 2016;**89**:447-454. DOI: 10.1016/j.chaos.2016.02.012. 8p

[42] Gómez-Aguilar J, López-López M, Alvarado-Martínez V, Baleanu D, Khan H. Chaos in a cancer model via fractional derivatives with exponential decay and Mittag-Leffler law. Entropy. 2017;**19**:681. DOI: 10.3390/ e19120681. 19p

[43] Baleanu D, Wu G, Zeng S. Chaos analysis and asymptotic stability of generalized caputo fractional differential equations. Chaos, Solitons and Fractals. 2017;**102**:99-105. DOI: 10.1016/j.chaos.2017.02.007

[44] Fabio M, Troparevsky M. An inverse problem for the caputo fractional derivative by means of the wavelet transform. Progress in Fractional Differentiation and Applications. 2018;**4**(1):15-26. DOI: 10.18576/pfda/040103

[45] Meyer Y. Ondelettes et Operateurs II. Operatteurs de Calderon Zygmund. Paris: Hermann et Cie; 1990. 381p. ISBN: 2705661263

[46] Mallat S. A Wavelet Tour of Signal Processing. Academic Press; 2009. 832p. ISBN: 978–0–12-374370-1. DOI: 10.1016/B978-0-12-374370-1.X0001-8

*A Review on Fractional Differential Equations and a Numerical Method to Solve Some… DOI: http://dx.doi.org/10.5772/intechopen.86273*

[47] Fabio M, Serrano E. Infinitely oscillating wavelets and an efficient implementation algorithm based on the FFT. Revista de Matemática: Teoría y Aplicaciones. 2015;**22**(1):61-69. ISSN: 1409–2433 (Print), 2215–3373 (Online)

of the Royal Society A - Mathematical Physical and Engineering Sciences. 2013;**371**:20120152. DOI: 10.1098/

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

[39] Craiem D, Rojo F, Atienza J, Guinea G, Armentano R. Fractional calculus applied to model arterial viscoelasticity. Latin American Applied Research. 2008;

[40] Yu Y, Perdikaris P, Karniadakis G. Fractional modeling of viscoelasticity in 3D cerebral arteries and aneurysms. Journal of Computational Physics. 2016;

**38**(2):141-145. ISNN: 0327–0793

**323**:219-242. DOI: 10.1016/j.

[41] Atangana A, Koca I. Chaos in a simple nonlinear system with Atangana–Baleanu derivatives with fractional order. Chaos, Solitons & Fractals. 2016;**89**:447-454. DOI: 10.1016/j.chaos.2016.02.012. 8p

[42] Gómez-Aguilar J, López-López M, Alvarado-Martínez V, Baleanu D, Khan H. Chaos in a cancer model via fractional derivatives with exponential

[43] Baleanu D, Wu G, Zeng S. Chaos analysis and asymptotic stability of generalized caputo fractional

differential equations. Chaos, Solitons and Fractals. 2017;**102**:99-105. DOI:

[45] Meyer Y. Ondelettes et Operateurs II. Operatteurs de Calderon Zygmund. Paris: Hermann et Cie; 1990. 381p.

[46] Mallat S. A Wavelet Tour of Signal Processing. Academic Press; 2009. 832p.

ISBN: 978–0–12-374370-1. DOI: 10.1016/B978-0-12-374370-1.X0001-8

decay and Mittag-Leffler law. Entropy. 2017;**19**:681. DOI: 10.3390/

10.1016/j.chaos.2017.02.007

10.18576/pfda/040103

ISBN: 2705661263

[44] Fabio M, Troparevsky M. An inverse problem for the caputo fractional derivative by means of the wavelet transform. Progress in Fractional Differentiation and Applications. 2018;**4**(1):15-26. DOI:

e19120681. 19p

jcp.2016.06.038

[32] Tavares D, Almeida R, Torres D. Caputo derivatives of fractional variable order: numerical approximations. Communications in Nonlinear Science and Numerical Simulation. 2016;**35**:69-87. DOI: 10.1016/j.

[33] Zhang J, Zhang X, Yang B. An approximation scheme for the time fractional convection–diffusion equation. Applied Mathematics and Computation. 2018;**335**:305-312. DOI:

[34] Lin S, Lu C. Laplace transform for solving some families of fractional differential equations and its

applications. Adv. Difference Equ. 2013; **2013**:137. DOI: 10.1186/1687-1847-

[35] Mainardi F, Paradisi P. Fractional

Computational Acoustics. 2001;**9**(4): 1417-1436. DOI: 10.1016/S0218-396X

[36] Parsian H. Time fractional wave equation: Caputo sense. Advanced Studies in Theoretical Physics. 2012;

[37] Turut V, Güzel N. On solving partial differential equations of fractional oder by using the variational iteration method and multivariate Padé approximations. European Journal of Pure and Applied Mathematics. 2013;

**6**(2):147-171. ISSN: 1307-5543

1-12. ISSN: 2090-5858

**20**

[38] Agarwal R, Sonal J. Mathematical modeling and analysis of dynamics of cytosolic calcium ion in astrocytes using fractional calculus. Journal of Fractional Calculus and Applications. 2018;**9**(2):

10.1016/j.amc.2018.04.019

diffusive waves. Journal of

rsta.2012.0152

cnsns.2015.10.027

2013-137. 9p

(01)00082-6

**6**(2):95-100

[48] Fabio M, Troparevsky M. Numerical solution to initial value problems for fractional differential equations. Progress in Fractional Differentiation and Applications. 2019; **5**(3):1-12. DOI: 10.18576/pfda/ INitialVP18NSP29nov. [in press]

[49] Troparevsky M, Fabio M. Approximate solutions to initial value problems with combined derivatives [in Spanish]. Mecánica Computacional. 2018;**XXXVI**(11):449-459. ISSN: 2591– 3522

**Chapter 2**

**Abstract**

examples are provided.

**1. Introduction**

**23**

Polynomials

Numerical Solutions to Some

Families of Fractional Order

*Adnan Khan, Kamal Shah and Danfeng Luo*

Differential Equations by Laguerre

This article is devoted to compute numerical solutions of some classes and families of fractional order differential equations (FODEs). For the required numerical analysis, we utilize Laguerre polynomials and establish some operational matrices regarding to fractional order derivatives and integrals without discretizing the data. Further corresponding to boundary value problems (BVPs), we establish a new operational matrix which is used to compute numerical solutions of boundary value problems (BVPs) of FODEs. Based on these operational matrices (OMs), we convert the proposed (FODEs) or their system to corresponding algebraic equation of Sylvester type or system of Sylvester type. The resulting algebraic equations are solved by MATLAB® using Gauss elimination method for the unknown coefficient matrix. To demonstrate the suggested scheme for numerical solution, many suitable

**Keywords:** FODEs, numerical solution, Laguerre polynomials, operational matrices

The theory of integrals as well as derivatives of arbitrary order is known by the special name "fractional calculus." It has an old history just like classical calculus. The chronicle of fractional calculus and encyclopedic book can be studied in [1, 2]. Researchers have now necessitated the use of fractional calculus due to its diverse applications in different fields, specially in electrical networks, signal and image processing and optics, etc. For conspicuous work on FODEs in the fields of dynamical systems, electrochemistry, advanced techniques of microorganisms culturing, weather forecasting, as well as statistics, we refer to peruse [3, 4]. Fractional derivatives show valid results in most cases where ordinary derivatives do not. Also annotating that fractional order derivatives as well as fractional integrals are global operators, while ordinary derivatives are local operators. Fractional order derivative provides greater degree of freedom. Therefore from different aspects, the aforesaid areas were investigated. For instance, many researchers have provide understanding to existence and uniqueness results about FODEs, for few results, we refer [5–7], and many others have actualized the instinctive framework of fractional differential equations in various problems [8–19] with many references included in them.

#### **Chapter 2**

## Numerical Solutions to Some Families of Fractional Order Differential Equations by Laguerre Polynomials

*Adnan Khan, Kamal Shah and Danfeng Luo*

### **Abstract**

This article is devoted to compute numerical solutions of some classes and families of fractional order differential equations (FODEs). For the required numerical analysis, we utilize Laguerre polynomials and establish some operational matrices regarding to fractional order derivatives and integrals without discretizing the data. Further corresponding to boundary value problems (BVPs), we establish a new operational matrix which is used to compute numerical solutions of boundary value problems (BVPs) of FODEs. Based on these operational matrices (OMs), we convert the proposed (FODEs) or their system to corresponding algebraic equation of Sylvester type or system of Sylvester type. The resulting algebraic equations are solved by MATLAB® using Gauss elimination method for the unknown coefficient matrix. To demonstrate the suggested scheme for numerical solution, many suitable examples are provided.

**Keywords:** FODEs, numerical solution, Laguerre polynomials, operational matrices

#### **1. Introduction**

The theory of integrals as well as derivatives of arbitrary order is known by the special name "fractional calculus." It has an old history just like classical calculus. The chronicle of fractional calculus and encyclopedic book can be studied in [1, 2]. Researchers have now necessitated the use of fractional calculus due to its diverse applications in different fields, specially in electrical networks, signal and image processing and optics, etc. For conspicuous work on FODEs in the fields of dynamical systems, electrochemistry, advanced techniques of microorganisms culturing, weather forecasting, as well as statistics, we refer to peruse [3, 4]. Fractional derivatives show valid results in most cases where ordinary derivatives do not. Also annotating that fractional order derivatives as well as fractional integrals are global operators, while ordinary derivatives are local operators. Fractional order derivative provides greater degree of freedom. Therefore from different aspects, the aforesaid areas were investigated. For instance, many researchers have provide understanding to existence and uniqueness results about FODEs, for few results, we refer [5–7], and many others have actualized the instinctive framework of fractional differential equations in various problems [8–19] with many references included in them.

Often it is very difficult to obtain the exact solution due to global nature of fractional derivatives in differential equations. Contrarily approximate solutions are obtained by numerical methods assorted in [20–22]. Various new numerical methods have been developed, among them is one famous method called "spectral method" which is used to solve problems in various realms [23]. In this method operational matrices are obtained by using orthogonal polynomials [24]. Many authors have successfully developed operational matrices by using Legender, Jacobi, and various other polynomials [25, 26]. For delay differential and various other related equations, Laguerre spectral methods have been used [27–32]. Bernstein polynomials and various classes of other polynomials were also used to obtain operational matrices corresponding to fractional integrals and derivatives [33–40]. Apart from them, operational matrices were also developed with the collocation method (see Refs. [41–43]). Since spectral methods are powerful tools to compute numerical solutions of both ODEs and FODEs. Therefore, we bring out numerical analysis via using Laguerre polynomials of some families and coupled systems of FODEs under initial as well as boundary conditions. In this regard we investigate the numerical solutions to the given families under initial conditions

$$\begin{cases} \,^c\_0 D\_t^\gamma z(t) \pm z(t) = 0, \quad 0 < \chi \le 1, \\ z(0) = z\_0, \quad z\_0 \in R, \end{cases} \tag{1}$$

**Definition 1.** The fractional integral of order *γ* >0 of a function *z* : ð Þ! 0, ∞ *R* is

*Numerical Solutions to Some Families of Fractional Order Differential Equations by Laguerre…*

ð*t* 0

provided the integral converges at the right sides. Further a simple and impor-

*z s*ð Þ ð Þ *t* � *s*

<sup>1</sup>� *<sup>γ</sup> ds*,

*t <sup>γ</sup>* <sup>þ</sup>*<sup>δ</sup> :*

> *f* ð Þ *<sup>n</sup>* ð Þ*<sup>s</sup> ds*,

*f* 0 ð Þ*s ds:*

ð Þ *<sup>t</sup>* � *<sup>s</sup> <sup>n</sup>*� *<sup>γ</sup>* �<sup>1</sup>

ð Þ *<sup>t</sup>* � *<sup>s</sup>* � *<sup>γ</sup>* �<sup>1</sup>

*n*�1

*n*�1 þ0*I γ <sup>t</sup> h t*ð Þ,

, *n* ¼ ½ �þ *γ* 1*:*

*<sup>i</sup>* ð Þ*t* and

0*I γ*

0*I γ t t*

**Definition 2.** Caputo fractional derivative is defined as

*<sup>t</sup> f t*ðÞ¼ <sup>1</sup>

*<sup>t</sup> f t*ðÞ¼ <sup>1</sup>

0< *γ* ≤ 1, then Caputo fractional derivative becomes

*f t*ðÞ¼ *d*<sup>0</sup> þ *d*1*t* þ *d*2*t*

*c* 0*D<sup>γ</sup>*

*f t*ðÞ¼ *d*<sup>0</sup> þ *d*1*t* þ *d*2*t*

where *di* for *i* ¼ 0, 1, 2, 3 … *n* � 1 are real constants.

*f t*ð Þ¼0*I γ*

Γð Þ *n* � *γ*

Γð Þ 1 � *γ*

*c* 0*D<sup>γ</sup>*

*<sup>t</sup>* is given by

*<sup>t</sup> z t*ðÞ¼ <sup>1</sup>

Γð Þ *γ*

*<sup>δ</sup>* <sup>¼</sup> <sup>Γ</sup>ð Þ *<sup>δ</sup>* <sup>þ</sup> <sup>1</sup> Γð Þ *δ* þ *γ* þ 1

> ð*t* 0

where *n* is a positive integer with the property that *n* � 1< *γ* ≤*n:* For example, if

ð*t* 0

*<sup>t</sup> f t*ðÞ¼ 0

<sup>2</sup> <sup>þ</sup> … <sup>þ</sup> *dn*�<sup>1</sup>*<sup>t</sup>*

*<sup>t</sup> f t*ðÞ¼ *h t*ð Þ, *n* � 1< *γ* ≤ *n*

*<sup>t</sup> h t*ðÞþX*<sup>n</sup>*�<sup>1</sup>

**Definition 3.** The famous Laguerre polynomials are represented by *L<sup>γ</sup>*

ð Þ �<sup>1</sup> *<sup>k</sup>*

<sup>2</sup> <sup>þ</sup> … <sup>þ</sup> *dn*�<sup>1</sup>*<sup>t</sup>*

*i*¼0

*f i* ð Þ 0 *<sup>i</sup>*! *<sup>t</sup> i :*

Γð Þ *i* þ *γ* þ 1 Γð Þ *k* þ 1 þ *γ* Γð Þ *i* � *k* þ 1 Γð Þ *k* þ 1

*t k :*

**Lemma 1.** Therefore in view of this result, if *<sup>h</sup>*∈*L<sup>n</sup>*½ � 0, *<sup>T</sup>* , then the unique

defined by

tant property of <sup>0</sup>*I*

*γ*

*DOI: http://dx.doi.org/10.5772/intechopen.90754*

*c* 0*D<sup>γ</sup>*

> *c* 0*D<sup>γ</sup>*

**Theorem 1.** The FODE given by

has a unique solution, such that

solution of nonhomogenous FODE

The above lemma is also stated as

*Lγ*

*<sup>i</sup>* ðÞ¼ *<sup>t</sup>* <sup>X</sup> *i*

*k*¼0

is written as

defined as

**25**

and subject to boundary conditions

$$\begin{cases} \,^c\_0 D\_t^\gamma z(t) \pm z(t) = 0, & 1 < \chi \le 2, \\ z(0) = z\_0, z(1) = z\_1, & z\_0, z\_1 \in R. \end{cases} \tag{2}$$

By similar numerical techniques, we also investigate the numerical solutions to the following systems with fractional order derivatives under initial and boundary conditions as

$$\begin{cases} \,^\epsilon\_0 D\_t^\mathcal{V} z(t) + az(t) + by(t) = f(t), \\\,^\epsilon\_0 D\_t^\mathcal{V} y(t) + cy(t) + dz(t) = g(t), \\\,^\epsilon z(\mathbf{0}) = z\_0, y(\mathbf{0}) = y\_0 \end{cases} \tag{3}$$

for 0< *γ* ≤ 1 and

$$\begin{cases} \,^c\_0 D\_t^\gamma z(t) az(t) + b\boldsymbol{\gamma}(t) = f(t), \\\,^c\_0 D\_t^\gamma \boldsymbol{\gamma}(t) + c\boldsymbol{\gamma}(t) + d\boldsymbol{z}(t) = \mathbf{g}(t), \\\,^c\_2 \mathbf{(0)} = \mathbf{z}\_0, \boldsymbol{\gamma}(\mathbf{0}) = \mathbf{y}\_0, \quad z(\mathbf{1}) = z\_1, \boldsymbol{\gamma}(\mathbf{1}) = \mathbf{y}\_1, \end{cases} \tag{4}$$

for 1<sup>&</sup>lt; *<sup>γ</sup>* <sup>≤</sup> 2 where *<sup>f</sup>*, *<sup>g</sup>* : ½ �� 0, 1 *<sup>R</sup>*<sup>2</sup> ! *<sup>R</sup>* and *<sup>z</sup>*0, *<sup>y</sup>*0, *<sup>z</sup>*1, *<sup>y</sup>*<sup>1</sup> <sup>∈</sup> *<sup>R</sup>:* We first obtain OMs for fractional derivatives and integrals by using Laguerre polynomials. Also corresponding to boundary conditions, we construct an operational matrix which is needed in numerical analysis of BVPs. With the help of the OMs we convert the considered problem of FODEs under initial/boundary conditions to Sylvester-type algebraic equations. Solving the mentioned matrix equations by using MATLAB®, we compute the numerical solutions of the considered problems.

#### **2. Preliminaries**

Here we recall some basic definition results that are needed in this work onward, keeping in mind that throughout the paper we use fractional derivative in Caputo sense. *Numerical Solutions to Some Families of Fractional Order Differential Equations by Laguerre… DOI: http://dx.doi.org/10.5772/intechopen.90754*

**Definition 1.** The fractional integral of order *γ* >0 of a function *z* : ð Þ! 0, ∞ *R* is defined by

$${}\_{0}I\_{t}^{\gamma}z(t) = \frac{1}{\Gamma(\gamma)} \int\_{0}^{t} \frac{z(s)}{(t-s)^{1-\gamma}}ds,$$

provided the integral converges at the right sides. Further a simple and important property of <sup>0</sup>*I γ <sup>t</sup>* is given by

$${}\_{0}I\_{t}^{\gamma}t^{\delta} = \frac{\Gamma(\delta+1)}{\Gamma(\delta+\gamma+1)}t^{\gamma+\delta}.$$

**Definition 2.** Caputo fractional derivative is defined as

$$\,\_0^\varepsilon D\_t^\gamma f(t) = \frac{1}{\Gamma(n-\gamma)} \int\_0^t (t-s)^{n-\gamma-1} f^{(n)}(s)ds,$$

where *n* is a positive integer with the property that *n* � 1< *γ* ≤*n:* For example, if 0< *γ* ≤ 1, then Caputo fractional derivative becomes

$$\,\_0^\varepsilon D\_t^\gamma f(t) = \frac{1}{\Gamma(1-\gamma)} \int\_0^t (t-s)^{-\gamma-1} f'(s)ds.$$

**Theorem 1.** The FODE given by

$$\,\_0^cD\_t^\circ f(t) = 0$$

has a unique solution, such that

$$f(t) = d\_0 + d\_1 t + d\_2 t^2 + \dots + d\_{n-1} t^{n-1}, \ n = \lceil \chi \rceil + 1.$$

**Lemma 1.** Therefore in view of this result, if *<sup>h</sup>*∈*L<sup>n</sup>*½ � 0, *<sup>T</sup>* , then the unique solution of nonhomogenous FODE

$$\,\_0^cD\_t^\gamma f(t) = h(t), \; n - 1 < \gamma \le n$$

is written as

Often it is very difficult to obtain the exact solution due to global nature of fractional derivatives in differential equations. Contrarily approximate solutions are obtained by numerical methods assorted in [20–22]. Various new numerical methods have been developed, among them is one famous method called "spectral method" which is used to solve problems in various realms [23]. In this method operational matrices are obtained by using orthogonal polynomials [24]. Many authors have successfully developed operational matrices by using Legender, Jacobi, and various other polynomials [25, 26]. For delay differential and various other related equations, Laguerre spectral methods have been used [27–32]. Bernstein polynomials and various classes of other polynomials were also used to obtain operational matrices corresponding to fractional integrals and derivatives [33–40]. Apart from them, operational matrices were also developed with the collocation method (see Refs. [41–43]). Since spectral methods are powerful tools to compute numerical solutions of both ODEs and FODEs. Therefore, we bring out numerical analysis via using Laguerre polynomials of some families and coupled systems of FODEs under initial as well as boundary conditions. In this regard we investigate

the numerical solutions to the given families under initial conditions

*z*ð Þ¼ 0 *z*0, *z*<sup>0</sup> ∈*R*,

*<sup>t</sup> z t*ðÞ� *z t*ðÞ¼ 0, 0 < *γ* ≤1,

*<sup>t</sup> z t*ð Þ� *z t*ðÞ¼ 0, 1< *γ* ≤ 2, *z*ð Þ¼ 0 *z*0, *z*ð Þ¼ 1 *z*1, *z*0, *z*<sup>1</sup> ∈*R:*

By similar numerical techniques, we also investigate the numerical solutions to the following systems with fractional order derivatives under initial and boundary

*<sup>t</sup> z t*ðÞþ *az t*ðÞþ *by t*ðÞ¼ *f t*ð Þ,

*<sup>t</sup> y t*ðÞþ *cy t*ðÞþ *dz t*ðÞ¼ *g t*ð Þ,

*z*ð Þ¼ 0 *z*0, *y*ð Þ¼ 0 *y*<sup>0</sup>

*z*ð Þ¼ 0 *z*0, *y*ð Þ¼ 0 *y*0, *z*ð Þ¼ 1 *z*1, *y*ð Þ¼ 1 *y*1,

for 1<sup>&</sup>lt; *<sup>γ</sup>* <sup>≤</sup> 2 where *<sup>f</sup>*, *<sup>g</sup>* : ½ �� 0, 1 *<sup>R</sup>*<sup>2</sup> ! *<sup>R</sup>* and *<sup>z</sup>*0, *<sup>y</sup>*0, *<sup>z</sup>*1, *<sup>y</sup>*<sup>1</sup> <sup>∈</sup> *<sup>R</sup>:* We first obtain OMs for fractional derivatives and integrals by using Laguerre polynomials. Also corresponding to boundary conditions, we construct an operational matrix which is needed in numerical analysis of BVPs. With the help of the OMs we convert the considered problem of FODEs under initial/boundary conditions to Sylvester-type algebraic equations. Solving the mentioned matrix equations by using MATLAB®,

Here we recall some basic definition results that are needed in this work onward, keeping in mind that throughout the paper we use fractional derivative in Caputo sense.

*<sup>t</sup> z t*ð Þ*az t*ðÞþ *by t*ðÞ¼ *f t*ð Þ,

we compute the numerical solutions of the considered problems.

*<sup>t</sup> y t*ðÞþ *cy t*ðÞþ *dz t*ðÞ¼ *g t*ð Þ,

(1)

(2)

(3)

(4)

*c* 0*D<sup>γ</sup>*

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

*c* 0*D<sup>γ</sup>*

> *c* 0*D<sup>γ</sup>*

(

*c* 0*D<sup>γ</sup>*

8 ><

>:

*c* 0*D<sup>γ</sup>* *c* 0*D<sup>γ</sup>*

�

�

and subject to boundary conditions

conditions as

for 0< *γ* ≤ 1 and

**2. Preliminaries**

**24**

$$f(t) = d\_0 + d\_1 t + d\_2 t^2 + \dots + d\_{n-1} t^{n-1} + {}\_0 I\_t^\gamma h(t),$$

where *di* for *i* ¼ 0, 1, 2, 3 … *n* � 1 are real constants. The above lemma is also stated as

$$f(t) = {}\_0I\_t^\gamma h(t) + \sum\_{i=0}^{n-1} \frac{f^i(0)}{i!} t^i.$$

**Definition 3.** The famous Laguerre polynomials are represented by *L<sup>γ</sup> <sup>i</sup>* ð Þ*t* and defined as

$$L\_i^{\gamma}(t) = \sum\_{k=0}^{i} \frac{(-1)^k \Gamma(i+\gamma+1)}{\Gamma(k+1+\gamma)\Gamma(i-k+1)\Gamma(k+1)} t^k.$$

They are orthogonal on 0, ½ � <sup>∞</sup> *:* If *<sup>L</sup><sup>γ</sup> <sup>i</sup>* ð Þ*<sup>t</sup>* and *<sup>L</sup><sup>γ</sup> <sup>j</sup>* ð Þ*t* are Laguerre polynomials, then the orthogonality condition is given as

$$\int\_0^\infty L\_i^\gamma \left( t \right) L\_j^\gamma \left( t \right) W^\gamma \left( t \right) dt = \delta\_{i,j} U\_{k,i}$$

where

$$W^r(t) = t^r e^{-t},$$

is the weight function and

$$U\_k = \begin{cases} \frac{\Gamma(\mathbf{1} + \mathbf{y} + k)}{\Gamma(\mathbf{1} + k)}, & i = j \\ \mathbf{0} & i \neq j. \end{cases}$$

Now let *Z t*ð Þ be any function, defined on the interval 0, ½ � ∞ *:* We express the function in terms of Laguerre polynomials as

$$\begin{aligned} Z(t) &= \sum\_{i=0}^{n} c\_i L\_i^{\gamma}(t). \\ &= c\_0 L\_0^{\gamma}(t) + c\_1 L\_1^{\gamma}(t) + \dots + c\_N L\_N^{\gamma}(t) \\ &= \begin{bmatrix} c\_0 & c\_1 & \dots & c\_N \end{bmatrix} \begin{bmatrix} L\_0^{\gamma}(t) \\ \vdots \\ L\_n^{\gamma}(t) \end{bmatrix}. \end{aligned} \tag{5}$$

*ci* <sup>¼</sup> <sup>1</sup> *hi* ð*L* 0

In vector form we can write Eq. (5) as

*DOI: http://dx.doi.org/10.5772/intechopen.90754*

considering whole function constant except *t*

*<sup>i</sup>* ðÞ¼ *<sup>t</sup>* <sup>X</sup> *i*

*k*¼0 *t*

<sup>∥</sup>*PM*,*az* � *z t*ð Þ∥*As*

*<sup>i</sup>* ð Þ*t* be given; then

fractional order derivative for *t*

*c* 0*D<sup>γ</sup> <sup>t</sup> L<sup>γ</sup>*

**Lemma 2.** Let *L<sup>β</sup>*

and 0≤*s* ≤*a*, and then

where *Aa*

*L*2

**integrals**

**27**

*c* 0*D<sup>γ</sup> <sup>t</sup> L<sup>β</sup>*

**2.2 Error analysis**

function vector.

**derivative**

*Z t*ð Þ*<sup>W</sup> <sup>γ</sup>* ð Þ*<sup>t</sup> <sup>L</sup><sup>γ</sup>*

*Numerical Solutions to Some Families of Fractional Order Differential Equations by Laguerre…*

where *M* = *m* þ 1, *cM* is the *M* terms coefficient vector and Ψ*M*ð Þ*t* is the *M* terms

If the Caputo fractional order derivative is applied to Laguerre polynomial, by

*<sup>k</sup>* to obtain (6) as

*<sup>k</sup>*� *<sup>γ</sup>* � � ð Þ �<sup>1</sup> *<sup>k</sup>*

The proof of the following results can be found with details in [20].

*<sup>α</sup>*,Λ<sup>≤</sup> *cMs*�*<sup>a</sup>*

*<sup>α</sup>*,ð Þ¼ <sup>Λ</sup> <sup>∥</sup>*∂<sup>a</sup>*

*<sup>α</sup>*,ð Þ¼ <sup>Λ</sup> <sup>X</sup>*<sup>a</sup>*

**3. Operational matrices corresponding to fractional derivatives and**

Here in this section, we provide the required OMs via Laguerre polynomials of

*<sup>α</sup>* ¼ f *<sup>z</sup>=<sup>z</sup>* is measurable on <sup>Λ</sup> and <sup>∥</sup>*z*<sup>∥</sup> *Aa*

∣*z*∣*Aa*

Now let Λ ¼ ϱ*=*0<ϱ < ∞ with *χ*ð Þϱ be a weight function. Then

∥*z*∥*A<sup>a</sup>*

*<sup>χ</sup>* ð Þ¼f Λ *κ = κ* is measurable on Λ and ∥*u*∥*L*<sup>2</sup>

ð Λ

with the following inner product and norm

ð Þ *u*, *v <sup>χ</sup>*,<sup>Λ</sup> ¼

fractional derivatives and integrals.

*<sup>i</sup>* ðÞ¼ *t* 0, *i* ¼ 0, 1, 2, ⋯, ½ �� *β* 1, *γ* >0*:*

<sup>2</sup> <sup>∣</sup>*z t*ð Þ∣*Aa*

*k*¼0 j j *z* 2 *Aa <sup>α</sup>*,ð Þ Λ

*pz*∥*wα*þ*a*,Λ,

!<sup>1</sup>

*χ*

*<sup>u</sup>*ð Þ<sup>ϱ</sup> *<sup>v</sup>*ð Þ<sup>ϱ</sup> *<sup>d</sup>*ϱ, <sup>∥</sup>*v*∥*χ*,<sup>Λ</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffi

**Theorem 2.** For error analysis, we state the theorem such that, *a* be any integer

*Z t*ðÞ¼ *c t <sup>M</sup>*Ψ*M*ð Þ*t :*

**2.1 Representation of Laguerre polynomial with Caputo fractional order**

*<sup>j</sup>* ð Þ*t dt:*

*<sup>k</sup>:* We use the definition of Caputo

Γð Þ *i* þ *γ* þ 1 <sup>Γ</sup>ð Þ *<sup>k</sup>* <sup>þ</sup> <sup>1</sup> <sup>þ</sup> *<sup>γ</sup>* <sup>Γ</sup>ð Þ *<sup>i</sup>* � *<sup>k</sup>* <sup>þ</sup> <sup>1</sup> <sup>Γ</sup>ð Þ <sup>1</sup> <sup>þ</sup> *<sup>k</sup>* � *<sup>γ</sup> :* (6)

*<sup>α</sup>*,Λ, <sup>∀</sup>*z t*ð Þϵ*Aa*

2 *:*

,Λ < ∞g.

*<sup>α</sup>*,ð Þ Λ < ∞ g and

h i *<sup>u</sup>*, *<sup>v</sup>* <sup>p</sup>

*<sup>χ</sup>*,Λ*:*

*<sup>α</sup>*ð Þ Λ ,

We set the above two vectors into their inner product and represent the column matrix by Ψð Þ*t* , so that

$$Z(t) = c^t \Psi(t).$$

Again as

$$\begin{aligned} Z(t) &= \sum\_{i=0}^{n} c\_i L\_i^{\mathcal{I}}(t), \\ \int\_0^L Z(t) W^{\mathcal{I}}(t) L\_{\boldsymbol{j}}^{\mathcal{I}}(t) dt &= \int\_0^L \sum\_{i=0}^n c\_i L\_i^{\mathcal{I}}(t) L\_{\boldsymbol{j}}^{\mathcal{I}}(t) W^{\mathcal{I}}(t) dt, \end{aligned}$$

which is written as

$$\sum\_{i=0}^{n} c\_i \int\_0^L L\_i^{\mathcal{I}}\left(t\right) L\_j^{\mathcal{I}}\left(t\right) W^{\mathcal{I}}\left(t\right) dt.$$

We call *hi* to the general term of integration

$$\int\_0^L Z(t)W^\gamma(t)L\_\dagger^\gamma(t)dt = \sum\_{i=0}^n c\_i h\_i.$$

Hence the coefficient *ci* is

*Numerical Solutions to Some Families of Fractional Order Differential Equations by Laguerre… DOI: http://dx.doi.org/10.5772/intechopen.90754*

$$c\_i = \frac{1}{h\_i} \int\_0^L Z(t) W^\gamma(t) L\_j^\gamma(t) dt.$$

In vector form we can write Eq. (5) as

$$Z(t) = c\_M^t \Psi\_M(t).$$

where *M* = *m* þ 1, *cM* is the *M* terms coefficient vector and Ψ*M*ð Þ*t* is the *M* terms function vector.

#### **2.1 Representation of Laguerre polynomial with Caputo fractional order derivative**

If the Caputo fractional order derivative is applied to Laguerre polynomial, by considering whole function constant except *t <sup>k</sup>:* We use the definition of Caputo fractional order derivative for *t <sup>k</sup>* to obtain (6) as

$$\,\_{0}^{c}D\_{t}^{\gamma}L\_{i}^{\gamma}(t) = \sum\_{k=0}^{i} \binom{k-\gamma}{t} \frac{(-1)^{k}\Gamma(i+\gamma+1)}{\Gamma(k+1+\gamma)\Gamma(i-k+1)\Gamma(1+k-\gamma)}.\tag{6}$$

#### **2.2 Error analysis**

They are orthogonal on 0, ½ � <sup>∞</sup> *:* If *<sup>L</sup><sup>γ</sup>*

ð<sup>∞</sup> 0 *Lγ <sup>i</sup>* ð Þ*<sup>t</sup> <sup>L</sup><sup>γ</sup>*

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

*Uk* ¼

function in terms of Laguerre polynomials as

8 < :

the orthogonality condition is given as

is the weight function and

matrix by Ψð Þ*t* , so that

ð*L* 0

Hence the coefficient *ci* is

**26**

which is written as

*Z t*ð Þ*<sup>W</sup> <sup>γ</sup>* ð Þ*<sup>t</sup> <sup>L</sup><sup>γ</sup>*

We call *hi* to the general term of integration

ð*L* 0

Again as

where

*<sup>i</sup>* ð Þ*<sup>t</sup>* and *<sup>L</sup><sup>γ</sup>*

*<sup>W</sup> <sup>γ</sup>* ðÞ¼ *<sup>t</sup> <sup>t</sup> <sup>γ</sup> <sup>e</sup>*

Γð Þ 1 þ *γ* þ *k*

Now let *Z t*ð Þ be any function, defined on the interval 0, ½ � ∞ *:* We express the

*i*¼0

<sup>¼</sup> *<sup>c</sup>*0*L<sup>γ</sup>*

*ciL<sup>γ</sup> <sup>i</sup>* ð Þ*t :*

¼ ½ � *c*<sup>0</sup> *c*<sup>1</sup> … *cN*

We set the above two vectors into their inner product and represent the column

*Z t*ðÞ¼ *c t* Ψð Þ*t :*

*Z t*ðÞ¼ <sup>X</sup>*<sup>n</sup>*

ð*L* 0

*<sup>j</sup>* ð Þ*t dt* ¼

*Z t*ð Þ*<sup>W</sup> <sup>γ</sup>* ð Þ*<sup>t</sup> <sup>L</sup><sup>γ</sup>*

X*n i*¼0 *ci* ð*L* 0 *Lγ <sup>i</sup>* ð Þ*<sup>t</sup> <sup>L</sup><sup>γ</sup>*

*i*¼0

X*n i*¼0

*ciL<sup>γ</sup> <sup>i</sup>* ð Þ*t* ,

> *ciL<sup>γ</sup> <sup>i</sup>* ð Þ*<sup>t</sup> <sup>L</sup><sup>γ</sup>*

*<sup>j</sup>* ð Þ*<sup>t</sup> <sup>W</sup> <sup>γ</sup>* ð Þ*<sup>t</sup> dt:*

*<sup>j</sup>* ð Þ*<sup>t</sup> dt* <sup>¼</sup> <sup>X</sup>*<sup>n</sup>*

*i*¼0

*cihi:*

<sup>0</sup> ðÞþ*<sup>t</sup> <sup>c</sup>*1*L<sup>γ</sup>*

<sup>1</sup> ðÞþ*<sup>t</sup>* … <sup>þ</sup> *cNL<sup>γ</sup>*

*<sup>j</sup>* ð Þ*<sup>t</sup> <sup>W</sup> <sup>γ</sup>* ð Þ*<sup>t</sup> dt*,

*Lγ* <sup>0</sup> ð Þ*t* ⋮ *Lγ <sup>n</sup>* ð Þ*t*

*<sup>N</sup>*ð Þ*t*

(5)

*Z t*ðÞ¼ <sup>X</sup>*<sup>n</sup>*

*<sup>j</sup>* ð Þ*<sup>t</sup> <sup>W</sup> <sup>γ</sup>* ð Þ*<sup>t</sup> dt* <sup>¼</sup> *<sup>δ</sup>i*,*jUk*,

�*t* ,

<sup>Γ</sup>ð Þ <sup>1</sup> <sup>þ</sup> *<sup>k</sup>* , *<sup>i</sup>* <sup>¼</sup> *<sup>j</sup>* 0 *i* 6¼ *j:*

*<sup>j</sup>* ð Þ*t* are Laguerre polynomials, then

The proof of the following results can be found with details in [20]. **Lemma 2.** Let *L<sup>β</sup> <sup>i</sup>* ð Þ*t* be given; then

$${}^{\xi}\_{0}D\_{t}^{\gamma}L\_{i}^{\beta}(t) = \mathbf{0}, \qquad i = \mathbf{0}, \mathbf{1}, \mathbf{2}, \cdots, [\beta] - \mathbf{1}, \gamma > \mathbf{0}.$$

**Theorem 2.** For error analysis, we state the theorem such that, *a* be any integer and 0≤*s* ≤*a*, and then

$$\|\|P\_{\mathcal{M},\mathfrak{a}\mathfrak{z}} - \mathfrak{z}(t)\|\|A\_a^s, \Lambda \le c\mathcal{M}^{\frac{s-4}{2}}\|\mathfrak{z}(t)\|A\_a^a, \Lambda, \forall \mathfrak{z}(t)\epsilon A\_a^a(\Lambda),$$

where *Aa <sup>α</sup>* ¼ f *<sup>z</sup>=<sup>z</sup>* is measurable on <sup>Λ</sup> and <sup>∥</sup>*z*<sup>∥</sup> *Aa <sup>α</sup>*,ð Þ Λ < ∞ g and

$$\begin{aligned} |z|A\_a^a, (\Lambda) &= \|\partial\_p^a z\|\_{\omega a + a, \Lambda}, \\ \|z\| A\_a^a, (\Lambda) &= \left(\sum\_{k=0}^a |z|\_{A\_a^a, (\Lambda)}^2\right)^{\frac{1}{2}}. \end{aligned}$$

Now let Λ ¼ ϱ*=*0<ϱ < ∞ with *χ*ð Þϱ be a weight function. Then *L*2 *<sup>χ</sup>* ð Þ¼f Λ *κ = κ* is measurable on Λ and ∥*u*∥*L*<sup>2</sup> *χ* ,Λ < ∞g. with the following inner product and norm

$$\left.u(\boldsymbol{u},\boldsymbol{v})\right|\_{\boldsymbol{\chi},\boldsymbol{\Lambda}} = \int\_{\boldsymbol{\Lambda}} \boldsymbol{u}(\boldsymbol{\varrho})\boldsymbol{v}(\boldsymbol{\varrho})d\boldsymbol{\varrho}, \qquad \|\boldsymbol{v}\|\boldsymbol{\varrho},\boldsymbol{\Lambda} = \sqrt{\langle \boldsymbol{u},\boldsymbol{v} \rangle}\_{\boldsymbol{\chi},\boldsymbol{\Lambda}}.$$

#### **3. Operational matrices corresponding to fractional derivatives and integrals**

Here in this section, we provide the required OMs via Laguerre polynomials of fractional derivatives and integrals.

**Lemma 3.** Let Ψ*M*ð Þ*t* be a function vector; the fractional integral of order *γ* for the function Ψ*M*ð Þ*t* can be generalized as

So Eq. (8) implies

*c* 0*I γ <sup>t</sup> L<sup>γ</sup> <sup>i</sup>* ðÞ¼ *t*

X*i k*¼0

X *j*

*r*¼0

where **W***<sup>γ</sup>*

**W***<sup>γ</sup> <sup>M</sup>*�*<sup>M</sup>* ¼

where

Θ*γ <sup>i</sup>*,*j*,*k*,*<sup>α</sup>* ¼

X *i*

X *i*

*r*¼0

where *Q <sup>γ</sup>*

**29**

*k*¼ *γ*

*c* 0*I γ <sup>t</sup> L<sup>γ</sup>*

which is the desired result.

Θ*γ*

Θ*γ*

Θ*γ*

d e *<sup>γ</sup>* ,0,*k*,*<sup>α</sup>* <sup>Θ</sup>*<sup>γ</sup>*

*<sup>i</sup>*,0,*k*,*<sup>α</sup>* <sup>Θ</sup>*<sup>γ</sup>*

*<sup>n</sup>*,0,*k*,*<sup>α</sup>* <sup>Θ</sup>*<sup>γ</sup>*

ð Þ �<sup>1</sup> *<sup>γ</sup>* <sup>þ</sup>*<sup>k</sup>*

*γ* for Ψ*M*ð Þ*t* is generalized as

*<sup>i</sup>* ðÞ¼ *<sup>t</sup>* <sup>X</sup> *i*

*DOI: http://dx.doi.org/10.5772/intechopen.90754*

*k*¼0

� X *j*

*r*¼0

*c* 0*D<sup>γ</sup>*

ð Þ �<sup>1</sup> *<sup>k</sup>* <sup>Γ</sup>ð Þ *<sup>i</sup>* <sup>þ</sup> *<sup>γ</sup>* <sup>þ</sup> <sup>1</sup>

*Numerical Solutions to Some Families of Fractional Order Differential Equations by Laguerre…*

ð Þ �<sup>1</sup> *<sup>k</sup>*þ*<sup>r</sup>* <sup>Γ</sup>ð Þ *<sup>j</sup>* <sup>þ</sup> <sup>1</sup> <sup>Γ</sup>ð Þ *<sup>i</sup>* <sup>þ</sup> *<sup>γ</sup>* <sup>þ</sup> <sup>1</sup> <sup>Γ</sup>ð Þ *<sup>k</sup>* <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> *<sup>r</sup>* <sup>þ</sup> *<sup>γ</sup>* <sup>þ</sup> <sup>1</sup>

*<sup>t</sup>* <sup>Ψ</sup>*M*ð Þ*<sup>t</sup>* <sup>≈</sup>**W***<sup>γ</sup>*

*<sup>M</sup>*�*<sup>M</sup>* is the OM of derivative of order *<sup>γ</sup>* , defined as in (9)

0 0 00 ⋯0 ⋮ ⋮ ⋮ ⋮⋮

d e *<sup>γ</sup>* ,1,*k*,*<sup>α</sup>* <sup>⋯</sup>Θ*<sup>γ</sup>*

*<sup>i</sup>*,1,*k*,*<sup>α</sup>* <sup>Θ</sup>*<sup>γ</sup>*

*<sup>n</sup>*,1,*k*,*<sup>α</sup>* <sup>Θ</sup>*<sup>γ</sup>*

⋮ ⋮ ⋮ ⋮⋮

*Proof*. Leaving the proof as it is very similar to the proof of the above lemma. **Lemma 5.** We consider a function *Z t*ð Þ defined on 0, ½ � <sup>∞</sup> and *y t*ðÞ¼ *KM*Ψ*<sup>T</sup>*

> *C*0,0, *C*0,1 ⋯ *C*0,*<sup>j</sup>* ⋯ *C*0,*<sup>m</sup> C*1,0 *C*1,1 ⋯ *C*1,*<sup>j</sup>* ⋯ *C*1,*<sup>m</sup>* ⋮ ⋮ ⋮ ⋮⋯ ⋮ *Ci*,0 *Ci*,1 ⋯ *Ci*,*<sup>j</sup>* ⋮ *Ci*,*<sup>m</sup>* ⋮ ⋮ ⋮ ⋮⋯ ⋮ *Cm*,0 *Cm*,1 ⋯ *Cm*,*<sup>j</sup>* ⋯ *Cm*,*<sup>m</sup>*

*Z t*ð Þ <sup>0</sup>*<sup>I</sup> <sup>γ</sup>* ½ �¼ *<sup>t</sup> y t*ð Þ *KMQ <sup>γ</sup>*

*<sup>M</sup>*�*<sup>M</sup>* is the operational matrix, given by

**Lemma 4.** Let Ψ*M*ð Þ*t* be a function vector; then the fractional derivative of order

Γð Þ *i* � *k* þ 1 Γð Þ *k* þ *γ* þ 1 Γð Þ 1 þ *k* � *γ*

ð Þ �<sup>1</sup> *<sup>r</sup>* <sup>Γ</sup>ð Þ *<sup>j</sup>* <sup>þ</sup> <sup>1</sup> <sup>Γ</sup>ð Þ *<sup>k</sup>* <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> *<sup>r</sup>* <sup>þ</sup> *<sup>γ</sup>* <sup>þ</sup> <sup>1</sup> Γð Þ *j* � *r* þ 1 Γð Þ *r* þ 1 Γð Þ *r* þ *γ* þ 1 *:*

<sup>Γ</sup>ð Þ <sup>1</sup> � *<sup>k</sup>* <sup>þ</sup> *<sup>i</sup>* <sup>Γ</sup>ð Þ *<sup>j</sup>* � *<sup>γ</sup>* <sup>þ</sup> <sup>1</sup> <sup>Γ</sup>ð Þ *<sup>γ</sup>* <sup>þ</sup> <sup>1</sup> <sup>Γ</sup>ð Þ *<sup>k</sup>* <sup>þ</sup> *<sup>γ</sup>* <sup>þ</sup> <sup>1</sup> <sup>Γ</sup>ð Þ *<sup>k</sup>* <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> <sup>Γ</sup>ð Þ *<sup>γ</sup>* <sup>þ</sup> *<sup>r</sup>* <sup>þ</sup> <sup>1</sup> *:*

*<sup>M</sup>*�*<sup>M</sup>*Ψ*M*ð Þ*<sup>t</sup>* ,

d e *<sup>γ</sup>* ,*j*,*k*,*<sup>α</sup>* ⋯ ⋯ <sup>Θ</sup>*<sup>γ</sup>*

*<sup>i</sup>*,*j*,*k*,*<sup>α</sup>* ⋯ ⋯ <sup>Θ</sup>*<sup>γ</sup>*

*<sup>n</sup>*,*j*,*k*,*<sup>α</sup>* ⋯ ⋯ <sup>Θ</sup>*<sup>γ</sup>*

Γð Þ *j* þ 1 Γð Þ *i* þ *α* þ 1 Γð Þ *k* þ *α* � *r* þ *γ* þ 1 <sup>Γ</sup>ð Þ *<sup>j</sup>* � *<sup>r</sup>* <sup>þ</sup> <sup>1</sup> <sup>Γ</sup>ð Þ *<sup>i</sup>* � *<sup>k</sup>* <sup>þ</sup> <sup>1</sup> <sup>Γ</sup>ð Þ *<sup>r</sup>* <sup>þ</sup> <sup>1</sup> <sup>Γ</sup>ð Þ *<sup>k</sup>* <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> <sup>Γ</sup>ð Þ *<sup>k</sup>* � *<sup>γ</sup>* <sup>þ</sup> <sup>1</sup> <sup>Γ</sup>ð Þ *<sup>α</sup>* <sup>þ</sup> *<sup>γ</sup>* <sup>þ</sup> <sup>1</sup> *:*

*<sup>M</sup>*�*<sup>M</sup>*Ψ*M*ð Þ*<sup>t</sup>* ,

,

d e *γ* ,*n*,*k*,*α*

, (9)

*<sup>M</sup>*ð Þ*t* ; then

*i*,*n*,*k*,*α*

*n*,*n*,*k*,*α*

$$\_0I\_t^\gamma \, \Psi\_M(t) \approx G\_{N \times N}^\gamma \Psi\_M(t),$$

where *G<sup>γ</sup> <sup>N</sup>*�*<sup>N</sup>* is the OM of integration of fractional order *<sup>γ</sup>* and given by

$$
\begin{bmatrix}
\mathsf{T}\_{0,0,k,r}^{\mathcal{I}} & \mathsf{T}\_{0,1,k,r}^{\mathcal{I}} & \cdots & \mathsf{T}\_{0,j,k,r}^{\mathcal{I}} & \cdots & \mathsf{T}\_{0,m,k,r}^{\mathcal{I}} \\
\mathsf{T}\_{1,0,k,r}^{\mathcal{I}} & \mathsf{T}\_{1,i,k,r}^{\mathcal{I}} & \cdots & \mathsf{T}\_{1,j,k,r}^{\mathcal{I}} & \cdots & \mathsf{T}\_{1,m,k,r}^{\mathcal{I}} \\
\vdots & \vdots & \vdots & \vdots & \cdots & \vdots \\
\mathsf{T}\_{i,0,k,r}^{\mathcal{I}} & \mathsf{T}\_{i,1,k,r}^{\mathcal{I}} & \cdots & \mathsf{T}\_{i,j,k,r}^{\mathcal{I}} & \vdots & \mathsf{T}\_{i,m,k,r}^{\mathcal{I}} \\
\vdots & \vdots & \vdots & \vdots & \cdots & \vdots \\
\mathsf{T}\_{m,0,k,r}^{\mathcal{I}} & \mathsf{T}\_{m,1,k,r}^{\mathcal{I}} & \cdots & \mathsf{T}\_{m,j,k,r}^{\mathcal{I}} & \cdots & \mathsf{T}\_{m,m,k,r}^{\mathcal{I}}
\end{bmatrix},
$$

where

$$\mathfrak{T}\_{ij,k,r}^{\mathsf{T}} = \sum\_{k=0}^{i} \sum\_{r=0}^{i} \frac{(-1)^{k+r} \Gamma(j+1) \Gamma(i+\gamma+1) \Gamma(k+\gamma+a+r+1)}{\Gamma(j-r+1) \Gamma(i-k+1) \Gamma(r+1) \Gamma(k+\gamma+1) \Gamma(k+a+1) \Gamma(\gamma+r+1)} \frac{1}{i!}$$

*Proof*. We apply the fractional order integral of order *γ* to the Laguerre polynomials

$$I\_0^c I\_t^\gamma L\_i^\gamma (t) = \sum\_{k=0}^i \frac{\Gamma(i+\gamma+1)}{\Gamma(i-k+1)\Gamma(k+\gamma+1)\Gamma(k+1)} I\_t^\gamma t^k. \tag{7}$$

Since from (7), we have

$$t\_0^{\epsilon} I\_t^{\gamma} t^k = \frac{\Gamma(k+1)}{\Gamma(1+k+a)} t^{k+\gamma} \cdot \frac{1}{t}$$

Therefore Eq. (7) implies that

$$\,\_0^\varepsilon I\_t^\gamma L\_i^\gamma (t) = \sum\_{k=0}^i t^{k+\gamma} \frac{\Gamma(i+\gamma+1)}{\Gamma(i-k+1)\Gamma(k+\gamma+1)\Gamma(k+1)} \frac{\Gamma(k+1)}{\Gamma(1+k+a)},$$

which is equal to

$$L\_0^r I\_t^\gamma L\_i^\gamma (t) = \sum\_{k=0}^i (-1)^k \frac{\Gamma(i+\gamma+1)}{\Gamma(i-k+1)\Gamma(k+\gamma+1)\Gamma(1+k-\gamma)} t^{k+\gamma}.\tag{8}$$

We approximate *t <sup>k</sup>*<sup>þ</sup> *<sup>γ</sup>* in (8) with Laguerre polynomials, i.e.

$$t^{k+\gamma} \approx \sum\_{j=0}^{n} H\_j L\_j^{\gamma}(t).$$

By using the relation of orthogonality, we can find coefficients

$$H\_j = \sum\_{r=0}^j (-\mathbf{1})^k \frac{\Gamma(j+1)\Gamma(k+a+r+\chi+\mathbf{1})}{\Gamma(\mathbf{1}+j-r)\Gamma(\mathbf{1}+r)\Gamma(\mathbf{1}+r+\chi)}.$$

*Numerical Solutions to Some Families of Fractional Order Differential Equations by Laguerre… DOI: http://dx.doi.org/10.5772/intechopen.90754*

So Eq. (8) implies

*c*

**Lemma 3.** Let Ψ*M*ð Þ*t* be a function vector; the fractional integral of order *γ* for

*<sup>N</sup>*�*<sup>N</sup>* is the OM of integration of fractional order *<sup>γ</sup>* and given by

*<sup>N</sup>*�*N*Ψ*M*ð Þ*<sup>t</sup>* ,

0,*j*,*k*,*<sup>r</sup>* <sup>⋯</sup> <sup>ℸ</sup>*<sup>γ</sup>*

1,*j*,*k*,*<sup>r</sup>* <sup>⋯</sup> <sup>ℸ</sup>*<sup>γ</sup>*

*<sup>i</sup>*,*j*,*k*,*<sup>r</sup>* <sup>⋮</sup> <sup>ℸ</sup>*<sup>γ</sup>*

*<sup>m</sup>*,*j*,*k*,*<sup>r</sup>* <sup>⋯</sup> <sup>ℸ</sup>*<sup>γ</sup>*

Γð Þ *j* þ 1 Γð Þ *i* þ *γ* þ 1 Γð Þ *k* þ *γ* þ *α* þ *r* þ 1 <sup>Γ</sup>ð Þ *<sup>j</sup>* � *<sup>r</sup>* <sup>þ</sup> <sup>1</sup> <sup>Γ</sup>ð Þ *<sup>i</sup>* � *<sup>k</sup>* <sup>þ</sup> <sup>1</sup> <sup>Γ</sup>ð Þ *<sup>r</sup>* <sup>þ</sup> <sup>1</sup> <sup>Γ</sup>ð Þ *<sup>k</sup>* <sup>þ</sup> *<sup>γ</sup>* <sup>þ</sup> <sup>1</sup> <sup>Γ</sup>ð Þ *<sup>k</sup>* <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> <sup>Γ</sup>ð Þ *<sup>γ</sup>* <sup>þ</sup> *<sup>r</sup>* <sup>þ</sup> <sup>1</sup> *:*

*<sup>k</sup>*<sup>þ</sup> *<sup>γ</sup> :*

Γð Þ *i* þ *γ* þ 1 Γð Þ *i* � *k* þ 1 Γð Þ *k* þ *γ* þ 1 Γð Þ *k* þ 1

0,*m*,*k*,*r*

,

*c*

0 *I γ t t k*

Γð Þ *k* þ 1 <sup>Γ</sup>ð Þ <sup>1</sup> <sup>þ</sup> *<sup>k</sup>* <sup>þ</sup> *<sup>α</sup>* ,

*t*

*<sup>k</sup>*<sup>þ</sup> *<sup>γ</sup> :* (8)

*:* (7)

1,*m*,*k*,*r*

*i*,*m*,*k*,*r*

*m*,*m*,*k*,*r*

*<sup>t</sup>* <sup>Ψ</sup>*M*ð Þ*<sup>t</sup>* <sup>≈</sup> *<sup>G</sup><sup>γ</sup>*

0,1,*k*,*<sup>r</sup>* <sup>⋯</sup> <sup>ℸ</sup>*<sup>γ</sup>*

1,*i*,*k*,*<sup>r</sup>* <sup>⋯</sup> <sup>ℸ</sup>*<sup>γ</sup>*

*<sup>i</sup>*,1,*k*,*<sup>r</sup>* <sup>⋯</sup> <sup>ℸ</sup>*<sup>γ</sup>*

*<sup>m</sup>*,1,*k*,*<sup>r</sup>* <sup>⋯</sup> <sup>ℸ</sup>*<sup>γ</sup>*

*Proof*. We apply the fractional order integral of order *γ* to the Laguerre

*<sup>k</sup>* <sup>¼</sup> <sup>Γ</sup>ð Þ *<sup>k</sup>* <sup>þ</sup> <sup>1</sup> <sup>Γ</sup>ð Þ <sup>1</sup> <sup>þ</sup> *<sup>k</sup>* <sup>þ</sup> *<sup>α</sup> <sup>t</sup>*

*<sup>k</sup>*<sup>þ</sup> *<sup>γ</sup>* <sup>Γ</sup>ð Þ *<sup>i</sup>* <sup>þ</sup> *<sup>γ</sup>* <sup>þ</sup> <sup>1</sup>

ð Þ �<sup>1</sup> *<sup>k</sup>* <sup>Γ</sup>ð Þ *<sup>i</sup>* <sup>þ</sup> *<sup>γ</sup>* <sup>þ</sup> <sup>1</sup>

*<sup>k</sup>*<sup>þ</sup> *<sup>γ</sup>* <sup>≈</sup> <sup>X</sup>*<sup>n</sup>*

*t*

By using the relation of orthogonality, we can find coefficients

Γð Þ *i* � *k* þ 1 Γð Þ *k* þ *γ* þ 1 Γð Þ *k* þ 1

Γð Þ *i* � *k* þ 1 Γð Þ *k* þ *γ* þ 1 Γð Þ 1 þ *k* � *γ*

*HjL<sup>γ</sup> <sup>j</sup>* ð Þ*t :*

ð Þ �<sup>1</sup> *<sup>k</sup>* <sup>Γ</sup>ð Þ *<sup>j</sup>* <sup>þ</sup> <sup>1</sup> <sup>Γ</sup>ð Þ *<sup>k</sup>* <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> *<sup>r</sup>* <sup>þ</sup> *<sup>γ</sup>* <sup>þ</sup> <sup>1</sup> <sup>Γ</sup>ð Þ <sup>1</sup> <sup>þ</sup> *<sup>j</sup>* � *<sup>r</sup>* <sup>Γ</sup>ð Þ <sup>1</sup> <sup>þ</sup> *<sup>r</sup>* <sup>Γ</sup>ð Þ <sup>1</sup> <sup>þ</sup> *<sup>r</sup>* <sup>þ</sup> *<sup>γ</sup> :*

*<sup>k</sup>*<sup>þ</sup> *<sup>γ</sup>* in (8) with Laguerre polynomials, i.e.

*j*¼0

⋮ ⋮ ⋮ ⋮⋯ ⋮

⋮ ⋮ ⋮ ⋮⋯ ⋮

the function Ψ*M*ð Þ*t* can be generalized as

ℸ*γ*

ℸ*γ*

ℸ*γ*

ℸ*γ*

0,0,*k*,*<sup>r</sup>* <sup>ℸ</sup>*<sup>γ</sup>*

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

1,0,*k*,*<sup>r</sup>* <sup>ℸ</sup>*<sup>γ</sup>*

*<sup>i</sup>*,0,*k*,*<sup>r</sup>* <sup>ℸ</sup>*<sup>γ</sup>*

*<sup>m</sup>*,0,*k*,*<sup>r</sup>* <sup>ℸ</sup>*<sup>γ</sup>*

*<sup>i</sup>* ðÞ¼ *<sup>t</sup>* <sup>X</sup> *i*

*k*¼0

*c* 0*I γ t t*

ð Þ �<sup>1</sup> *<sup>k</sup>*þ*<sup>r</sup>*

where *G<sup>γ</sup>*

where

*<sup>i</sup>*,*j*,*k*,*<sup>r</sup>* <sup>¼</sup> <sup>X</sup> *i*

polynomials

*c* 0*I γ <sup>t</sup> L<sup>γ</sup>*

*c* 0*I γ <sup>t</sup> L<sup>γ</sup>*

**28**

which is equal to

We approximate *t*

*k*¼0

X *i*

*r*¼0

*c* 0*I γ <sup>t</sup> L<sup>γ</sup>*

Since from (7), we have

Therefore Eq. (7) implies that

*<sup>i</sup>* ðÞ¼ *<sup>t</sup>* <sup>X</sup> *i*

*<sup>i</sup>* ðÞ¼ *<sup>t</sup>* <sup>X</sup> *i*

*k*¼0

*Hj* <sup>¼</sup> <sup>X</sup> *j*

*r*¼0

*k*¼0 *t*

ℸ*γ*

0*I γ*

$$\begin{split} \, ^{\leq}\_{0}I\_{i}^{r}L\_{i}^{r}(t) &= \sum\_{k=0}^{i}(-1)^{k} \frac{\Gamma(i+\gamma+1)}{\Gamma(i-k+1)\Gamma(k+\gamma+1)\Gamma(1+k-\gamma)} \\ &\times \sum\_{r=0}^{j}(-1)^{r} \frac{\Gamma(j+1)\Gamma(k+a+r+\gamma+1)}{\Gamma(j-r+1)\Gamma(r+1)\Gamma(r+\gamma+1)} \\ \, ^{\leq}\_{0}I\_{i}^{r}L\_{i}^{r}(t) &= \\ \sum\_{k=0}^{i} \sum\_{r=0}^{j}(-1)^{k+r} \frac{\Gamma(j+1)\Gamma(i+\gamma+1)\Gamma(k+a+r+\gamma+1)}{\Gamma(1-k+i)\Gamma(j-\gamma+1)\Gamma(\gamma+1)\Gamma(k+\gamma+1)\Gamma(k+a+1)\Gamma(\gamma+r+1)}. \end{split}$$

which is the desired result.

**Lemma 4.** Let Ψ*M*ð Þ*t* be a function vector; then the fractional derivative of order *γ* for Ψ*M*ð Þ*t* is generalized as

$$
\epsilon\_0^\epsilon D\_t^\gamma \Psi\_M(t) \approx \mathbf{W}\_{M \times M}^\gamma \Psi\_M(t),
$$

where **W***<sup>γ</sup> <sup>M</sup>*�*<sup>M</sup>* is the OM of derivative of order *<sup>γ</sup>* , defined as in (9)

$$\mathbf{W}\_{M\times M}^{\boldsymbol{\tau}} = \begin{bmatrix} \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \cdots \mathbf{0} \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ \boldsymbol{\Theta}\_{\lceil \boldsymbol{\tau} \rceil, 0, k, a}^{\boldsymbol{\tau}} & \boldsymbol{\Theta}\_{\lceil \boldsymbol{\tau} \rceil, 1, k, a}^{\boldsymbol{\tau}} & \cdots \boldsymbol{\Theta}\_{\lceil \boldsymbol{\tau} \rceil, j, k, a}^{\boldsymbol{\tau}} & \cdots & \cdots & \boldsymbol{\Theta}\_{\lceil \boldsymbol{\tau} \rceil, \boldsymbol{\nu}, k, a}^{\boldsymbol{\tau}} \\ \boldsymbol{\Theta}\_{i,0,k,a}^{\boldsymbol{\tau}} & \boldsymbol{\Theta}\_{i,1,k,a}^{\boldsymbol{\tau}} & \boldsymbol{\Theta}\_{i,j,k,a}^{\boldsymbol{\tau}} & \cdots & \cdots & \boldsymbol{\Theta}\_{\lceil i,n,k,a}^{\boldsymbol{\tau}} \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ \boldsymbol{\Theta}\_{n,0,k,a}^{\boldsymbol{\tau}} & \boldsymbol{\Theta}\_{n,1,k,a}^{\boldsymbol{\tau}} & \boldsymbol{\Theta}\_{n,j,k,a}^{\boldsymbol{\tau}} & \cdots & \cdots & \boldsymbol{\Theta}\_{n,n,k,a}^{\boldsymbol{\tau}} \end{bmatrix}, \tag{9}$$

where

$$\begin{aligned} \Theta\_{i,j,k,a}^{\gamma} &= \\ \sum\_{k=\gamma}^{i} \sum\_{r=0}^{i} \frac{(-1)^{r+k} \Gamma(j+1) \Gamma(i+a+1) \Gamma(k+a-r+\gamma+1)}{\Gamma(j-r+1) \Gamma(i-k+1) \Gamma(r+1) \Gamma(k+a+1) \Gamma(k-\gamma+1) \Gamma(a+\gamma+1)} \frac{1}{\Gamma(j+1) \Gamma(k+a+1) \Gamma(a+\gamma+1)} \end{aligned}$$

*Proof*. Leaving the proof as it is very similar to the proof of the above lemma. **Lemma 5.** We consider a function *Z t*ð Þ defined on 0, ½ � <sup>∞</sup> and *y t*ðÞ¼ *KM*Ψ*<sup>T</sup> <sup>M</sup>*ð Þ*t* ; then

$$[Z(t)]\_0[I\_t^\gamma \mathcal{Y}(t)] = K\_M Q\_{M \times M}^\gamma \Psi\_M(t),$$

where *Q <sup>γ</sup> <sup>M</sup>*�*<sup>M</sup>* is the operational matrix, given by

$$
\begin{bmatrix}
\mathbf{C}\_{0,0}, & \mathbf{C}\_{0,1} & \cdots & \mathbf{C}\_{0,j} & \cdots & \mathbf{C}\_{0,m} \\
\mathbf{C}\_{1,0} & \mathbf{C}\_{1,1} & \cdots & \mathbf{C}\_{1,j} & \cdots & \mathbf{C}\_{1,m} \\
\vdots & \vdots & \vdots & \vdots & \cdots & \vdots \\
\mathbf{C}\_{i,0} & \mathbf{C}\_{i,1} & \cdots & \mathbf{C}\_{ij} & \vdots & \mathbf{C}\_{i,m} \\
\vdots & \vdots & \vdots & \vdots & \cdots & \vdots \\
\mathbf{C}\_{m,0} & \mathbf{C}\_{m,1} & \cdots & \mathbf{C}\_{m,j} & \cdots & \mathbf{C}\_{m,m}
\end{bmatrix},
$$

where

$$C\_{i\_{\vec{j}}} = \frac{1}{h\_i} \int\_0^1 \Delta\_{i,\gamma,k} Z(t) L\_{\vec{j}}^{\gamma}(t) dt,$$

method is compared with the exact solution. Similarly we investigate numerical

*Numerical Solutions to Some Families of Fractional Order Differential Equations by Laguerre…*

**Case 1.** In the first case, we consider the fractional order differential equation

*<sup>t</sup> z t*ð Þ� *z t*ðÞ¼ 0, 0 < *γ* ⩽1,

*<sup>t</sup> z t*ðÞ¼ <sup>Ł</sup>*Mψ<sup>T</sup>*

*γ <sup>t</sup>* Ł*Mψ<sup>T</sup> <sup>M</sup>*ð Þ*<sup>t</sup>* ,

Using the initial condition to get *<sup>e</sup>*<sup>0</sup> <sup>¼</sup> *<sup>z</sup>*<sup>0</sup> and approximate *<sup>z</sup>*<sup>0</sup> as *<sup>z</sup>*<sup>0</sup> <sup>≈</sup>*FM <sup>ψ</sup><sup>T</sup>*

*<sup>M</sup>*�*<sup>M</sup> <sup>ψ</sup><sup>T</sup>*

*<sup>M</sup>*�*<sup>M</sup> <sup>ψ</sup><sup>T</sup>*

Solving the Sylvester matrix for Ł*M*, we get the numerical value for *z t*ð Þ.

*z*ð Þ¼ 0 1, *z*<sup>0</sup> ∈ *R:*

*<sup>t</sup> z t*ðÞ� *z t*ðÞ¼ 0, 0 < *γ* ≤ 1,

*z t*ðÞ¼ *<sup>E</sup><sup>γ</sup>* �*<sup>t</sup> <sup>γ</sup>* ð Þ,

Approximating the solution through the proposed method and plotting the exact

*<sup>t</sup> z t*ðÞþ *z t*ðÞ¼ 0, 1< *γ* ⩽2, *z*ð Þ¼ 0 *z*0, *z*ð Þ¼ 1 *z*1, *z*0, *z*<sup>1</sup> ∈ *R:*

*<sup>t</sup> z t*ðÞ¼ *KMψ<sup>T</sup>*

where *<sup>E</sup><sup>γ</sup>* is the Mittag-Leffler representation, and at *<sup>γ</sup>* <sup>¼</sup> 1, *z t*ðÞ¼ *<sup>e</sup>*�*<sup>t</sup>*

as well as numerical solution by using scale *M* ¼ 8 corresponding to *γ* ¼ 1 in

**Figure 1**, we see that the proposed method works very well.

*c* 0*D<sup>γ</sup>*

*c* 0*D<sup>γ</sup>*

*<sup>M</sup>*ð Þ*t :*

*<sup>M</sup>*ðÞþ*<sup>t</sup> FM <sup>ψ</sup><sup>T</sup>*

*<sup>M</sup>*ðÞþ*t FM* ¼ 0*:*

*<sup>M</sup>*ðÞ¼ *t* 0*:*

(12)

*<sup>M</sup>*ð Þ*t* ,

*:*

*<sup>M</sup>*ð Þ*t :* (14)

(13)

solutions to various coupled systems under some initial conditions

**4.1 Treatment of FODEs under initial and boundary conditions**

*z*ð Þ¼ 0 *z*0, *z*<sup>0</sup> ∈*R*

*<sup>t</sup>* by the Lemma 1, on (12) we write

*z t*ðÞ¼ *e*<sup>0</sup> þ <sup>0</sup>*I*

*c* 0*D<sup>γ</sup>*

*<sup>M</sup>*ðÞþ*<sup>t</sup>* <sup>Ł</sup>*<sup>M</sup> <sup>G</sup><sup>γ</sup>*

Finally the Sylvester-type algebraic equation is obtained as

<sup>Ł</sup>*<sup>M</sup>* <sup>þ</sup> <sup>Ł</sup>*<sup>M</sup> <sup>G</sup><sup>γ</sup>*

*c* 0*D<sup>γ</sup>*

Since the exact solution is given by

as well as boundary conditions.

we see that

and applying <sup>0</sup>*I*

Eq. (12) implies

**Example 1.**

**Case 2**.

We take

**31**

Here we discuss different cases.

*DOI: http://dx.doi.org/10.5772/intechopen.90754*

*γ*

Ł*<sup>M</sup> ψ<sup>T</sup>*

*c* 0*D<sup>γ</sup>*

with

$$w\_i = \sum\_{k=0}^i \frac{(-1)^{i+1} \Gamma(i+1+\chi)}{\Gamma(k+\chi+1)\Gamma(1-k+i)\Gamma(k+\chi)}.$$

*Proof*. By considering the general term of Ψ*M*ð Þ*t*

$${}\_{0}I\_{1}^{\prime}L\_{i}(t) = \frac{1}{\Gamma(\chi)}\int\_{0}^{1}(1-s)^{\gamma-1}L\_{i}(s)ds.$$

$${}\_{0}I\_{1}^{\prime}L\_{i}(t) = \frac{1}{\Gamma(\chi)}\int\_{0}^{1}(1-s)^{\gamma-1}\sum\_{k=0}^{i}(s)^{k}\frac{(-1)^{k}\Gamma(i+1+\gamma)}{\Gamma(-k+1+i)\Gamma(k+1+\gamma)\Gamma(1+k)}ds.$$

$${}\_{0}I\_{1}^{\prime}L\_{i}(t) = \sum\_{k=0}^{i}\frac{(-1)^{k}\Gamma(i+1+\gamma)}{\Gamma(\chi)\Gamma(-k+1+i)\Gamma(k+1+\gamma)\Gamma(1+k)}\int\_{0}^{1}(1-s)^{\gamma-1}(s)^{k}ds.\tag{10}$$

Using the famous Laplace transform, we have from (10)

$$\begin{aligned} \varepsilon(\int\_0^1 (1-s)^{\gamma-1} s^k ds &= \frac{\Gamma(\gamma)\Gamma(k+1)}{\Gamma(\gamma+k)}. \\\ \_0I\_1^{\gamma}L\_i(t) &= \sum\_{k=0}^i \frac{(-1)^k \Gamma(i+1+\gamma)}{\Gamma(\gamma)\Gamma(-k+1+i)\Gamma(k+1+\gamma)\Gamma(1+k)} \frac{\Gamma(\gamma)\Gamma(k+1)}{\Gamma(\gamma+k)}. \\\ &\sum\_{k=0}^i \frac{(-1)^k \Gamma(i+1+\gamma)}{\Gamma(-k+1+i)\Gamma(k+\gamma+1)\Gamma(1+k)} = \Delta\_{i,\gamma,k}. \end{aligned}$$

Now using Laguerre polynomials, we have

$$
\Delta\_{i,\mathcal{Y},k}\mathcal{Z}(t) = \sum\_{j=0}^{m} C\_{i,j} L\_i(t),
$$

where *Ci*,*<sup>j</sup>* is calculated by using orthogonality as

$$C\_{i,j} = \frac{1}{hi} \int\_0^1 \Delta\_{i,r,k} x(t) L\_j^\gamma(t) dt. \tag{11}$$

To get the desired result, we evaluate the above (11) relation for *i* ¼ 0, 1, … , *m* and *j* ¼ 0, 1, … , *m*.

#### **4. Main result**

In this section, we discuss some cases of FODEs with initial condition as well as boundary conditions. The approximate solution obtained through desired

*Numerical Solutions to Some Families of Fractional Order Differential Equations by Laguerre… DOI: http://dx.doi.org/10.5772/intechopen.90754*

method is compared with the exact solution. Similarly we investigate numerical solutions to various coupled systems under some initial conditions as well as boundary conditions.

#### **4.1 Treatment of FODEs under initial and boundary conditions**

Here we discuss different cases.

**Case 1.** In the first case, we consider the fractional order differential equation

$$\begin{cases} \,^c\_0 D\_t^\gamma z(t) \pm z(t) = 0, & 0 < \chi \lessapprox 1, \\ z(0) = z\_0, & z\_0 \in \mathbb{R} \end{cases} \tag{12}$$

we see that

where

with

0*I γ* <sup>1</sup> *Li*ðÞ¼ *t*

0*I γ*

> 0*I γ*

and *j* ¼ 0, 1, … , *m*.

**4. Main result**

**30**

*Ci*,*<sup>j</sup>* <sup>¼</sup> <sup>1</sup> *hi* ð1 0

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

*wi* <sup>¼</sup> <sup>X</sup> *i*

*k*¼0

*Proof*. By considering the general term of Ψ*M*ð Þ*t*

ð Þ <sup>1</sup> � *<sup>s</sup> <sup>γ</sup>* �<sup>1</sup><sup>X</sup>

0*I γ* <sup>1</sup> *Li*ðÞ¼ *t*

*£*ð ð1 0

Now using Laguerre polynomials, we have

where *Ci*,*<sup>j</sup>* is calculated by using orthogonality as

*Ci*,*<sup>j</sup>* <sup>¼</sup> <sup>1</sup> *hi* ð1 0

1 Γð Þ *γ*

*i*

*k*¼0

<sup>1</sup> *Li*ðÞ¼ *<sup>t</sup>* <sup>X</sup>

<sup>1</sup> *Li*ðÞ¼ *<sup>t</sup>* <sup>X</sup>

*i*

*k*¼0

X *i*

*k*¼0

ð1 0 <sup>Δ</sup>*i*, *<sup>γ</sup>* ,*kZ t*ð Þ*L<sup>γ</sup>*

ð Þ �<sup>1</sup> *<sup>i</sup>*þ<sup>1</sup>

1 Γð Þ *γ*

*i*

*k*¼0 ð Þ*s*

ð Þ �<sup>1</sup> *<sup>k</sup>*

Using the famous Laplace transform, we have from (10)

ð Þ <sup>1</sup> � *<sup>s</sup> <sup>γ</sup>* �<sup>1</sup>

ð Þ �<sup>1</sup> *<sup>k</sup>*

ð Þ �<sup>1</sup> *<sup>k</sup>*

<sup>Δ</sup>*<sup>i</sup>*, *<sup>γ</sup>* ,*kz t*ðÞ¼ <sup>X</sup>*<sup>m</sup>*

*s k* ð1 0

Γð Þ *i* þ 1 þ *γ* Γð Þ *γ* Γð Þ �*k* þ 1 þ *i* Γð Þ *k* þ 1 þ *γ* Γð Þ 1 þ *k*

*<sup>j</sup>* ð Þ*t dt*,

Γð Þ *i* þ 1 þ *γ* <sup>Γ</sup>ð Þ *<sup>k</sup>* <sup>þ</sup> *<sup>γ</sup>* <sup>þ</sup> <sup>1</sup> <sup>Γ</sup>ð Þ <sup>1</sup> � *<sup>k</sup>* <sup>þ</sup> *<sup>i</sup>* <sup>Γ</sup>ð Þ *<sup>k</sup>* <sup>þ</sup> *<sup>γ</sup> :*

ð Þ <sup>1</sup> � *<sup>s</sup> <sup>γ</sup>* �<sup>1</sup>

*<sup>k</sup>* ð Þ �<sup>1</sup> *<sup>k</sup>*

*ds* <sup>¼</sup> <sup>Γ</sup>ð Þ *<sup>γ</sup>* <sup>Γ</sup>ð Þ *<sup>k</sup>* <sup>þ</sup> <sup>1</sup> <sup>Γ</sup>ð Þ *<sup>γ</sup>* <sup>þ</sup> *<sup>k</sup> :*

*Ci*,*jLi*ð Þ*t* ,

Γð Þ *i* þ 1 þ *γ* Γð Þ *γ* Γð Þ �*k* þ 1 þ *i* Γð Þ *k* þ 1 þ *γ* Γð Þ 1 þ *k*

Γð Þ *i* þ 1 þ *γ* <sup>Γ</sup>ð Þ �*<sup>k</sup>* <sup>þ</sup> <sup>1</sup> <sup>þ</sup> *<sup>i</sup>* <sup>Γ</sup>ð Þ *<sup>k</sup>* <sup>þ</sup> *<sup>γ</sup>* <sup>þ</sup> <sup>1</sup> <sup>Γ</sup>ð Þ <sup>1</sup> <sup>þ</sup> *<sup>k</sup>* <sup>¼</sup> <sup>Δ</sup>*<sup>i</sup>*, *<sup>γ</sup>* ,*<sup>k</sup>:*

*j*¼0

<sup>Δ</sup>*<sup>i</sup>*, *<sup>γ</sup>* ,*kz t*ð Þ*L<sup>γ</sup>*

To get the desired result, we evaluate the above (11) relation for *i* ¼ 0, 1, … , *m*

In this section, we discuss some cases of FODEs with initial condition as well as

boundary conditions. The approximate solution obtained through desired

*Li*ð Þ*s ds:*

Γð Þ *i* þ 1 þ *γ* <sup>Γ</sup>ð Þ �*<sup>k</sup>* <sup>þ</sup> <sup>1</sup> <sup>þ</sup> *<sup>i</sup>* <sup>Γ</sup>ð Þ *<sup>k</sup>* <sup>þ</sup> <sup>1</sup> <sup>þ</sup> *<sup>γ</sup>* <sup>Γ</sup>ð Þ <sup>1</sup> <sup>þ</sup> *<sup>k</sup> ds:*

> ð1 0

ð Þ <sup>1</sup> � *<sup>s</sup> <sup>γ</sup>* �<sup>1</sup>

Γð Þ *γ* Γð Þ *k* þ 1 <sup>Γ</sup>ð Þ *<sup>γ</sup>* <sup>þ</sup> *<sup>k</sup> :*

*<sup>j</sup>* ð Þ*t dt:* (11)

ð Þ*s k ds:*

(10)

$$
\xi\_0 D\_t^\gamma z(t) = \mathbf{h}\_M \boldsymbol{\mu}\_M^T(t).
$$

and applying <sup>0</sup>*I γ <sup>t</sup>* by the Lemma 1, on (12) we write

$$z(t) = e\_0 + \ \_0I\_t^{\gamma} \left[ \mathbf{E}\_M \boldsymbol{\mu}\_M^T(t) \right],$$

Using the initial condition to get *<sup>e</sup>*<sup>0</sup> <sup>¼</sup> *<sup>z</sup>*<sup>0</sup> and approximate *<sup>z</sup>*<sup>0</sup> as *<sup>z</sup>*<sup>0</sup> <sup>≈</sup>*FM <sup>ψ</sup><sup>T</sup> <sup>M</sup>*ð Þ*t* , Eq. (12) implies

$$\mathbf{E}\_{\mathcal{M}} \cdot \boldsymbol{\mu}\_{\mathcal{M}}^T(t) + \mathbf{E}\_{\mathcal{M}} \cdot \mathbf{G}\_{\mathcal{M} \times \mathcal{M}}^\gamma \cdot \boldsymbol{\mu}\_{\mathcal{M}}^T(t) + F\_{\mathcal{M}} \cdot \boldsymbol{\mu}\_{\mathcal{M}}^T(t) = \mathbf{0}.$$

Finally the Sylvester-type algebraic equation is obtained as

$$\mathbf{E}\_{\mathcal{M}} + \mathbf{E}\_{\mathcal{M}} \; \mathcal{G}\_{\mathcal{M} \times \mathcal{M}}^{\mathcal{I}} \; \boldsymbol{\Psi}\_{\mathcal{M}}^{T}(t) + F\_{\mathcal{M}} = \mathbf{0}.$$

Solving the Sylvester matrix for Ł*M*, we get the numerical value for *z t*ð Þ. **Example 1.**

$$\begin{cases} \,^c\_0 D\_t^\gamma z(t) \pm z(t) = 0, & 0 < \gamma \le 1, \\ z(0) = 1, & z\_0 \in R. \end{cases}$$

Since the exact solution is given by

$$z(t) = E\_r(-t^r),$$

where *<sup>E</sup><sup>γ</sup>* is the Mittag-Leffler representation, and at *<sup>γ</sup>* <sup>¼</sup> 1, *z t*ðÞ¼ *<sup>e</sup>*�*<sup>t</sup> :*

Approximating the solution through the proposed method and plotting the exact as well as numerical solution by using scale *M* ¼ 8 corresponding to *γ* ¼ 1 in **Figure 1**, we see that the proposed method works very well.

**Case 2**.

$$\begin{cases} \ \ \_0^cD\_t^\gamma z(t) + z(t) = 0, & 1 < \chi \lessapprox 2, \\\ z(0) = z\_0, \ z(1) = z\_1, & z\_0, \ z\_1 \in R. \end{cases} \tag{13}$$

We take

$$\prescript{\epsilon}{}{0}\_{t}\prescript{\gamma}{}{z}(t) = K\_{M}\mathscr{y}\_{M}^{T}(t). \tag{14}$$

**Figure 1.** *Plots of both approximate and exact solution for the Example 1 for Case 1.*

Applying Lemma 1 to Eq. (14), we get

$$
\omega(t) = \epsilon\_0 + \epsilon\_1(t) + {}\_0I\_t^{\gamma}K\_M \boldsymbol{\upmu}\_M^T(t). \tag{15}
$$

Upon using the suggested method, we see from the subplot at the left of **Figure 2** that exact and numerical solutions are very close to each other for very low scale

*Numerical Solutions to Some Families of Fractional Order Differential Equations by Laguerre…*

level. Also, the absolute error is given in subplot at the right of **Figure 2**.

with the initials as well as boundary conditions.

*c* 0*D<sup>γ</sup>*

*The plot of exact and approximate solution for Example 2 for Case 2.*

*DOI: http://dx.doi.org/10.5772/intechopen.90754*

�

*c* 0*D<sup>γ</sup>*

(

We take approximation as

Applying Lemma 1 to Eq. (20), we get

(

*z t*ðÞ¼ *<sup>F</sup>*<sup>1</sup>

*y t*ðÞ¼ *<sup>y</sup>*<sup>0</sup> <sup>≈</sup>*F*<sup>2</sup>

with the conditions

Let

**Figure 2.**

and

**33**

*c* 0*D<sup>γ</sup>*

**Case 1.** First we take the coupled system of FODEs as

*<sup>t</sup> z t*ðÞ¼ <sup>Ł</sup>*Mψ<sup>T</sup>*

**4.2 Coupled systems of linear FODEs under initial and boundary conditions**

In this subsection, we consider different forms of coupled systems of FODEs

*<sup>t</sup> z t*ðÞþ *az t*ð Þþ *by t*ðÞ¼ *f t*ð Þ

*<sup>t</sup> y t*ð Þþ *cy t*ðÞþ *dz t*ðÞ¼ *g t*ð Þ,

*<sup>M</sup>*ð Þ*<sup>t</sup>* , *<sup>c</sup>*

*z t*ðÞ¼ *<sup>e</sup>*<sup>0</sup> <sup>þ</sup> <sup>Ł</sup>*MG<sup>γ</sup>*

*y t*ðÞ¼ *<sup>d</sup>*<sup>0</sup> <sup>þ</sup> *KMG<sup>γ</sup>*

Using the initial conditions given in Eq. (19), from Eq. (21), we get

*Mψ<sup>T</sup>*

*z*<sup>0</sup> ≈*F*<sup>1</sup>

*<sup>M</sup>*ð Þþ*<sup>t</sup>* <sup>Ł</sup>*MG<sup>γ</sup>*

*Mψ<sup>T</sup>*

0*D<sup>γ</sup>*

*<sup>M</sup>*�*<sup>M</sup>ψ<sup>T</sup>*

*<sup>M</sup>*�*<sup>M</sup>ψ<sup>T</sup>*

*<sup>M</sup>*�*<sup>M</sup>ψ<sup>T</sup>*

*<sup>M</sup>*ðÞþ*<sup>t</sup> KMG<sup>γ</sup>*

*Mψ<sup>T</sup> <sup>M</sup>*ð Þ*t* ,

*z*ð Þ¼ 0 *z*0, *y*ð Þ¼ 0 *y*0, *z*0, *y*<sup>0</sup> ∈ *R:* (19)

*<sup>t</sup> y t*ðÞ¼ *KMψ<sup>T</sup>*

*<sup>M</sup>*ð Þ*t* ,

*<sup>M</sup>*ð Þ*t :*

*<sup>M</sup>*ð Þ*t* ,

*<sup>M</sup>*�*<sup>M</sup>ψ<sup>T</sup>*

*<sup>M</sup>*ð Þ*t :*

(18)

(21)

(22)

*<sup>M</sup>*ð Þ*t :* (20)

Using the conditions by putting *t* ¼ 0 and *t* ¼ 1 to get *e*<sup>0</sup> ¼ *z*<sup>0</sup> and

$$\mathbf{z}\_1 = \mathbf{z}\_1 - \mathbf{z}\_0 - K\_{M0} I\_1^\gamma \boldsymbol{\mu}\_M^T(\mathbf{t}) /\_{t=1} \mathbf{z}\_1$$

Equation (15) implies

$$z(t) = z\_0 + (z\_1 - z\_0)t - tK\_{M0}I\_1^\gamma \varphi\_M^T(t)/\_{t=1} + {}\_0I\_t^\gamma K\_M \varphi\_M^T(t).$$

where *z*<sup>0</sup> þ ð Þ *z*<sup>1</sup> � *z*<sup>0</sup> *t* is the smooth function of *t* and constants; we approximate it as

$$z\_0 + (z\_1 - z\_0)t \approx G\_{M \times M}^{\gamma} \mu\_M^T(t)$$

and

$$tK\_{M0}I\_1^\gamma \boldsymbol{\nu}\_M^T(\mathbf{1}) \approx K\_M Q\_{M \times M}^\gamma \boldsymbol{\nu}\_M^T(t) \,.$$

Hence

$$z(t) = \mathbf{G}\_{\mathcal{M}\times\mathcal{M}}^{\mathcal{I}}\boldsymbol{\mu}\_{\mathcal{M}}^{\mathcal{T}}(t) - \mathbf{K}\_{\mathcal{M}}\mathbf{Q}\_{\mathcal{M}\times\mathcal{M}}^{\mathcal{I}}\boldsymbol{\mu}\_{\mathcal{M}}^{\mathcal{T}}(t) + \mathbf{K}\_{\mathcal{M}}\mathbf{G}\_{\mathcal{M}\times\mathcal{M}}^{\mathcal{I}}\boldsymbol{\mu}\_{\mathcal{M}}^{\mathcal{T}}(t)$$

So Eq. (13) implies

$$K\_M \boldsymbol{\upmu}\_M^T(\mathbf{t}) + \mathbf{G}\_{M \times M}^\mathcal{I} \boldsymbol{\upmu}\_M^T(\mathbf{t}) - K\_M \mathbf{Q}\_{M \times M}^\mathcal{I} \boldsymbol{\upmu}\_M^T(\mathbf{t}) + K\_M \mathbf{G}\_{M \times M}^\mathcal{I} \boldsymbol{\upmu}\_M^T(\mathbf{t}) = \mathbf{0}$$

which is further solved for *KM* to get the required numerical solution. For Case 2, we give the following example. **Example 2.**

$$\begin{cases} \,^c\_0 D\_t^\gamma z(t) + z(t) = 0, & 0 < \gamma \le 2, \\ z(0) = -1, \; z(1) = 1. \end{cases} \tag{16}$$

At *γ* ¼ 2, we get the exact solution as of (16) as given by (17)

$$z(t) = \mathbf{1} \mathbf{1} \mathbf{4}. \mathbf{58} \sin \left( \mathbf{x} \right) - \cos \left( \mathbf{x} \right) \tag{17}$$

*Numerical Solutions to Some Families of Fractional Order Differential Equations by Laguerre… DOI: http://dx.doi.org/10.5772/intechopen.90754*

**Figure 2.** *The plot of exact and approximate solution for Example 2 for Case 2.*

Upon using the suggested method, we see from the subplot at the left of **Figure 2** that exact and numerical solutions are very close to each other for very low scale level. Also, the absolute error is given in subplot at the right of **Figure 2**.

#### **4.2 Coupled systems of linear FODEs under initial and boundary conditions**

In this subsection, we consider different forms of coupled systems of FODEs with the initials as well as boundary conditions.

**Case 1.** First we take the coupled system of FODEs as

$$\begin{cases} \,^c\_0 D\_t^\gamma z(t) + az(t) + by(t) = f(t) \\ \,^c\_0 D\_t^\gamma y(t) + cy(t) + dz(t) = \mathbf{g}(t), \end{cases} \tag{18}$$

with the conditions

$$z(\mathbf{0}) = z\_0, \qquad y(\mathbf{0}) = y\_0, z\_0, y\_0 \in \mathcal{R}. \tag{19}$$

Let

Applying Lemma 1 to Eq. (14), we get

*Plots of both approximate and exact solution for the Example 1 for Case 1.*

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

Equation (15) implies

it as

**Figure 1.**

and

Hence

*z t*ðÞ¼ *e*<sup>0</sup> þ *e*1ð Þþ*t* <sup>0</sup>*I*

Using the conditions by putting *t* ¼ 0 and *t* ¼ 1 to get *e*<sup>0</sup> ¼ *z*<sup>0</sup> and

*e*<sup>1</sup> ¼ *z*<sup>1</sup> � *z*<sup>0</sup> � *KM*0*I*

*<sup>z</sup>*<sup>0</sup> <sup>þ</sup> ð Þ *<sup>z</sup>*<sup>1</sup> � *<sup>z</sup>*<sup>0</sup> *<sup>t</sup>*<sup>≈</sup> *<sup>G</sup><sup>γ</sup>*

*<sup>M</sup>*ðÞ�*<sup>t</sup> KMQ <sup>γ</sup>*

*<sup>M</sup>*ðÞ�*<sup>t</sup> KMQ <sup>γ</sup>*

*z*ð Þ¼� 0 1, *z*ð Þ¼ 1 1*:*

At *γ* ¼ 2, we get the exact solution as of (16) as given by (17)

which is further solved for *KM* to get the required numerical solution.

*z t*ðÞ¼ *<sup>z</sup>*<sup>0</sup> <sup>þ</sup> ð Þ *<sup>z</sup>*<sup>1</sup> � *<sup>z</sup>*<sup>0</sup> *<sup>t</sup>* � *tKM*0*<sup>I</sup> <sup>γ</sup>*

*tKM*0*I <sup>γ</sup>* <sup>1</sup> *ψ<sup>T</sup>*

*<sup>M</sup>*�*<sup>M</sup>ψ<sup>T</sup>*

*<sup>M</sup>*�*<sup>M</sup>ψ<sup>T</sup>*

For Case 2, we give the following example.

*c* 0*D<sup>γ</sup>*

*z t*ðÞ¼ *<sup>G</sup><sup>γ</sup>*

*<sup>M</sup>*ðÞþ*<sup>t</sup> <sup>G</sup><sup>γ</sup>*

So Eq. (13) implies

*KMψ<sup>T</sup>*

**Example 2.**

**32**

*γ <sup>t</sup> KMψ<sup>T</sup>*

*γ* <sup>1</sup> *ψ<sup>T</sup>*

<sup>1</sup> *ψ<sup>T</sup>*

where *z*<sup>0</sup> þ ð Þ *z*<sup>1</sup> � *z*<sup>0</sup> *t* is the smooth function of *t* and constants; we approximate

*<sup>M</sup>*ð Þ<sup>1</sup> <sup>≈</sup>*KMQ <sup>γ</sup>*

*<sup>M</sup>*�*<sup>M</sup>ψ<sup>T</sup>*

*<sup>M</sup>*�*<sup>M</sup>ψ<sup>T</sup>*

*<sup>t</sup> z t*ðÞþ *z t*ðÞ¼ 0, 0 < *γ* ≤2,

*<sup>M</sup>*ð Þ*t =<sup>t</sup>*¼<sup>1</sup>*:*

*<sup>M</sup>*ð Þ*t =<sup>t</sup>*¼<sup>1</sup>þ0*I*

*<sup>M</sup>*�*<sup>M</sup>ψ<sup>T</sup> <sup>M</sup>*ð Þ*t*

*<sup>M</sup>*�*<sup>M</sup>ψ<sup>T</sup>*

*<sup>M</sup>*ð Þ*t :*

*<sup>M</sup>*ð Þþ*<sup>t</sup> KMG<sup>γ</sup>*

*<sup>M</sup>*ðÞþ*<sup>t</sup> KMG<sup>γ</sup>*

*z t*ðÞ¼ 114*:*58 sin ð Þ� *x* cosð Þ *x* (17)

*<sup>M</sup>*�*<sup>M</sup>ψ<sup>T</sup> <sup>M</sup>*ð Þ*t*

*<sup>M</sup>*�*<sup>M</sup>ψ<sup>T</sup>*

*<sup>M</sup>*ðÞ¼ *t* 0

(16)

*γ <sup>t</sup> KMψ<sup>T</sup>*

*<sup>M</sup>*ð Þ*t :* (15)

*<sup>M</sup>*ð Þ*t* ,

$$\mathbf{t}\_0^c D\_t^\gamma z(t) = \mathbf{E}\_M \boldsymbol{\mu}\_M^T(t), \ \ \_0^c D\_t^\gamma \boldsymbol{\mathcal{y}}(t) = \mathbf{K}\_M \boldsymbol{\mu}\_M^T(t). \tag{20}$$

Applying Lemma 1 to Eq. (20), we get

$$\begin{cases} \boldsymbol{z}(t) = \boldsymbol{\sigma}\_0 + \mathbf{L}\_M \mathbf{G}\_{M \times M}^{\boldsymbol{r}} \boldsymbol{\mu}\_M^T(t), \\ \boldsymbol{\nu}(t) = \boldsymbol{d}\_0 + \mathbf{K}\_M \mathbf{G}\_{M \times M}^{\boldsymbol{r}} \boldsymbol{\mu}\_M^T(t). \end{cases} \tag{21}$$

Using the initial conditions given in Eq. (19), from Eq. (21), we get

$$\begin{cases} \boldsymbol{z}(t) = \boldsymbol{F}\_{\boldsymbol{M}}^{1} \boldsymbol{\mu}\_{\boldsymbol{M}}^{T}(t) + \mathbb{E}\_{\boldsymbol{M}} \mathbf{G}\_{\boldsymbol{M} \times \boldsymbol{M}}^{\boldsymbol{\gamma}} \boldsymbol{\mu}\_{\boldsymbol{M}}^{T}(t), \\ \boldsymbol{\mathcal{y}}(t) = \boldsymbol{\mathcal{y}}\_{0} \approx \boldsymbol{F}\_{\boldsymbol{M}}^{2} \boldsymbol{\mu}\_{\boldsymbol{M}}^{T}(t) + \mathbb{K}\_{\boldsymbol{M}} \mathbf{G}\_{\boldsymbol{M} \times \boldsymbol{M}}^{\boldsymbol{\gamma}} \boldsymbol{\mu}\_{\boldsymbol{M}}^{T}(t). \end{cases} \tag{22}$$

We take approximation as

*z*<sup>0</sup> ≈*F*<sup>1</sup> *Mψ<sup>T</sup> <sup>M</sup>*ð Þ*t* ,

and

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

$$\mathcal{Y}\_0 \approx F\_M^2 \mathcal{Y}\_M^T(t),$$

By taking *γ* ¼ 1, the exact solution is obtained as

*DOI: http://dx.doi.org/10.5772/intechopen.90754*

describes the efficiency of the proposed method.

*c* 0*D<sup>γ</sup>*

(

*c* 0*D<sup>γ</sup>*

*Absolute error at M* ¼ 5, *γ* ¼ 0*:*9*, for different values of t in Example 3.*

*Plots of exact and approximate solution of Example 3.*

**Figure 4**.

we consider

**Table 1.**

**Figure 3.**

**35**

*z t*ðÞ¼ cosðÞþ*t e*

*t*

*Numerical Solutions to Some Families of Fractional Order Differential Equations by Laguerre…*

where the external source functions are given by *f t*ðÞ¼ cosðÞþ*<sup>t</sup> <sup>e</sup>*�*<sup>t</sup>* <sup>þ</sup> <sup>2</sup>*e<sup>t</sup>* and *g t*ðÞ¼ *<sup>e</sup>*�*<sup>t</sup>* <sup>þ</sup> sin ðÞþ*<sup>t</sup>* 2 cosð Þ*<sup>t</sup> :* The exact solution *zex*, *yex* can be computed by any method of ODEs. Approximating the problem by the considered method, we see that the computed numerical and exact solutions have close agreement at very small-scale level. The corresponding accuracy has been recorded in **Table 1**. Further the comparison between exact and numerical solution and the results about absolute error have been demonstrated in **Figures 3** and **4**, respectively. In **Figure 3** we are given the comparison between exact solution and approximate solutions by using proposed method. Similarly the absolute errors have been described in

By comparing the exact and numerical solution through the proposed method, we observe that our numerical solution does not show any disagreement with the exact solution as can be seen in **Figure 3**. The absolute errors ∥*zapp* � *zex*∥ and ∥*yapp* � *yex*∥ plotted at the scale *M* ¼ 5 are very low as given in **Figure 4**, which

**Case 2.** Similarly for the coupled system of FODEs with boundary conditions,

*<sup>t</sup> z t*ðÞþ *az t*ðÞþ *by t*ðÞ¼ *f t*ð Þ,

*<sup>t</sup> y t*ðÞþ *cy t*ðÞþ *dz t*ðÞ¼ *g t*ð Þ,

*z*ð Þ¼ 0 *z*0, *y*ð Þ¼ 0 *y*0, *z*ð Þ¼ 1 *z*1, *y*ð Þ¼ 1 *y*1*:*

*t* **CPU time (s) Absolute error ∥***z*app � *z*ex**∥ Absolute error ∥***y*app � *y*ex**∥ CPU time (s)** 0 30.5 0.00003 0.000006 32.5 0.15 32.7 0.000016 0.000034 33.3 o.35 35.8 0.000013 0.00003 33.9 0.65 33.6 0.000012 0.00003 35.6 0.87 34.8 0.000018 0.000036 36.5 1 35.9 0.00003 0.000006 36.8

, *y* ¼ sin ðÞþ*t e*

�*t* ,

(23)

while source functions are approximated as

$$f(t) \approx F\_M^3 \Psi\_M^T(t),$$

and

$$\mathbf{g}(t) \approx F\_M^4 \Psi\_M^T(t).$$

Therefore the consider system on using (19)–(22), (18) becomes

$$\begin{cases} \mathbb{E}\_{M} \boldsymbol{\upmu}\_{M}^{T} + a \left( \boldsymbol{F}\_{M}^{1} \boldsymbol{\upmu}\_{M}^{T}(t) + \mathbb{E}\_{M} \mathbf{G}\_{M \times M}^{\boldsymbol{\upgamma}} \boldsymbol{\upmu}\_{M}^{T}(t) \right) \\ \quad + b \left( \boldsymbol{F}\_{M}^{2} \boldsymbol{\upmu}\_{M}^{T}(t) + \boldsymbol{K}\_{M} \mathbf{G}\_{M \times M}^{\boldsymbol{\upgamma}} \boldsymbol{\upmu}\_{M}^{T}(t) = \boldsymbol{F}\_{M}^{3} \boldsymbol{\upmu}\_{M}^{T}(t) \right) \\ \quad \left\{ \boldsymbol{K}\_{M} \boldsymbol{\upmu}\_{M}^{T} + c \left( \boldsymbol{F}\_{M}^{2} \boldsymbol{\upmu}\_{M}^{T}(t) + \boldsymbol{K}\_{M} \mathbf{G}\_{M \times M}^{\boldsymbol{\upgamma}} \boldsymbol{\upmu}\_{M}^{T}(t) \right) \right. \\ \left. \quad + d \left( \boldsymbol{F}\_{M}^{1} \boldsymbol{\upmu}\_{M}^{T}(t) + \boldsymbol{\upmu}\_{M} \mathbf{G}\_{M \times M}^{\boldsymbol{\upgamma}} \boldsymbol{\upmu}\_{M}^{T}(t) = \boldsymbol{F}\_{M}^{4} \boldsymbol{\upmu}\_{M}^{T}(t) \right) . \end{cases}$$

On further rearrangement we have

$$\begin{cases} \mathsf{E}\_{M} + a \left( F\_{M}^{1} + \mathsf{E}\_{M} G\_{M \times M}^{\operatorname{\gamma}} \right) + b \left( F\_{M}^{2} + K\_{M} G\_{M \times M}^{\operatorname{\gamma}} = F\_{M}^{3} \right) \\ K\_{M} + c \left( F\_{M}^{2} + K\_{M} G\_{M \times M}^{\operatorname{\gamma}} \right) + d \left( F\_{M}^{1} + \mathsf{E}\_{M} G\_{M \times M}^{\operatorname{\gamma}} = F\_{M}^{4} \right) \end{cases}$$

which further can be written as

$$\begin{cases} \mathbb{E}\_{\mathcal{M}} \Big( I\_{M \times \mathcal{M}} + a \mathcal{G}\_{M \times \mathcal{M}}^{\mathcal{I}} \Big) + \mathcal{K}\_{\mathcal{M}} \Big( b \mathcal{G}\_{M \times \mathcal{M}}^{\mathcal{I}} \Big) + \Big( a \mathcal{F}\_{M}^{1} + b \mathcal{F}\_{M}^{2} - \mathcal{F}\_{\mathcal{M}}^{3} \Big) = \mathbf{0}, \\\ \mathcal{K}\_{\mathcal{M}} \Big( I\_{M \times \mathcal{M}} + c \mathcal{G}\_{M \times \mathcal{M}}^{\mathcal{I}} \Big) + \mathbb{E}\_{\mathcal{M}} \Big( d \mathcal{G}\_{M \times \mathcal{M}}^{\mathcal{I}} \Big) + \Big( c \mathcal{F}\_{M}^{2} + d \mathcal{F}\_{M}^{1} - \mathcal{F}\_{\mathcal{M}}^{4} \Big) = \mathbf{0}. \end{cases}$$

In matrix form we write as

$$
\begin{split}
\begin{bmatrix}
\begin{bmatrix} \mathbf{E}\_{M} & \mathbf{K}\_{M} \end{bmatrix} \begin{bmatrix} I\_{M \times M} + a \mathbf{G}\_{M \times M}^{\prime} & \mathbf{0} \\\\ \mathbf{0} & I\_{M \times M} + c \mathbf{G}\_{M \times M}^{\prime} \end{bmatrix} + \begin{bmatrix} \mathbf{E}\_{M} & \mathbf{K}\_{M} \\\\ bG\_{M \times M}^{\prime} & \mathbf{0} \end{bmatrix} \begin{bmatrix} \mathbf{0} & dG\_{M \times M}^{\prime} \\\\ bG\_{M \times M}^{\prime} & \mathbf{0} \end{bmatrix} \\ + \begin{bmatrix} aF\_{M}^{1} + bF\_{M}^{2} - F\_{M}^{3} \\\\ cF\_{M}^{2} + dF\_{M}^{1} - F\_{M}^{4} \end{bmatrix} = \mathbf{0}.
\end{split}
$$

We solve this system of matrix equation for ½ � Ł*<sup>M</sup> KM* by using Gaussian's elimination method. The considered system is in the form of *XA* þ *XB* þ *C* ¼ 0, .

$$\begin{aligned} \text{where } & X = \begin{bmatrix} \mathbb{E}\_{M} & K\_{M} \end{bmatrix} \overline{A} = \begin{bmatrix} I\_{M \times M} + aG\_{M \times M}^{\mathbb{T}} & 0 \\ 0 & I\_{M \times M} + cG\_{M \times M}^{\mathbb{T}} \end{bmatrix}, \\\ \overline{B} &= \begin{bmatrix} 0 & dG\_{M \times M}^{\mathbb{T}} \\ bG\_{M \times M}^{\mathbb{T}} & 0 \end{bmatrix} \text{ and } \overline{C} = \begin{bmatrix} aF\_{M}^{1} + bF\_{M}^{2} - F\_{M}^{3} \\ cF\_{M}^{2} + dF\_{M}^{1} - F\_{M}^{4} \end{bmatrix}. \end{aligned}$$

Upon computation of matrices Ł*M*, *KM* by using MATLAB®, we put these matrices in Eq. (22) to find *zapp* and *yapp*, respectively.

**Example 3.** We now provide its example by considering the system of FODEs:

$$\begin{cases} \,^c\_0 D\_t^\gamma z(t) + z(t) + \mathfrak{y}(t) = f(t) \\ \,^c\_0 D\_t^\gamma \mathfrak{y}(t) + \mathfrak{y}(t) + z(t) = \mathfrak{g}(t), \\ z(\mathbf{0}) = \mathbf{2}, \ \mathfrak{y}(\mathbf{0}) = \mathbf{1}. \end{cases}$$

*Numerical Solutions to Some Families of Fractional Order Differential Equations by Laguerre… DOI: http://dx.doi.org/10.5772/intechopen.90754*

By taking *γ* ¼ 1, the exact solution is obtained as

$$z(t) = \cos\left(t\right) + e^t, \qquad \mathcal{y} = \sin\left(t\right) + e^{-t}, \dots$$

where the external source functions are given by *f t*ðÞ¼ cosðÞþ*<sup>t</sup> <sup>e</sup>*�*<sup>t</sup>* <sup>þ</sup> <sup>2</sup>*e<sup>t</sup>* and *g t*ðÞ¼ *<sup>e</sup>*�*<sup>t</sup>* <sup>þ</sup> sin ðÞþ*<sup>t</sup>* 2 cosð Þ*<sup>t</sup> :* The exact solution *zex*, *yex* can be computed by any method of ODEs. Approximating the problem by the considered method, we see that the computed numerical and exact solutions have close agreement at very small-scale level. The corresponding accuracy has been recorded in **Table 1**. Further the comparison between exact and numerical solution and the results about absolute error have been demonstrated in **Figures 3** and **4**, respectively. In **Figure 3** we are given the comparison between exact solution and approximate solutions by using proposed method. Similarly the absolute errors have been described in **Figure 4**.

By comparing the exact and numerical solution through the proposed method, we observe that our numerical solution does not show any disagreement with the exact solution as can be seen in **Figure 3**. The absolute errors ∥*zapp* � *zex*∥ and ∥*yapp* � *yex*∥ plotted at the scale *M* ¼ 5 are very low as given in **Figure 4**, which describes the efficiency of the proposed method.

**Case 2.** Similarly for the coupled system of FODEs with boundary conditions, we consider

$$\begin{cases} \,^c\_0 D\_t^\gamma z(t) + az(t) + by(t) = f(t), \\\,^c\_0 D\_t^\gamma y(t) + cy(t) + dz(t) = g(t), \\\,^z\_1 \mathbf{(0)} = \mathbf{z}\_0, y(\mathbf{0}) = \mathbf{y}\_0, z(\mathbf{1}) = \mathbf{z}\_1, y(\mathbf{1}) = \mathbf{y}\_1. \end{cases} \tag{23}$$


*y*<sup>0</sup> ≈*F*<sup>2</sup> *Mψ<sup>T</sup> <sup>M</sup>*ð Þ*t* ,

*f t*ð Þ≈*F*<sup>3</sup>

*g t*ð Þ<sup>≈</sup> *<sup>F</sup>*<sup>4</sup>

Therefore the consider system on using (19)–(22), (18) becomes

*Mψ<sup>T</sup>*

*<sup>M</sup>*ðÞþ*<sup>t</sup> KMG<sup>γ</sup>*

*Mψ<sup>T</sup>*

*<sup>M</sup>*ðÞþ*<sup>t</sup>* <sup>Ł</sup>*MG<sup>γ</sup>*

*M*�*M* � � <sup>þ</sup> *<sup>b</sup>*ð*F*<sup>2</sup>

*M*�*M* � � <sup>þ</sup> *d F*<sup>1</sup>

*M*Ψ*<sup>T</sup> <sup>M</sup>*ð Þ*t* ,

*M*Ψ*<sup>T</sup> <sup>M</sup>*ð Þ*t :*

*<sup>M</sup>*ðÞþ*<sup>t</sup>* <sup>Ł</sup>*MG<sup>γ</sup>*

*<sup>M</sup>*ðÞþ*<sup>t</sup> KMG<sup>γ</sup>*

*<sup>M</sup>*ð Þ*<sup>t</sup>* � �

*<sup>M</sup>*�*Mψ<sup>T</sup>*

*<sup>M</sup>*�*<sup>M</sup>ψ<sup>T</sup>*

*<sup>M</sup>*ð Þ*<sup>t</sup> :* �

*M*�*M* � � <sup>þ</sup> *aF*<sup>1</sup>

*M*�*M* � � <sup>þ</sup> *cF*<sup>2</sup>

*M*�*M*

*<sup>M</sup>* � *<sup>F</sup>*<sup>3</sup> *M*

*<sup>M</sup>* � *<sup>F</sup>*<sup>4</sup> *M*

We solve this system of matrix equation for ½ � Ł*<sup>M</sup> KM* by using Gaussian's elim-

þ

*<sup>M</sup>*�*<sup>M</sup>* <sup>0</sup> <sup>0</sup> *IM*�*<sup>M</sup>* <sup>þ</sup> *cG<sup>γ</sup>*

" #

*<sup>M</sup>* � *<sup>F</sup>*<sup>3</sup> *M*

*<sup>M</sup>* � *<sup>F</sup>*<sup>4</sup> *M:*

� �

*<sup>M</sup>* <sup>þ</sup> *bF*<sup>2</sup>

*<sup>M</sup>*ð Þ*<sup>t</sup>* � �

*<sup>M</sup>*�*Mψ<sup>T</sup>*

*<sup>M</sup>*ðÞ¼ *<sup>t</sup> <sup>F</sup>*<sup>3</sup>

*<sup>M</sup>*�*<sup>M</sup>ψ<sup>T</sup>*

*<sup>M</sup>*ðÞ¼ *<sup>t</sup> <sup>F</sup>*<sup>4</sup>

*<sup>M</sup>* <sup>þ</sup> *KMG<sup>γ</sup>*

*<sup>M</sup>* <sup>þ</sup> <sup>Ł</sup>*MG<sup>γ</sup>*

*<sup>M</sup>:* �

*<sup>M</sup>* <sup>þ</sup> *bF*<sup>2</sup>

*<sup>M</sup>* <sup>þ</sup> *dF*<sup>1</sup>

Ł*<sup>M</sup> KM*

¼ 0*:*

*Mψ<sup>T</sup> <sup>M</sup>*ð Þ*t :*

*Mψ<sup>T</sup>*

*<sup>M</sup>*�*<sup>M</sup>* <sup>¼</sup> *<sup>F</sup>*<sup>3</sup>

*<sup>M</sup>*�*<sup>M</sup>* <sup>¼</sup> *<sup>F</sup>*<sup>4</sup>

*<sup>M</sup>* � *<sup>F</sup>*<sup>3</sup> *M*

*<sup>M</sup>* � *<sup>F</sup>*<sup>4</sup> *M*

" # 0 *dG<sup>γ</sup>*

*M*�*M*

*:*

*bG<sup>γ</sup>*

, .

*<sup>M</sup>*�*<sup>M</sup>* <sup>0</sup>

" #

*M*�*M*

� � <sup>¼</sup> <sup>0</sup>

� � <sup>¼</sup> <sup>0</sup>*:*

*M*

while source functions are approximated as

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

Ł*Mψ<sup>T</sup>*

(

(

<sup>þ</sup>*b*ð*F*<sup>2</sup> *Mψ<sup>T</sup>*

*KMψ<sup>T</sup>*

<sup>þ</sup>*d F*<sup>1</sup> *Mψ<sup>T</sup>*

On further rearrangement we have

<sup>Ł</sup>*<sup>M</sup>* <sup>þ</sup> *a F*<sup>1</sup>

(

(

½ � Ł*<sup>M</sup> KM*

*KM* <sup>þ</sup> *c F*<sup>2</sup>

which further can be written as

<sup>Ł</sup>*<sup>M</sup> IM*�*<sup>M</sup>* <sup>þ</sup> *aG<sup>γ</sup>*

*KM IM*�*<sup>M</sup>* <sup>þ</sup> *cG<sup>γ</sup>*

In matrix form we write as

*IM*�*<sup>M</sup>* <sup>þ</sup> *aG<sup>γ</sup>*

*<sup>M</sup>* <sup>þ</sup> *a F*<sup>1</sup>

*<sup>M</sup>* <sup>þ</sup> *c F*<sup>2</sup>

*<sup>M</sup>* <sup>þ</sup> <sup>Ł</sup>*MG<sup>γ</sup>*

*<sup>M</sup>* <sup>þ</sup> *KMG<sup>γ</sup>*

*M*�*M* � � <sup>þ</sup> *KM bG<sup>γ</sup>*

*M*�*M* � � <sup>þ</sup> <sup>Ł</sup>*<sup>M</sup> dG<sup>γ</sup>*

> *<sup>M</sup>*�*<sup>M</sup>* <sup>0</sup> <sup>0</sup> *IM*�*<sup>M</sup>* <sup>þ</sup> *cG<sup>γ</sup>*

> > *aF*<sup>1</sup>

*cF*<sup>2</sup> *<sup>M</sup>* <sup>þ</sup> *dF*<sup>1</sup>

*<sup>M</sup>* <sup>þ</sup> *bF*<sup>2</sup>

ination method. The considered system is in the form of *XA* þ *XB* þ *C* ¼ 0, .

*cF*<sup>2</sup> *<sup>M</sup>* <sup>þ</sup> *dF*<sup>1</sup>

Upon computation of matrices Ł*M*, *KM* by using MATLAB®, we put these

**Example 3.** We now provide its example by considering the system of FODEs:

*<sup>t</sup> z t*ðÞþ *z t*ðÞþ *y t*ðÞ¼ *f t*ð Þ

*<sup>t</sup> y t*ðÞþ *y t*ðÞþ *z t*ðÞ¼ *g t*ð Þ,

*z*ð Þ¼ 0 2, *y*ð Þ¼ 0 1*:*

and *<sup>C</sup>* <sup>¼</sup> *aF*<sup>1</sup>

" #

" #

þ

where *<sup>X</sup>* <sup>¼</sup> ½ � <sup>Ł</sup>*<sup>M</sup> KM <sup>A</sup>* <sup>¼</sup> *IM*�*<sup>M</sup>* <sup>þ</sup> *aG<sup>γ</sup>*

*M*�*M*

matrices in Eq. (22) to find *zapp* and *yapp*, respectively.

*c* 0*D<sup>γ</sup>*

8 ><

>:

*c* 0*D<sup>γ</sup>*

*<sup>B</sup>* <sup>¼</sup> <sup>0</sup> *dG<sup>γ</sup>*

*<sup>M</sup>*�*<sup>M</sup>* <sup>0</sup> � �

*bG<sup>γ</sup>*

**34**

and

*Absolute error at M* ¼ 5, *γ* ¼ 0*:*9*, for different values of t in Example 3.*

**Figure 3.** *Plots of exact and approximate solution of Example 3.*

**Figure 4.** *Plots of absolute error of Example 3.*

Let us assume

$$\begin{cases} \,^\epsilon\_0 D\_t^\gamma x(t) = \mathbf{E}\_M \boldsymbol{\mu}\_M^T(t), \\\,^\epsilon\_0 D\_t^\gamma y(t) = \mathbf{K}\_M \boldsymbol{\mu}\_M^T(t). \end{cases} \tag{24}$$

approximating *f t*ð Þ and *g t*ð Þ such that

*DOI: http://dx.doi.org/10.5772/intechopen.90754*

*<sup>M</sup>*ðÞþ*<sup>t</sup> a F*<sup>1</sup>

*<sup>M</sup>*ðÞþ*<sup>t</sup> c F*<sup>2</sup>

<sup>Ł</sup>*<sup>M</sup> IM*�*<sup>M</sup>* � *aQ <sup>γ</sup>* ,*<sup>z</sup>*

*<sup>M</sup>* <sup>þ</sup> *bF*<sup>2</sup>

*KM IM*�*<sup>M</sup>* � *cQ <sup>γ</sup>* ,*<sup>y</sup>*

*<sup>M</sup>* <sup>þ</sup> *dF*<sup>1</sup>

In matrix form, we can write

*LM*Ψ*<sup>T</sup>*

8 >>>><

>>>>:

8 >>>><

>>>>:

<sup>þ</sup>*b F*<sup>2</sup> *M*Ψ*<sup>T</sup>*

*KM*Ψ*<sup>T</sup>*

<sup>þ</sup>*d F*<sup>1</sup> *M*Ψ*<sup>T</sup>*

<sup>þ</sup> *aF*<sup>1</sup>

<sup>þ</sup> *cF*<sup>2</sup>

½ � Ł*<sup>M</sup> KM*

þ½ � *LMKM*

*aF*<sup>1</sup>

*cF*<sup>2</sup> *<sup>M</sup>* <sup>þ</sup> *dF*<sup>1</sup>

*<sup>M</sup>* <sup>þ</sup> *bF*<sup>2</sup>

" #

*<sup>L</sup>* <sup>¼</sup> *IM*�*<sup>M</sup>* � *aQ <sup>γ</sup>* ,*<sup>z</sup>*

so that the system is of the form

*IM*�*<sup>M</sup>* � *bQ <sup>γ</sup>* ,*<sup>y</sup>*

þ

required solution.

**37**

*f t*ð Þ≈*F*<sup>3</sup>

(

*<sup>M</sup>*ðÞ�*<sup>t</sup> LMQ <sup>γ</sup>* ,*<sup>z</sup>*

*<sup>M</sup>*ðÞ�*<sup>t</sup> KMQ <sup>γ</sup>* ,*<sup>y</sup>*

*<sup>M</sup>*�*M*Ψ*<sup>T</sup>*

*<sup>M</sup>*�*M*Ψ*<sup>T</sup>*

On rearrangement of terms, the above equations give

*<sup>M</sup>*�*<sup>M</sup>* <sup>þ</sup> *aG<sup>γ</sup>*

*<sup>M</sup>* ¼ 0

*<sup>M</sup>*�*<sup>M</sup>* <sup>þ</sup> *cG<sup>γ</sup>*

*<sup>M</sup>* ¼ 0*:*

On using (24)–(29), system (23) can be written as

*M*Ψ*<sup>T</sup>*

*M*Ψ*<sup>T</sup>*

*<sup>M</sup>*ðÞ�*<sup>t</sup> KMQ <sup>γ</sup>*

*<sup>M</sup>*ð Þ�*<sup>t</sup>* <sup>Ł</sup>*MQ <sup>γ</sup>* ,*<sup>z</sup>*

*<sup>M</sup>* � *<sup>F</sup>*<sup>3</sup>

*<sup>M</sup>* � *<sup>F</sup>*<sup>4</sup>

*IM*�*<sup>M</sup>* � *aQ <sup>γ</sup>* ,*<sup>z</sup>*

*IM*�*<sup>M</sup>* � *bQ <sup>γ</sup>* ,*<sup>y</sup>*

*<sup>M</sup>* � *<sup>F</sup>*<sup>3</sup> *M*

*<sup>M</sup>* � *<sup>F</sup>*<sup>4</sup> *M* *g t*ð Þ≈*F*<sup>4</sup>

*<sup>M</sup>*ð Þ*<sup>t</sup>* � � � *<sup>F</sup>*<sup>3</sup>

*<sup>M</sup>*ð Þ*<sup>t</sup>* � � � *<sup>F</sup>*<sup>4</sup>

*M*�*M* � � <sup>þ</sup> *KM IM*�*<sup>M</sup>* � *bQ <sup>γ</sup>* ,*<sup>y</sup>*

*M*�*M* � � <sup>þ</sup> <sup>Ł</sup>*<sup>M</sup> IM*�*<sup>M</sup>* � *dQ <sup>γ</sup>* ,*<sup>z</sup>*

*<sup>M</sup>*�*<sup>M</sup>* <sup>þ</sup> *aG<sup>γ</sup>*

*<sup>M</sup>*�*<sup>M</sup>* <sup>þ</sup> *bG<sup>γ</sup>*

¼ 0*:*

We convert the system to algebraic equation by considering

*<sup>M</sup>*�*<sup>M</sup>* <sup>þ</sup> *aG<sup>γ</sup>*

*<sup>M</sup>* <sup>¼</sup> <sup>0</sup> *IM*�*<sup>M</sup>* � *dQ <sup>γ</sup>* ,*<sup>z</sup>*

*<sup>M</sup>*�*<sup>M</sup>* <sup>þ</sup> *bG<sup>γ</sup>*

and *<sup>N</sup>* <sup>¼</sup> *aF*<sup>1</sup>

tion for the coupled system with the boundary conditions as

*M*Ψ*<sup>T</sup> <sup>M</sup>*ð Þ*t*

*Numerical Solutions to Some Families of Fractional Order Differential Equations by Laguerre…*

(29)

*M*Ψ*<sup>T</sup> <sup>M</sup>*ð Þ*t :*

*<sup>M</sup>*�*M*Ψ*<sup>T</sup>*

*<sup>M</sup>*ð Þþ*<sup>t</sup> KMG<sup>γ</sup>*

*<sup>M</sup>*ðÞþ*<sup>t</sup>* <sup>Ł</sup>*MG<sup>γ</sup>*

*<sup>M</sup>*�*M*Ψ*<sup>T</sup>*

*<sup>M</sup>*ð Þ*<sup>t</sup>* � �

*<sup>M</sup>*ð Þ*<sup>t</sup>* � �

*<sup>M</sup>*ðÞþ*<sup>t</sup> LMG<sup>γ</sup>*

*<sup>M</sup>*�*M*Ψ*<sup>T</sup>*

*<sup>M</sup>*�*M*Ψ*<sup>T</sup>*

*<sup>M</sup>*�*<sup>M</sup>* <sup>0</sup>

*<sup>M</sup>*�*<sup>M</sup>* <sup>0</sup>

<sup>0</sup> *IM*�*<sup>M</sup>* � *cQ <sup>γ</sup>* ,*<sup>y</sup>*

<sup>0</sup> *IM*�*<sup>M</sup>* � *dQ <sup>γ</sup>* ,*<sup>z</sup>*

*<sup>M</sup>*�*<sup>M</sup>* <sup>0</sup>

*<sup>M</sup>*�*<sup>M</sup>* <sup>0</sup>

*<sup>M</sup>* � *<sup>F</sup>*<sup>3</sup> *M*

*:*

*<sup>M</sup>* � *<sup>F</sup>*<sup>4</sup> *M*

<sup>0</sup> *IM*�*<sup>M</sup>* � *cQ <sup>γ</sup>* ,*<sup>y</sup>*

" #

*<sup>M</sup>* <sup>þ</sup> *bF*<sup>2</sup>

" #

*cF*<sup>2</sup> *<sup>M</sup>* <sup>þ</sup> *dF*<sup>1</sup>

*XL* þ *XM* þ *N* ¼ 0,

and solving the given equation for the unknown matrix *X* ¼ ½ � *LMKM* , we get the

**Example 4.** As an example, we consider the Caputo fractional differential equa-

" #

" #

" #

*<sup>M</sup>*ðÞþ*<sup>t</sup> KMG<sup>γ</sup>*

*<sup>M</sup>*�*M*Ψ*<sup>T</sup>*

*<sup>M</sup>*�*M*Ψ*<sup>T</sup>*

� �

� �

*M*Ψ*<sup>T</sup>*

*M*Ψ*<sup>T</sup>*

*<sup>M</sup>*�*<sup>M</sup>* <sup>þ</sup> *bG<sup>γ</sup>*

*<sup>M</sup>*�*<sup>M</sup>* <sup>þ</sup> *dG<sup>γ</sup>*

*<sup>M</sup>*�*<sup>M</sup>* <sup>þ</sup> *cG<sup>γ</sup>*

*<sup>M</sup>*�*<sup>M</sup>* <sup>þ</sup> *cG<sup>γ</sup>*

*<sup>M</sup>*�*<sup>M</sup>* <sup>þ</sup> *dG<sup>γ</sup>*

*M*�*M*

*M*�*M*

*<sup>M</sup>*�*<sup>M</sup>* <sup>þ</sup> *dG<sup>γ</sup>*

*<sup>M</sup>*ðÞ¼ *t* 0

*<sup>M</sup>*ðÞ¼ *t* 0*:*

*M*�*M*

*M*�*M*

*M*�*M*

*M*�*M*

Applying Lemma 1 to Eq. (24), we get

$$\begin{cases} z(t) = \mathbf{e}\_0 + \mathbf{e}\_1(t) + \mathbf{k}\_M \mathbf{G}\_{M \times M}^{\gamma} \boldsymbol{\Psi}\_M^T(t) \\ y(t) = d\_0 + d\_1(t) + \mathbf{K}\_M \mathbf{G}\_{M \times M}^{\star \gamma} \boldsymbol{\Psi}\_M^T(t), \end{cases} \tag{25}$$

where *d*0, *d*1,*e*0,*e*<sup>1</sup> ∈*R:* Using the initial conditions in Eq. (25), we have *e*<sup>0</sup> ¼ *z*0, *d*<sup>0</sup> ¼ *y*0*:* On using boundary conditions, we have from Eq. (25)

$$\begin{aligned} z(\mathbf{1}) &= z\_0 + \mathbf{e}\_1 + \mathbf{E}\_M \mathbf{G}\_{M \times M}^{\mathcal{I}} \Psi\_M^T(t) \big|\_{t=1}, \\ z(\mathbf{1}) &- z\_0 - \mathbf{E}\_M \mathbf{G}\_{M \times M}^{\mathcal{I}} \Psi\_M^T(t) \big|\_{t=1} = \mathbf{e}\_1. \end{aligned}$$

Similarly

$$\begin{aligned} \boldsymbol{\mathcal{Y}}(\mathbf{1}) &= \boldsymbol{\mathcal{Y}}\_0 + \boldsymbol{d}\_1 + \boldsymbol{K}\_M \boldsymbol{G}\_{M \times M}^{\star \boldsymbol{\mathcal{Y}}} \boldsymbol{\Psi}\_M^T(t) \big|\_{t=1}, \\ \boldsymbol{\mathcal{Y}}(\mathbf{1}) &= \boldsymbol{\mathcal{Y}}\_0 - \boldsymbol{K}\_M \boldsymbol{G}\_{M \times M}^{\star \boldsymbol{\mathcal{Y}}} \boldsymbol{\Psi}\_M^T(t) \big|\_{t=1} = \boldsymbol{d}\_1. \end{aligned}$$

Equation (25) implies that

$$\begin{cases} z(t) = z\_0 + t(z\_1 - z\_0) - t(L\_M G\_{M \times M}^{\gamma} \Psi\_M^T(t)|\_{t=1}) + L\_M G\_{M \times M}^{\gamma} \Psi\_M^T(t) \\ y(t) = y\_0 + t(y\_1 - y\_0) - t(K\_M G\_{M \times M}^{\star \gamma} \Psi\_M^T(t)|\_{t=1}) + K\_M G\_{M \times M}^{\star \gamma} \Psi\_M^T(t) . \end{cases} \tag{26}$$

Let *<sup>z</sup>*<sup>0</sup> <sup>þ</sup> *t z*ð Þ <sup>1</sup> � *<sup>z</sup>*<sup>0</sup> <sup>≈</sup>*F*<sup>1</sup> *M*Ψ*<sup>T</sup> <sup>M</sup>*ð Þ*t* and *y*<sup>0</sup> þ *t y*<sup>1</sup> � *y*<sup>0</sup> � �≈*F*<sup>2</sup> *Mψ<sup>T</sup> <sup>M</sup>*ð Þ*t* , with

$$\begin{aligned} \mathsf{E}\_{\mathsf{M}} \mathbf{G}\_{\mathsf{M}\times\mathsf{M}}^{\gamma} \boldsymbol{\Psi}\_{\mathsf{M}}^{T}(t) &= \mathsf{E}\_{\mathsf{M}} \mathbf{Q}\_{\mathcal{M}\times\mathsf{M}}^{\gamma,x} \boldsymbol{\Psi}\_{\mathsf{M}}^{T}(t) \\ t \mathsf{K}\_{\mathsf{M}} \mathbf{G}\_{\mathsf{M}\times\mathsf{M}}^{\star,\gamma} \boldsymbol{\Psi}\_{\mathsf{M}}^{T}(t) &= \mathsf{K}\_{\mathsf{M}} \mathbf{Q}\_{\mathcal{M}\times\mathsf{M}}^{\gamma,y} \boldsymbol{\Psi}\_{\mathsf{M}}^{T}(t) . \end{aligned} \tag{27}$$

Hence Eq. (26) implies

$$\begin{cases} \boldsymbol{z}(t) = \boldsymbol{F}\_{\boldsymbol{M}}^{1} \boldsymbol{\Psi}\_{\boldsymbol{M}}^{T}(t) - L\_{\boldsymbol{M}} \boldsymbol{Q}\_{\boldsymbol{M}\times\boldsymbol{M}}^{\boldsymbol{r},\boldsymbol{x}} \boldsymbol{\Psi}\_{\boldsymbol{M}}^{T}(t) + L\_{\boldsymbol{M}} \boldsymbol{G}\_{\boldsymbol{M}\times\boldsymbol{M}}^{\boldsymbol{r}} \boldsymbol{\Psi}\_{\boldsymbol{M}}^{T}(t) \\ \boldsymbol{y}(t) = \boldsymbol{F}\_{\boldsymbol{M}}^{2} \boldsymbol{\Psi}\_{\boldsymbol{M}}^{T}(t) - K\_{\boldsymbol{M}} \boldsymbol{Q}\_{\boldsymbol{M}\times\boldsymbol{M}}^{\boldsymbol{r},\boldsymbol{y}} \boldsymbol{\Psi}\_{\boldsymbol{M}}^{T}(t) + K\_{\boldsymbol{M}} \boldsymbol{G}\_{\boldsymbol{M}\times\boldsymbol{M}}^{\boldsymbol{\star}\boldsymbol{\gamma}} \boldsymbol{\Psi}\_{\boldsymbol{M}}^{T}(t) . \end{cases} \tag{28}$$

*Numerical Solutions to Some Families of Fractional Order Differential Equations by Laguerre… DOI: http://dx.doi.org/10.5772/intechopen.90754*

approximating *f t*ð Þ and *g t*ð Þ such that

$$\begin{cases} f(t) \approx F\_M^3 \Psi\_M^T(t) \\ g(t) \approx F\_M^4 \Psi\_M^T(t). \end{cases} \tag{29}$$

On using (24)–(29), system (23) can be written as

$$\begin{cases} L\_M \Psi\_M^T(t) + a \left( F\_M^1 \Psi\_M^T(t) - L\_M \mathbf{Q}\_{M \times M}^{\gamma, x} \Psi\_M^T(t) + L\_M \mathbf{G}\_{M \times M}^{\gamma} \Psi\_M^T(t) \right) \\ \quad + b \left( F\_M^2 \Psi\_M^T(t) - K\_M \mathbf{Q}\_{M \times M}^{\gamma} \Psi\_M^T(t) + K\_M \mathbf{G}\_{M \times M}^{\gamma} \Psi\_M^T(t) \right) - F\_M^3 \Psi\_M^T(t) = \mathbf{0} \\ K\_M \Psi\_M^T(t) + c \left( F\_M^2 \Psi\_M^T(t) - K\_M \mathbf{Q}\_{M \times M}^{\gamma, y} \Psi\_M^T(t) + K\_M \mathbf{G}\_{M \times M}^{\gamma} \Psi\_M^T(t) \right) \\ \quad + d \left( F\_M^1 \Psi\_M^T(t) - \mathbf{E}\_M \mathbf{Q}\_{M \times M}^{\gamma, x} \Psi\_M^T(t) + \mathbf{E}\_M \mathbf{G}\_{M \times M}^{\gamma} \Psi\_M^T(t) \right) - F\_M^4 \Psi\_M^T(t) = \mathbf{0}. \end{cases}$$

On rearrangement of terms, the above equations give

$$\begin{cases} \mathsf{E}\_{M}(I\_{M\times M}-a\mathsf{Q}\_{M\times M}^{\gamma,x}+a\mathsf{G}\_{M\times M}^{\gamma}) + \mathsf{K}\_{M}(I\_{M\times M}-b\mathsf{Q}\_{M\times M}^{\gamma,y}+b\mathsf{G}\_{M\times M}^{\gamma})) \\ \quad + a\mathsf{F}\_{M}^{1}+b\mathsf{F}\_{M}^{2}-\mathsf{F}\_{M}^{3}=\mathsf{0} \\ \quad \begin{aligned} &K\_{M}(I\_{M\times M}-c\mathsf{Q}\_{M\times M}^{\gamma,y}+c\mathsf{G}\_{M\times M}^{\gamma})+\mathsf{L}\_{M}(I\_{M\times M}-d\mathsf{Q}\_{M\times M}^{\gamma,x}+d\mathsf{G}\_{M\times M}^{\gamma}) \\ &+c\mathsf{F}\_{M}^{2}+d\mathsf{F}\_{M}^{1}-\mathsf{F}\_{M}^{4}=\mathsf{0}. \end{aligned} \end{cases}$$

In matrix form, we can write

Let us assume

*Plots of absolute error of Example 3.*

**Figure 4.**

Similarly

(

Equation (25) implies that

*y t*ðÞ¼ *y*<sup>0</sup> þ *t y*<sup>1</sup> � *y*<sup>0</sup>

Let *<sup>z</sup>*<sup>0</sup> <sup>þ</sup> *t z*ð Þ <sup>1</sup> � *<sup>z</sup>*<sup>0</sup> <sup>≈</sup>*F*<sup>1</sup>

Hence Eq. (26) implies

(

**36**

*z t*ðÞ¼ *<sup>F</sup>*<sup>1</sup>

*y t*ðÞ¼ *<sup>F</sup>*<sup>2</sup>

*M*Ψ*<sup>T</sup>*

*M*Ψ*<sup>T</sup>*

*z t*ðÞ¼ *<sup>z</sup>*<sup>0</sup> <sup>þ</sup> *t z*ð Þ� <sup>1</sup> � *<sup>z</sup>*<sup>0</sup> *<sup>t</sup>*ð*LMG<sup>γ</sup>*

*c* 0*D<sup>γ</sup>*

(

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

Applying Lemma 1 to Eq. (24), we get

(

*c* 0*D<sup>γ</sup>*

*d*<sup>0</sup> ¼ *y*0*:* On using boundary conditions, we have from Eq. (25)

*<sup>z</sup>*ð Þ� <sup>1</sup> *<sup>z</sup>*<sup>0</sup> � <sup>Ł</sup>*MG<sup>γ</sup>*

*z t*ðÞ¼ *<sup>e</sup>*<sup>0</sup> <sup>þ</sup> *<sup>e</sup>*1ð Þþ*<sup>t</sup>* <sup>Ł</sup>*MG<sup>γ</sup>*

*<sup>z</sup>*ð Þ¼ <sup>1</sup> *<sup>z</sup>*<sup>0</sup> <sup>þ</sup> *<sup>e</sup>*<sup>1</sup> <sup>þ</sup> <sup>Ł</sup>*MG<sup>γ</sup>*

*<sup>y</sup>*ð Þ¼ <sup>1</sup> *<sup>y</sup>*<sup>0</sup> <sup>þ</sup> *<sup>d</sup>*<sup>1</sup> <sup>þ</sup> *KMG*<sup>⋆</sup>*<sup>γ</sup>*

*<sup>y</sup>*ð Þ� <sup>1</sup> *<sup>y</sup>*<sup>0</sup> � *KMG*<sup>⋆</sup>*<sup>γ</sup>*

� � � *t KMG*<sup>⋆</sup>*<sup>γ</sup>*

*M*Ψ*<sup>T</sup>*

Ł*MG<sup>γ</sup>*

*tKMG*<sup>⋆</sup>*<sup>γ</sup>*

*<sup>M</sup>*ðÞ�*<sup>t</sup> LMQ <sup>γ</sup>* ,*<sup>z</sup>*

*<sup>M</sup>*ðÞ�*<sup>t</sup> KMQ <sup>γ</sup>* ,*<sup>y</sup>*

*<sup>M</sup>*�*<sup>M</sup>*Ψ*<sup>T</sup>*

*<sup>M</sup>*�*<sup>M</sup>*Ψ*<sup>T</sup>*

*y t*ðÞ¼ *<sup>d</sup>*<sup>0</sup> <sup>þ</sup> *<sup>d</sup>*1ðÞþ*<sup>t</sup> KMG*<sup>⋆</sup>*<sup>γ</sup>*

*<sup>t</sup> z t*ðÞ¼ <sup>Ł</sup>*Mψ<sup>T</sup>*

*<sup>t</sup> y t*ðÞ¼ *KMψ<sup>T</sup>*

where *d*0, *d*1,*e*0,*e*<sup>1</sup> ∈*R:* Using the initial conditions in Eq. (25), we have *e*<sup>0</sup> ¼ *z*0,

*<sup>M</sup>*�*<sup>M</sup>*Ψ*<sup>T</sup> <sup>M</sup>*ð Þ*t* � � *<sup>t</sup>*¼<sup>1</sup> <sup>¼</sup> *<sup>e</sup>*1*:*

*<sup>M</sup>*�*<sup>M</sup>*Ψ*<sup>T</sup> <sup>M</sup>*ð Þ*t* � � *<sup>t</sup>*¼<sup>1</sup> <sup>¼</sup> *<sup>d</sup>*1*:*

*<sup>M</sup>*�*<sup>M</sup>*Ψ*<sup>T</sup> <sup>M</sup>*ð Þ*t* � � *t*¼1

*<sup>M</sup>*�*<sup>M</sup>*Ψ*<sup>T</sup> <sup>M</sup>*ð Þ*<sup>t</sup>* � �

*<sup>M</sup>*ð Þ*t* and *y*<sup>0</sup> þ *t y*<sup>1</sup> � *y*<sup>0</sup>

*<sup>M</sup>*�*<sup>M</sup>*Ψ*<sup>T</sup>*

*<sup>M</sup>*�*<sup>M</sup>*Ψ*<sup>T</sup>*

*<sup>M</sup>*ðÞ¼ *<sup>t</sup>* <sup>Ł</sup>*MQ <sup>γ</sup>* ,*<sup>z</sup>*

*<sup>M</sup>*ðÞ¼ *<sup>t</sup> KMQ <sup>γ</sup>* ,*<sup>y</sup>*

*<sup>M</sup>*ðÞþ*<sup>t</sup> LMG<sup>γ</sup>*

*<sup>M</sup>*ðÞþ*<sup>t</sup> KMG*<sup>⋆</sup>*<sup>γ</sup>*

*<sup>M</sup>*ð Þ*t* ,

*<sup>M</sup>*ð Þ*t :*

*<sup>M</sup>*�*<sup>M</sup>*Ψ*<sup>T</sup> <sup>M</sup>*ð Þ*t*

*<sup>M</sup>*�*<sup>M</sup>*Ψ*<sup>T</sup> <sup>M</sup>*ð Þ*t* � � *t*¼1 ,

*<sup>M</sup>*�*<sup>M</sup>*Ψ*<sup>T</sup> <sup>M</sup>*ð Þ*t* � � *t*¼1 ,

> � *t*¼1

� �≈*F*<sup>2</sup>

*<sup>M</sup>*�*<sup>M</sup>*Ψ*<sup>T</sup> <sup>M</sup>*ð Þ*t*

*<sup>M</sup>*�*<sup>M</sup>*Ψ*<sup>T</sup>*

*<sup>M</sup>*�*<sup>M</sup>*Ψ*<sup>T</sup> <sup>M</sup>*ð Þ*t*

*<sup>M</sup>*�*<sup>M</sup>*Ψ*<sup>T</sup>*

*<sup>M</sup>*ð Þ*t :*

Þ þ *LMG<sup>γ</sup>*

Þ þ *KMG*<sup>⋆</sup>*<sup>γ</sup>*

*Mψ<sup>T</sup>*

*<sup>M</sup>*�*<sup>M</sup>*Ψ*<sup>T</sup> <sup>M</sup>*ð Þ*t*

*<sup>M</sup>*�*<sup>M</sup>*Ψ*<sup>T</sup>*

*<sup>M</sup>*ð Þ*t* , with

*<sup>M</sup>*ð Þ*t :*

*<sup>M</sup>*ð Þ*<sup>t</sup> :* (27)

*<sup>M</sup>*�*<sup>M</sup>*Ψ*<sup>T</sup>*

*<sup>M</sup>*ð Þ*t* ,

(24)

(25)

(26)

(28)

$$\begin{aligned} \begin{bmatrix} \mathbf{E}\_{M} & \mathbf{K}\_{M} \end{bmatrix} \begin{bmatrix} I\_{M \times M} - a \mathbf{Q}\_{M \times M}^{\mathrm{r},x} + a \mathbf{G}\_{M \times M}^{\mathrm{r}} & \mathbf{0} \\ \mathbf{0} & I\_{M \times M} - c \mathbf{Q}\_{M \times M}^{\mathrm{r},y} + c \mathbf{G}\_{M \times M}^{\mathrm{r}} \end{bmatrix} \\ + [L\_{M} \mathbf{K}\_{M}] \begin{bmatrix} \mathbf{0} & I\_{M \times M} - d \mathbf{Q}\_{M \times M}^{\mathrm{r},x} + d \mathbf{G}\_{M \times M}^{\mathrm{r}} \\ I\_{M \times M} - b \mathbf{Q}\_{M \times M}^{\mathrm{r},y} + b \mathbf{G}\_{M \times M}^{\mathrm{r}} & \mathbf{0} \end{bmatrix} \\ + \begin{bmatrix} a \mathbf{F}\_{M}^{1} + b \mathbf{F}\_{M}^{2} - \mathbf{F}\_{M}^{3} \\ c \mathbf{F}\_{M}^{2} + d \mathbf{F}\_{M}^{1} - \mathbf{F}\_{M}^{4} \end{bmatrix} = \mathbf{0}. \end{aligned}$$

We convert the system to algebraic equation by considering

$$
\overline{\mathcal{L}} = \begin{bmatrix}
I\_{M \times M} - aQ\_{M \times M}^{\gamma, x} + aG\_{M \times M}^{\gamma} & \mathbf{0} \\
\mathbf{0} & I\_{M \times M} - cQ\_{M \times M}^{\gamma, y} + cG\_{M \times M}^{\gamma}
\end{bmatrix},
$$

$$
\overline{\mathcal{M}} = \begin{bmatrix}
\mathbf{0} & I\_{M \times M} - dQ\_{M \times M}^{\gamma, x} + dG\_{M \times M}^{\gamma} \\
I\_{M \times M} - bQ\_{M \times M}^{\gamma, y} + bG\_{M \times M}^{\gamma} & \mathbf{0} \\
\mathbf{0} & I\_{M \times M}^{\gamma}
\end{bmatrix},
$$

$$
\text{and } \overline{\mathcal{N}} = \begin{bmatrix}
aF\_M^1 + bF\_M^2 - F\_M^3 \\
cF\_M^2 + dF\_M^1 - F\_M^4
\end{bmatrix}.
$$

so that the system is of the form

$$X\overline{L} + X\overline{M} + \overline{N} = 0,$$

and solving the given equation for the unknown matrix *X* ¼ ½ � *LMKM* , we get the required solution.

**Example 4.** As an example, we consider the Caputo fractional differential equation for the coupled system with the boundary conditions as

$$\begin{cases} \,^c\_0 D\_t^\gamma z(t) + 2z(t) - 2y(t) - f(t) = \mathbf{0}, \\ \,^c\_0 D\_t^\gamma y(t) - 3y(t) + 2z(t) - g(t) = \mathbf{0}, \\ z(0) = 4 \quad z(1) = -4, \\ y(0) = 2, \quad y(1) = -2. \end{cases}$$

At *γ* ¼ 2, the exact solutions are

$$\begin{cases} z(t) = t^6 + t^5 + t^4 - t^3 + t + \mathbf{1}, \\ y(t) = t^7 - t^6 + t^5 + t^4 + t^3 - t^2 - t + \mathbf{1}. \end{cases}$$

where the source functions are given by

$$\begin{cases} f(t) = -2t^7 + 4t^6 + 30t^4 + 16t^3 + 12t^2 - 2t + 2\\ g(t) = -3t^7 + 12t^6 + 35t^5 - 27t^4 - 19t^3 + 20t^2 + 9t - 4. \end{cases}$$

comparison of errors for exact and approximate solutions for fixed scale level *M* ¼ 5 and order *γ* ¼ 1*:*9*:* Further the absolute error has been recorded at different values of space variable in **Table 2** which provides the information about efficiency of the

*t* **Absolute error ∥***z*app � *z*ex**∥ CPU time (s) Absolute error ∥***y*app � *y*ex**∥ CPU time (s)** 0 0.011 49.4 0.010 50.0 0.15 0.0062 50.3 0.0052 52.5 0.35 0.0058 51.2 0.0047 54.6 0.65 0.006 51.5 0.005 55.5 0.85 0.0075 52.6 0.007 56.4 1 0.011 53.8 0.010 56.2

*Numerical Solutions to Some Families of Fractional Order Differential Equations by Laguerre…*

We have successfully used the class of orthogonal polynomials of Laguerre polynomials to establish a numerical method to compute the numerical solution of FODEs and their coupled systems under some initial and boundary conditions. By

corresponding to fractional order derivatives and integration. Also we have computed a new matrix corresponding to boundary conditions for boundary value problems of FODEs. Using the aforementioned matrices, we have converted the considered problem of FODEs to Sylvester-type algebraic equations. To obtain the numerical solution, we easily solved the desired algebraic equations by taking help from MATLAB®. Corresponding to the established procedure, we have provided numbers of examples to demonstrate our results. Also some error analyses have been provided along with graphical representations. By increasing the scale level, the accuracy is increased and vice versa. On the other hand, when the fractional order is approaching to integer value, the solutions tend to the exact solutions of the considered FODE. Therefore in each example, we have compared the exact and approximate solution and found that both the solutions were in closure contact with each other. Hence the established method can be very helpful in solving many classes and systems of FODEs under both initial and boundary conditions. In future the shifted Laguerre polynomials can be used to compute numerical solutions of

All authors equally contributed this paper and approved the final version.

We declare that no competing interests exist regarding this manuscript.

using these polynomials, we have obtained some operational matrices

partial differential equations of fractional order.

**Author contribution**

**Competing interests**

**39**

proposed method.

*Absolute error at different values of t for Example 4.*

*DOI: http://dx.doi.org/10.5772/intechopen.90754*

**Table 2.**

**5. Conclusion**

We approximate the solution at the considered method by taking scale level *M* ¼ 5*:* One can see that numerical plot and exact solution plot coincide very well as shown in **Figure 5**. Similarly the absolute error has been plotted at the given scale *M* ¼ 5 in **Figure 6**, which is very low. The lowest value of absolute error ∥*zapp* � *zex*∥ and ∥*yapp* � *yex*∥ indicates efficiency of the proposed method. The table shows the

**Figure 5.** *Plots of exact and approximate solution for Case 4, boundary value problem.*

**Figure 6.** *Plots of absolute error for Case 4, boundary value problem.*

*Numerical Solutions to Some Families of Fractional Order Differential Equations by Laguerre… DOI: http://dx.doi.org/10.5772/intechopen.90754*


**Table 2.**

*c* 0*D<sup>γ</sup>*

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

8 >>><

>>>:

At *γ* ¼ 2, the exact solutions are

*f t*ðÞ¼�2*t*

(

**Figure 5.**

**Figure 6.**

**38**

*g t*ðÞ¼�3*t*

(

*c* 0*D<sup>γ</sup>*

*z t*ðÞ¼ *t*

*y t*ðÞ¼ *t*

<sup>7</sup> <sup>þ</sup> <sup>4</sup>*<sup>t</sup>*

<sup>7</sup> <sup>þ</sup> <sup>12</sup>*<sup>t</sup>*

*Plots of exact and approximate solution for Case 4, boundary value problem.*

*Plots of absolute error for Case 4, boundary value problem.*

where the source functions are given by

<sup>6</sup> <sup>þ</sup> *<sup>t</sup>* <sup>5</sup> <sup>þ</sup> *<sup>t</sup>* <sup>4</sup> � *<sup>t</sup>*

<sup>7</sup> � *<sup>t</sup>* <sup>6</sup> <sup>þ</sup> *<sup>t</sup>* <sup>5</sup> <sup>þ</sup> *<sup>t</sup>* <sup>4</sup> <sup>þ</sup> *<sup>t</sup>* <sup>3</sup> � *<sup>t</sup>*

<sup>6</sup> <sup>þ</sup> <sup>30</sup>*<sup>t</sup>*

<sup>6</sup> <sup>þ</sup> <sup>35</sup>*<sup>t</sup>*

<sup>4</sup> <sup>þ</sup> <sup>16</sup>*<sup>t</sup>*

<sup>5</sup> � <sup>27</sup>*<sup>t</sup>*

We approximate the solution at the considered method by taking scale level *M* ¼ 5*:* One can see that numerical plot and exact solution plot coincide very well as shown in **Figure 5**. Similarly the absolute error has been plotted at the given scale *M* ¼ 5 in **Figure 6**, which is very low. The lowest value of absolute error ∥*zapp* � *zex*∥ and ∥*yapp* � *yex*∥ indicates efficiency of the proposed method. The table shows the

*<sup>t</sup> z t*ðÞþ 2*z t*ð Þ� 2*y t*ð Þ� *f t*ðÞ¼ 0,

*<sup>t</sup> y t*ðÞ� 3*y t*ðÞþ 2*z t*ð Þ� *g t*ðÞ¼ 0,

<sup>3</sup> <sup>þ</sup> *<sup>t</sup>* <sup>þ</sup> 1,

<sup>3</sup> <sup>þ</sup> <sup>12</sup>*<sup>t</sup>*

<sup>4</sup> � <sup>19</sup>*<sup>t</sup>*

<sup>2</sup> � *<sup>t</sup>* <sup>þ</sup> <sup>1</sup>*:*

<sup>2</sup> � <sup>2</sup>*<sup>t</sup>* <sup>þ</sup> <sup>2</sup>

<sup>2</sup> <sup>þ</sup> <sup>9</sup>*<sup>t</sup>* � <sup>4</sup>*:*

<sup>3</sup> <sup>þ</sup> <sup>20</sup>*<sup>t</sup>*

*z*ð Þ¼ 0 4 *z*ð Þ¼� 1 4, *y*ð Þ¼ 0 2, *y*ð Þ¼� 1 2*:*

*Absolute error at different values of t for Example 4.*

comparison of errors for exact and approximate solutions for fixed scale level *M* ¼ 5 and order *γ* ¼ 1*:*9*:* Further the absolute error has been recorded at different values of space variable in **Table 2** which provides the information about efficiency of the proposed method.

#### **5. Conclusion**

We have successfully used the class of orthogonal polynomials of Laguerre polynomials to establish a numerical method to compute the numerical solution of FODEs and their coupled systems under some initial and boundary conditions. By using these polynomials, we have obtained some operational matrices corresponding to fractional order derivatives and integration. Also we have computed a new matrix corresponding to boundary conditions for boundary value problems of FODEs. Using the aforementioned matrices, we have converted the considered problem of FODEs to Sylvester-type algebraic equations. To obtain the numerical solution, we easily solved the desired algebraic equations by taking help from MATLAB®. Corresponding to the established procedure, we have provided numbers of examples to demonstrate our results. Also some error analyses have been provided along with graphical representations. By increasing the scale level, the accuracy is increased and vice versa. On the other hand, when the fractional order is approaching to integer value, the solutions tend to the exact solutions of the considered FODE. Therefore in each example, we have compared the exact and approximate solution and found that both the solutions were in closure contact with each other. Hence the established method can be very helpful in solving many classes and systems of FODEs under both initial and boundary conditions. In future the shifted Laguerre polynomials can be used to compute numerical solutions of partial differential equations of fractional order.

#### **Author contribution**

All authors equally contributed this paper and approved the final version.

#### **Competing interests**

We declare that no competing interests exist regarding this manuscript.

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#### **Author details**

Adnan Khan1 , Kamal Shah1,2\* and Danfeng Luo3

1 Department of Mathematics, University of Malakand, Dir(L), Khyber Pakhtunkhwa, Pakistan

2 Department of Mathematics and General Sciences, Prince Sultan University, Riyadh, Saudi Arabia

3 Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education), School of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan, P.R. China

\*Address all correspondence to: kamalshah408@gmail.com

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Numerical Solutions to Some Families of Fractional Order Differential Equations by Laguerre… DOI: http://dx.doi.org/10.5772/intechopen.90754*

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**Author details**

Riyadh, Saudi Arabia

Khyber Pakhtunkhwa, Pakistan

Changsha, Hunan, P.R. China

provided the original work is properly cited.

, Kamal Shah1,2\* and Danfeng Luo3

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

1 Department of Mathematics, University of Malakand, Dir(L),

\*Address all correspondence to: kamalshah408@gmail.com

2 Department of Mathematics and General Sciences, Prince Sultan University,

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

3 Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education), School of Mathematics and Statistics, Hunan Normal University,

Adnan Khan1

**40**

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[21] Ray SS, Bera RK. Solution of an extraordinary differential equation by adomian decomposition method. Journal of Applied Mathematics. 2004; **4**:331338

[22] Hashim I, Abdulaziz O, Momani S. Homotopy analysis method for fractional IVPs. Communications in Nonlinear Science and Numerical Simulation. 2009;**14**:674-684

[23] Bengochea G. Operational solution of fractional differential equations. Applied Mathematics Letters. 2014;**32**: 48-52

[24] Khalil H, Khan RA. The use of Jacobi polynomials in the numerical solution of coupled system of fractional differential equations. International Journal of Computer Mathematics. 2015;**92**(7): 1452-1472

[25] Doha EH, Bhrawy AH, Ezz-Eldien SS. Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equations. Applied Mathematical Modelling. 2011; **35**:5662-5672

[26] Esmaeili S, Shamsi M, Luchko Y. Numerical solution of fractional differential equations with a collocation method based on Muntz polynomials. Computers & Mathematics with Applications. 2011; **62**:918-929

[27] Odibat Z, Momani S, Erturk VS. Generalized differential transform method an application to differential equations of fractional order. Applied Mathematics and Computation. 2008; **197**:467-477

[28] Baleanu D, Mustafa OG, Agarwal RP. An existence result for a superlinear fractional differential equation. Applied Mathematics Letters. 2010;**23**:1129-1132

International Journal of Computer Mathematics. 2014;**91**(12):2554-2567

[37] Khalil H, Khan RA. A new method based on Legendre polynomials for solutions of the fractional two-

*DOI: http://dx.doi.org/10.5772/intechopen.90754*

*Numerical Solutions to Some Families of Fractional Order Differential Equations by Laguerre…*

dimensional heat conduction equation. Computers & Mathematics with Applications. 2014;**67**:1938-1953

pseudospectral approximation. Journal

[38] Guo BY, Wang LL. Modified Laguerre pseudospectral method refined

by multidomain Legendre

of Computational and Applied Mathematics. 2006;**190**:304-324

[40] Bhrawy AH, Taha TM,

**42**(1):490-500

**43**

[39] Gulsu M, Gurbuz B, Ozturk Y, Sezer M. Laguerre polynomial approach for solving linear delay difference equations. Applied Mathematics and Computation. 2011;**217**:6765-6776

Machado JAT. A review of operational matrices and spectral techniques for fractional calculus. Nonlinear Dynamics. 2015;**81**(3):1023-1052

[41] Diethelm K, Ford NJ. Numerical solution of the Bagley Torvik equation. BIT Numerical Mathematics. 2002;

[42] Akyuz-Dascioglu A, Isler N.

nonlinear differential equations. Mathematical and Computational Applications. 2013;**18**:293-300

Bernstein collocation method for solving

[43] Shah K. Using a numerical method by omitting discretization of data to study numerical solutions for boundary value problems of fractional order differential equations. Mathematical Methods in the Applied Sciences. 2019; **42**:6944-6959. DOI: 10.1002/mma.5800

[29] Baleanu D, Mustafa OG, Agarwal RP. On the solution set for a class of sequential fractional differential equations. Journal of Physics A. 2010; **43**:385-209

[30] Doha EH, Abd-Elhameed WM. Efficient solutions of multidimensional sixth-order boundary F value problems using symmetric generalized Jacobi-Galerkin method. Abstract and Applied Analysis. 2012;**2012**:12

[31] Bhrawy AH, Al-Shomrani MM. A Jacobi dual-Petrov Galerkin-Jacobi collocation method for solving Korteweg-de Vries equations. Abstract and Applied Analysis. 2012;**2012**:14

[32] Singh AK, Singh VK, Singh VK. The Bernstein operational matrix of integration. Applied Mathematical Sciences. 2009;**3**:2427-2436

[33] Bhrawy AH, Alofi AS, Ezz-Eldien SS. A quadrature tau method for fractional differential equations with variable coefficients. Applied Mathematics Letters. 2011;**24**:2146-2152

[34] Bhrawy AH, Mohammed MA. A shifted Legendre spectral method for fractional-order multi-point boundary value problems. Advances in Difference Equations. 2012;**2012**:8

[35] Khalil H, Khan RA. New operational matrix of integration and coupled system of Fredholm integral equations. Chinese Journal of Mathematics. 2014; **16**:12

[36] Khan RA, Khalil H. A new method based on Legendre polynomials for solution of system of fractional order partial differential equations.

*Numerical Solutions to Some Families of Fractional Order Differential Equations by Laguerre… DOI: http://dx.doi.org/10.5772/intechopen.90754*

International Journal of Computer Mathematics. 2014;**91**(12):2554-2567

[20] Yang S, Xiao A, Su H. Convergence of the variational iteration method for

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

[28] Baleanu D, Mustafa OG,

[29] Baleanu D, Mustafa OG,

Analysis. 2012;**2012**:12

2010;**23**:1129-1132

**43**:385-209

Agarwal RP. An existence result for a superlinear fractional differential equation. Applied Mathematics Letters.

Agarwal RP. On the solution set for a class of sequential fractional differential equations. Journal of Physics A. 2010;

[30] Doha EH, Abd-Elhameed WM. Efficient solutions of multidimensional sixth-order boundary F value problems using symmetric generalized Jacobi-Galerkin method. Abstract and Applied

[31] Bhrawy AH, Al-Shomrani MM. A Jacobi dual-Petrov Galerkin-Jacobi collocation method for solving

Korteweg-de Vries equations. Abstract and Applied Analysis. 2012;**2012**:14

[32] Singh AK, Singh VK, Singh VK. The

[33] Bhrawy AH, Alofi AS, Ezz-Eldien SS. A quadrature tau method for fractional differential equations with

Mathematics Letters. 2011;**24**:2146-2152

[35] Khalil H, Khan RA. New operational matrix of integration and coupled system of Fredholm integral equations. Chinese Journal of Mathematics. 2014;

[36] Khan RA, Khalil H. A new method based on Legendre polynomials for solution of system of fractional order

partial differential equations.

[34] Bhrawy AH, Mohammed MA. A shifted Legendre spectral method for fractional-order multi-point boundary value problems. Advances in Difference

Bernstein operational matrix of integration. Applied Mathematical Sciences. 2009;**3**:2427-2436

variable coefficients. Applied

Equations. 2012;**2012**:8

**16**:12

solving multi-order fractional differential equations. Computers & Mathematics with Applications. 2010;

[21] Ray SS, Bera RK. Solution of an extraordinary differential equation by adomian decomposition method. Journal of Applied Mathematics. 2004;

[22] Hashim I, Abdulaziz O, Momani S.

[23] Bengochea G. Operational solution of fractional differential equations. Applied Mathematics Letters. 2014;**32**:

[24] Khalil H, Khan RA. The use of Jacobi polynomials in the numerical solution of coupled system of fractional differential equations. International Journal of Computer Mathematics. 2015;**92**(7):

[25] Doha EH, Bhrawy AH, Ezz-Eldien SS. Efficient Chebyshev spectral methods for solving multi-term

fractional orders differential equations. Applied Mathematical Modelling. 2011;

[26] Esmaeili S, Shamsi M, Luchko Y. Numerical solution of fractional differential equations with a

collocation method based on Muntz

Mathematics with Applications. 2011;

[27] Odibat Z, Momani S, Erturk VS. Generalized differential transform method an application to differential equations of fractional order. Applied Mathematics and Computation. 2008;

polynomials. Computers &

Homotopy analysis method for fractional IVPs. Communications in Nonlinear Science and Numerical Simulation. 2009;**14**:674-684

**60**:2871-2879

**4**:331338

48-52

1452-1472

**35**:5662-5672

**62**:918-929

**197**:467-477

**42**

[37] Khalil H, Khan RA. A new method based on Legendre polynomials for solutions of the fractional twodimensional heat conduction equation. Computers & Mathematics with Applications. 2014;**67**:1938-1953

[38] Guo BY, Wang LL. Modified Laguerre pseudospectral method refined by multidomain Legendre pseudospectral approximation. Journal of Computational and Applied Mathematics. 2006;**190**:304-324

[39] Gulsu M, Gurbuz B, Ozturk Y, Sezer M. Laguerre polynomial approach for solving linear delay difference equations. Applied Mathematics and Computation. 2011;**217**:6765-6776

[40] Bhrawy AH, Taha TM, Machado JAT. A review of operational matrices and spectral techniques for fractional calculus. Nonlinear Dynamics. 2015;**81**(3):1023-1052

[41] Diethelm K, Ford NJ. Numerical solution of the Bagley Torvik equation. BIT Numerical Mathematics. 2002; **42**(1):490-500

[42] Akyuz-Dascioglu A, Isler N. Bernstein collocation method for solving nonlinear differential equations. Mathematical and Computational Applications. 2013;**18**:293-300

[43] Shah K. Using a numerical method by omitting discretization of data to study numerical solutions for boundary value problems of fractional order differential equations. Mathematical Methods in the Applied Sciences. 2019; **42**:6944-6959. DOI: 10.1002/mma.5800

**Chapter 3**

**Abstract**

Equations

*and Umar Audu Omesa*

analysis of the proposed methods.

nonlinear equations, Jacobian matrix

system of nonlinear equations:

The above system of equations (1) can be written as

**1. Introduction**

**45**

A Shamanskii-Like Accelerated

*Ibrahim Mohammed Sulaiman, Mustafa Mamat*

Scheme for Nonlinear Systems of

Newton-type methods with diagonal update to the Jacobian matrix are regarded as one most efficient and low memory scheme for system of nonlinear equations. One of the main advantages of these methods is solving nonlinear system of equations having singular Fréchet derivative at the root. In this chapter, we present a Jacobian approximation to the Shamanskii method, to obtain a convergent and accelerated scheme for systems of nonlinear equations. Precisely, we will focus on the efficiency of our proposed method and compare the performance with other existing methods. Numerical examples illustrate the efficiency and the theoretical

**Keywords:** Newton method, Shamanskii method, diagonal updating scheme,

A large aspect of scientific and management problems is often formulated by obtaining the values of *x* of which the function evaluation of that variable is equal to zero [1]. The above description can be represented mathematically by the following

*f* <sup>1</sup>ð Þ¼ *x*1*; x*2*;* …*; xn* 0

⋮ ⋮⋮ ¼ ⋮ *f <sup>n</sup>*ð Þ¼ *x*1*; x*2*;* …*; xn* 0

where *<sup>x</sup>*1*, x*2*,* …*, xn* <sup>∈</sup> <sup>R</sup>*<sup>n</sup>* are vectors and *fi* is nonlinear functions for *<sup>i</sup>* <sup>¼</sup> <sup>1</sup>*,* <sup>2</sup>*,* …*, n*.

where *<sup>F</sup>* : *Rn* ! *<sup>R</sup><sup>n</sup>* is continuously differentiable in an open neighborhood of the solution *x*<sup>∗</sup> . These systems are seen as natural description of observed phenomenon of numerous real-life problems whose solutions are seen as an important goal in

*f* <sup>2</sup>ð Þ¼ *x*1*; x*2*;* …*; xn* 0 (1)

*F x*ð Þ¼ 0 (2)

#### **Chapter 3**

## A Shamanskii-Like Accelerated Scheme for Nonlinear Systems of Equations

*Ibrahim Mohammed Sulaiman, Mustafa Mamat and Umar Audu Omesa*

#### **Abstract**

Newton-type methods with diagonal update to the Jacobian matrix are regarded as one most efficient and low memory scheme for system of nonlinear equations. One of the main advantages of these methods is solving nonlinear system of equations having singular Fréchet derivative at the root. In this chapter, we present a Jacobian approximation to the Shamanskii method, to obtain a convergent and accelerated scheme for systems of nonlinear equations. Precisely, we will focus on the efficiency of our proposed method and compare the performance with other existing methods. Numerical examples illustrate the efficiency and the theoretical analysis of the proposed methods.

**Keywords:** Newton method, Shamanskii method, diagonal updating scheme, nonlinear equations, Jacobian matrix

#### **1. Introduction**

A large aspect of scientific and management problems is often formulated by obtaining the values of *x* of which the function evaluation of that variable is equal to zero [1]. The above description can be represented mathematically by the following system of nonlinear equations:

$$\begin{aligned} \boldsymbol{f}\_1(\mathbf{x}\_1, \mathbf{x}\_2, \dots, \mathbf{x}\_n) &= \mathbf{0} \\ \boldsymbol{f}\_2(\mathbf{x}\_1, \mathbf{x}\_2, \dots, \mathbf{x}\_n) &= \mathbf{0} \\ \vdots \quad \vdots \quad \vdots &= \quad \vdots \\ \boldsymbol{f}\_n(\mathbf{x}\_1, \mathbf{x}\_2, \dots, \mathbf{x}\_n) &= \mathbf{0} \end{aligned} \tag{1}$$

where *<sup>x</sup>*1*, x*2*,* …*, xn* <sup>∈</sup> <sup>R</sup>*<sup>n</sup>* are vectors and *fi* is nonlinear functions for *<sup>i</sup>* <sup>¼</sup> <sup>1</sup>*,* <sup>2</sup>*,* …*, n*. The above system of equations (1) can be written as

$$F(\mathbf{x}) = \mathbf{0} \tag{2}$$

where *<sup>F</sup>* : *Rn* ! *<sup>R</sup><sup>n</sup>* is continuously differentiable in an open neighborhood of the solution *x*<sup>∗</sup> . These systems are seen as natural description of observed phenomenon of numerous real-life problems whose solutions are seen as an important goal in

mathematical study. Recently, this area has been studied extensively [2, 3]. The most powerful techniques for handling nonlinear systems of equations are to linearize the equations and proceed to iterate on the linearized set of equations until an accurate solution is obtained [4]. This can be achieved by obtaining the derivative or gradient of the equations. Various scholars stress that the derivatives should be obtained analytically rather than using numerical approach. However, this is usually not always convenient and, in most cases, not even possible as equations may be generated simply by a computer algorithm [2]. For one variable problem, system of nonlinear equation defined in (2) represents a function *F* : *R* ! *R* where *f* is continuous in the interval *f* ∈½ � *a; b* .

**Definition 4:** Suppose *<sup>F</sup>* : *<sup>R</sup><sup>n</sup>* ! *<sup>R</sup><sup>n</sup>* is continuously differentiable at the point *x*∈*R<sup>n</sup>* and each component function *f* <sup>1</sup>*, f* <sup>2</sup>*,* …*, f <sup>m</sup>* is also continuously differentiable

> *∂f* 1 *∂x*<sup>2</sup> *∂f* 2 *∂x*<sup>2</sup> ⋮ *∂f n ∂x*<sup>2</sup>

Most of the algorithms employ for obtaining the solution of Eq. (1) centered on approximating the Jacobian matrix which often provides a linear map *T x*ð Þ : *<sup>R</sup><sup>n</sup>* ! *<sup>R</sup><sup>n</sup>*

Also, if *F* is differentiable at point *x*<sup>∗</sup> , then the affine function *A x*ð Þ¼ *f x*ð Þþ<sup>∗</sup>

*F x*ð Þ� *F x*ð Þ� <sup>∗</sup> *J* k k <sup>∗</sup> ð Þ *x* � *x*<sup>∗</sup> k k *x* � *x*<sup>∗</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ *<sup>x</sup>*<sup>1</sup> � *<sup>x</sup>*<sup>∗</sup> <sup>2</sup> <sup>þ</sup> ð Þ *<sup>x</sup>*<sup>2</sup> � *<sup>x</sup>*<sup>∗</sup> <sup>2</sup> <sup>þ</sup> … <sup>þ</sup> ð Þ *xn* � *<sup>x</sup>*<sup>∗</sup> <sup>2</sup>

If all the given component functions *f* <sup>1</sup>*, f* <sup>2</sup>*,* …*, f <sup>m</sup>* of *Jx*ð Þ *F* are continuous, then we

The most famous method for solving nonlinear systems of equations *F x*ð Þ¼ 0 is the Newton method which generates a sequence f g *xk* from any given initial point *x*<sup>0</sup>

ð Þ *xk* �<sup>1</sup>

ð Þ *xk* is the Jacobian for *F x*ð Þ*<sup>k</sup>* . The above sequence Eq. (5) is said to converge quadratically to the solution *x*<sup>∗</sup> if *x*<sup>0</sup> is sufficiently near the solution point

ð Þ *<sup>x</sup>*<sup>0</sup> �<sup>1</sup>

outstanding among other numerical methods. However, Jacobian evaluation and

out the iteration process for finite dimensional problem as presented in Eq. (6),

The rate of convergence is linear and improves as the initial point gets better. Suppose *<sup>x</sup>*<sup>0</sup> is sufficiently chosen near solution point *<sup>x</sup>*<sup>∗</sup> and *F x*<sup>∗</sup> ð Þ is nonsingular;

*xk*þ<sup>1</sup> ¼ *xk* � *F*<sup>0</sup>

time-consuming [9]. This led to the study of different variants of Newton methods for systems of nonlinear equations. One of the simplest and low-cost variants of the Newton method that almost entirely evades derivate evaluation at every iteration is

ð Þ *xk* is nonsingular [7, 8]. This convergent rate makes the method

ð Þ *xn* �<sup>1</sup>

*xn*þ<sup>1</sup> � *<sup>x</sup>*<sup>∗</sup> k k<sup>≤</sup> *Kc <sup>x</sup>*<sup>0</sup> � *<sup>x</sup>*<sup>∗</sup> k k *xn* � *<sup>x</sup>*<sup>∗</sup> k k (7)

*J* <sup>∗</sup> ð Þ *x* � *x*<sup>∗</sup> is a good approximation to *F x*ð Þ near *x* ¼ *x*<sup>∗</sup> in such a way that

*xk*þ<sup>1</sup> ¼ *xk* � *F*<sup>0</sup>

the chord method. This scheme computes the Jacobian matrix *F*<sup>0</sup>

… *<sup>∂</sup><sup>f</sup>* <sup>1</sup> *∂xm*

1

CCCCCCCCCCCA

<sup>⋯</sup> *<sup>∂</sup><sup>f</sup>* <sup>2</sup> *∂xm* ⋱ ⋮

<sup>⋯</sup> *<sup>∂</sup><sup>f</sup> <sup>n</sup> ∂xm*

*T x*ð Þ¼ *JxF x*ð Þ <sup>∀</sup> *<sup>x</sup>*<sup>∈</sup> *<sup>R</sup><sup>n</sup>* (3)

(4)

.

*F x*ð Þ*<sup>k</sup>* (5)

*F x*ð Þ*<sup>n</sup>* are expensive and

*F x*ð Þ*<sup>k</sup>* (6)

ð Þ *x*<sup>0</sup> once through-

*∂f* 1 *∂x*<sup>1</sup> *∂f* 2 *∂x*<sup>1</sup> ⋮ *∂f n ∂x*<sup>1</sup>

0

*A Shamanskii-Like Accelerated Scheme for Nonlinear Systems of Equations*

BBBBBBBBBBB@

at *x*; then the derivative of *F x* is defined as

*DOI: http://dx.doi.org/10.5772/intechopen.87246*

defined by Eq. (3)

where k k *x* � *x*<sup>∗</sup> ¼

via the following:

where *F*<sup>0</sup>

and the Jacobian *F*<sup>0</sup>

then, for some *Kc* >0, we have

**47**

say the function *F* is differentiable.

*Jx*ð Þ¼ *F*

lim*<sup>x</sup>*!*x*<sup>∗</sup>

q

solving the linear system for the step *s x*ð Þ¼� *<sup>n</sup> F*<sup>0</sup>

**Definition 1:** Consider a system of equations *f* <sup>1</sup>*, f* <sup>2</sup>*,* …*, f <sup>n</sup>*; the solution of this system in one variable, two variables, and *n* variable is referred to as a point ð Þ *<sup>a</sup>*1*; <sup>a</sup>*2*;* …*; an* <sup>∈</sup>*R<sup>n</sup>* such that *<sup>f</sup>* <sup>1</sup>ð Þ¼ *<sup>a</sup>*1*; <sup>a</sup>*2*;* …*; an <sup>f</sup>* <sup>2</sup>ð Þ¼ *<sup>a</sup>*1*; <sup>a</sup>*2*;* …*; an* … <sup>¼</sup> *f <sup>n</sup>*ð Þ¼ *a*1*; a*2*;* …*; an* 0.

In general, the problem to be considered is that for some function *f x*ð Þ, one wishes to evaluate the derivative at some points *x*, i.e.,

$$\text{Given}\\
f(x)\text{, Evaluate;}\\
\text{div}\\
\text{v} = \frac{df}{d\mathbf{x}}$$

This often used to represent an instantaneous change of the function at some given points [5].

**Definition 2:** For a function *f x*ð Þ that is smooth, then there exists, at any point *x*, a vector of first-order partial derivative or gradient vector:

$$\nabla f(\mathbf{x}) = \begin{bmatrix} \frac{\partial f}{\partial \mathbf{x}\_1} \\ \frac{\partial f}{\partial \mathbf{x}\_2} \\ \cdot \\ \cdot \\ \cdot \\ \cdot \\ \cdot \\ \cdot \\ \frac{\partial f}{\partial \mathbf{x}\_n} \end{bmatrix} = \mathbf{g}(\mathbf{x}).$$

The Taylor's series expansion of the function *f x*ð Þ about point *x*<sup>0</sup> is an ideal starting point for this discussion [1].

**Definition 3:** Let *f* be a differentiable function; the Taylor's *f x*ð Þ around a point *a* is the infinite sum:

$$f(\mathbf{x}) = f(a) + f'(a)(\mathbf{x} - a) + \frac{f''(a)}{2}(\mathbf{x} - a)^2 + \frac{f^{\prime\prime}(a)}{3!}(\mathbf{x} - a)^3 + \dots$$

However, continuous differentiable vector valued function does not satisfy the mean value theorem (MVT), an essential tool in calculus [6]. Hence, academics suggested the use of the following theorem to replace the mean valued theorem.

**Theorem 1:** Let *<sup>F</sup>* : <sup>R</sup>*<sup>n</sup>* ! <sup>R</sup>*<sup>m</sup>* be continuously differentiable in an open convex set *<sup>D</sup>* <sup>⊂</sup> <sup>R</sup>*<sup>n</sup>*. For any *x, x* <sup>þ</sup> *<sup>s</sup>*<sup>∈</sup> *<sup>D</sup>*

$$F(\varkappa + s) - F(\varkappa) = \int\_0^1 f(\varkappa + t\varkappa)s dt \equiv \int\_{\varkappa}^{\varkappa + s} F'(z) dz$$

*A Shamanskii-Like Accelerated Scheme for Nonlinear Systems of Equations DOI: http://dx.doi.org/10.5772/intechopen.87246*

**Definition 4:** Suppose *<sup>F</sup>* : *<sup>R</sup><sup>n</sup>* ! *<sup>R</sup><sup>n</sup>* is continuously differentiable at the point *x*∈*R<sup>n</sup>* and each component function *f* <sup>1</sup>*, f* <sup>2</sup>*,* …*, f <sup>m</sup>* is also continuously differentiable at *x*; then the derivative of *F x* is defined as

$$f\_{\mathbf{x}}(F) = \begin{pmatrix} \frac{\partial f\_1}{\partial \mathbf{x}\_1} & \frac{\partial f\_1}{\partial \mathbf{x}\_2} & \cdots & \frac{\partial f\_1}{\partial \mathbf{x}\_m} \\ \frac{\partial f\_2}{\partial \mathbf{x}\_1} & \frac{\partial f\_2}{\partial \mathbf{x}\_2} & \cdots & \frac{\partial f\_2}{\partial \mathbf{x}\_m} \\ \vdots & \vdots & \ddots & \vdots \\ \frac{\partial f\_n}{\partial \mathbf{x}\_1} & \frac{\partial f\_n}{\partial \mathbf{x}\_2} & \cdots & \frac{\partial f\_n}{\partial \mathbf{x}\_m} \end{pmatrix}.$$

Most of the algorithms employ for obtaining the solution of Eq. (1) centered on approximating the Jacobian matrix which often provides a linear map *T x*ð Þ : *<sup>R</sup><sup>n</sup>* ! *<sup>R</sup><sup>n</sup>* defined by Eq. (3)

$$T(\mathfrak{x}) = f\_{\mathfrak{x}} F(\mathfrak{x}) \,\forall \mathfrak{x} \in \mathbb{R}^n \tag{3}$$

Also, if *F* is differentiable at point *x*<sup>∗</sup> , then the affine function *A x*ð Þ¼ *f x*ð Þþ<sup>∗</sup> *J* <sup>∗</sup> ð Þ *x* � *x*<sup>∗</sup> is a good approximation to *F x*ð Þ near *x* ¼ *x*<sup>∗</sup> in such a way that

$$\lim\_{\mathbf{x}\to\mathbf{x}\_{\*}}\frac{||F(\mathbf{x}) - F(\mathbf{x}\_{\*}) - J\_{\*}(\mathbf{x} - \mathbf{x}\_{\*})||}{||\mathbf{x} - \mathbf{x}\_{\*}||}\tag{4}$$

where k k *x* � *x*<sup>∗</sup> ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ *<sup>x</sup>*<sup>1</sup> � *<sup>x</sup>*<sup>∗</sup> <sup>2</sup> <sup>þ</sup> ð Þ *<sup>x</sup>*<sup>2</sup> � *<sup>x</sup>*<sup>∗</sup> <sup>2</sup> <sup>þ</sup> … <sup>þ</sup> ð Þ *xn* � *<sup>x</sup>*<sup>∗</sup> <sup>2</sup> q .

If all the given component functions *f* <sup>1</sup>*, f* <sup>2</sup>*,* …*, f <sup>m</sup>* of *Jx*ð Þ *F* are continuous, then we say the function *F* is differentiable.

The most famous method for solving nonlinear systems of equations *F x*ð Þ¼ 0 is the Newton method which generates a sequence f g *xk* from any given initial point *x*<sup>0</sup> via the following:

$$\mathbf{x}\_{k+1} = \mathbf{x}\_k - F'(\mathbf{x}\_k)^{-1} F(\mathbf{x}\_k) \tag{5}$$

where *F*<sup>0</sup> ð Þ *xk* is the Jacobian for *F x*ð Þ*<sup>k</sup>* . The above sequence Eq. (5) is said to converge quadratically to the solution *x*<sup>∗</sup> if *x*<sup>0</sup> is sufficiently near the solution point and the Jacobian *F*<sup>0</sup> ð Þ *xk* is nonsingular [7, 8]. This convergent rate makes the method outstanding among other numerical methods. However, Jacobian evaluation and solving the linear system for the step *s x*ð Þ¼� *<sup>n</sup> F*<sup>0</sup> ð Þ *xn* �<sup>1</sup> *F x*ð Þ*<sup>n</sup>* are expensive and time-consuming [9]. This led to the study of different variants of Newton methods for systems of nonlinear equations. One of the simplest and low-cost variants of the Newton method that almost entirely evades derivate evaluation at every iteration is the chord method. This scheme computes the Jacobian matrix *F*<sup>0</sup> ð Þ *x*<sup>0</sup> once throughout the iteration process for finite dimensional problem as presented in Eq. (6),

$$\varkappa\_{k+1} = \varkappa\_k - F'(\varkappa\_0)^{-1} F(\varkappa\_k) \tag{6}$$

The rate of convergence is linear and improves as the initial point gets better. Suppose *<sup>x</sup>*<sup>0</sup> is sufficiently chosen near solution point *<sup>x</sup>*<sup>∗</sup> and *F x*<sup>∗</sup> ð Þ is nonsingular; then, for some *Kc* >0, we have

$$\left| \left| \mathfrak{x}\_{n+1} - \mathfrak{x}^\* \right| \right| \le K\_{\mathfrak{c}} \left| \left| \mathfrak{x}\_0 - \mathfrak{x}^\* \right| \right| \left| \left| \mathfrak{x}\_n - \mathfrak{x}^\* \right| \right| \tag{7}$$

mathematical study. Recently, this area has been studied extensively [2, 3]. The most powerful techniques for handling nonlinear systems of equations are to linearize the equations and proceed to iterate on the linearized set of equations until an accurate solution is obtained [4]. This can be achieved by obtaining the derivative or gradient of the equations. Various scholars stress that the derivatives should be obtained analytically rather than using numerical approach. However, this is usually not always convenient and, in most cases, not even possible as equations may be generated simply by a computer algorithm [2]. For one variable problem, system of nonlinear equation defined in (2) represents a function *F* : *R* ! *R* where *f* is

**Definition 1:** Consider a system of equations *f* <sup>1</sup>*, f* <sup>2</sup>*,* …*, f <sup>n</sup>*; the solution of this system in one variable, two variables, and *n* variable is referred to as a point

In general, the problem to be considered is that for some function *f x*ð Þ, one

Given *f x*ð Þ*,* Evaluate*;* deriv <sup>¼</sup> *df*

This often used to represent an instantaneous change of the function at some

**Definition 2:** For a function *f x*ð Þ that is smooth, then there exists, at any point *x*,

*∂f ∂x*<sup>1</sup> *∂f ∂x*<sup>2</sup> *: :*

¼ *g x*ð Þ*:*

2 þ *f* 000 ð Þ *a*

> ð*<sup>x</sup>*þ*<sup>s</sup> x F*0 ð Þ*z dz*

<sup>3</sup>! ð Þ *<sup>x</sup>* � *<sup>a</sup>* <sup>3</sup> <sup>þ</sup> …

*: ∂f ∂xn*

The Taylor's series expansion of the function *f x*ð Þ about point *x*<sup>0</sup> is an ideal

00 ð Þ *a* <sup>2</sup> ð Þ *<sup>x</sup>* � *<sup>a</sup>*

ð1 0

**Definition 3:** Let *f* be a differentiable function; the Taylor's *f x*ð Þ around a point

However, continuous differentiable vector valued function does not satisfy the mean value theorem (MVT), an essential tool in calculus [6]. Hence, academics suggested the use of the following theorem to replace the mean valued theorem. **Theorem 1:** Let *<sup>F</sup>* : <sup>R</sup>*<sup>n</sup>* ! <sup>R</sup>*<sup>m</sup>* be continuously differentiable in an open convex

*J x*ð Þ þ *ts sdt* �

*dx*

ð Þ *<sup>a</sup>*1*; <sup>a</sup>*2*;* …*; an* <sup>∈</sup>*R<sup>n</sup>* such that *<sup>f</sup>* <sup>1</sup>ð Þ¼ *<sup>a</sup>*1*; <sup>a</sup>*2*;* …*; an <sup>f</sup>* <sup>2</sup>ð Þ¼ *<sup>a</sup>*1*; <sup>a</sup>*2*;* …*; an* … <sup>¼</sup>

wishes to evaluate the derivative at some points *x*, i.e.,

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

a vector of first-order partial derivative or gradient vector:

∇*f x*ð Þ¼

continuous in the interval *f* ∈½ � *a; b* .

starting point for this discussion [1].

*f x*ð Þ¼ *f a*ð Þþ *f*

set *<sup>D</sup>* <sup>⊂</sup> <sup>R</sup>*<sup>n</sup>*. For any *x, x* <sup>þ</sup> *<sup>s</sup>*<sup>∈</sup> *<sup>D</sup>*

0

*F x*ð Þ� þ *s F x*ð Þ¼

ð Þ *<sup>a</sup>* ð Þþ *<sup>x</sup>* � *<sup>a</sup> <sup>f</sup>*

*a* is the infinite sum:

**46**

*f <sup>n</sup>*ð Þ¼ *a*1*; a*2*;* …*; an* 0.

given points [5].

The convergence theorems and proof of Eq. (7) can be referred to [9, 10]. Motivated by the excellent convergence of Newton method and low cost of Jacobian evaluation of chord method, a method due originally to Shamanskii [11, 12] that lies between Newton method and chord method was proposed and has been analyzed in Kelly [9, 13–15]. Other variants of Newton methods with different Jacobian approximation schemes include [9, 14, 16–18]. However, most of these methods require the computation and storage of the full or approximate Jacobian, which become very difficult and time-consuming as the dimension of systems increases [10, 19].

superior performance compared to Newton method in terms of efficiency whenever work is measured in terms of function evaluations [9]. Also, if the value of *t* is sufficiently chosen, then, as the dimension increases, the performance of the Shamanskii method improves and thus reduces the limit of complexity of factoring the approximate Jacobian for two pseudo-Newton iterations [14]. Please refer to

**Theorem 2** [15]: Let *<sup>F</sup>* : *<sup>D</sup>* <sup>⊂</sup> *Rn* ! *<sup>R</sup><sup>n</sup>* conform hypotheses H1 2ð Þ, H2, and H3. Then the solution point *x*<sup>∗</sup> is a point of attraction of the Shamanskii iteration, i.e.,

Evaluation or inversion of the Jacobian matrix at every iteration or after few iterations does not seem relevant even though the computational cost has generally been reduced as in Shamanskii method [14, 25–28]. As a matter of fact, it can be easily shown that by adding a diagonal updating scheme to a method, we would have a new low memory iterative approach which would approximate the Jacobian

ð Þ *xk* into nonsingular diagonal matrix that can be updated in every iteration [29–31]. Indeed, using the Shamanskii procedure, the proposed method avoids the main complexity of the Newton-type methods by reusing the evaluated Jacobian during the iteration process. This is the basic idea of the Shamanskii-like method

Given an initial approximation *x*0, we compute Eq. (2) to obtain the Jacobian

ð Þ *xk*þ<sup>1</sup> ≈ *Dk*þ<sup>1</sup> (12)

*Dk*þ<sup>1</sup>*sk* ≈*yk* (13)

ð Þ *xk sk* ≈*yk* (14)

*<sup>k</sup> yk* (15)

*<sup>F</sup>* (16)

<sup>Ω</sup>*<sup>k</sup>* (17)

ð Þ *xk* and present a diagonal approximation to the Jacobian say *Dk* as follows:

Suppose *sk* ¼ *xk*þ<sup>1</sup> � *xk* and *yk* ¼ *F x*ð Þ� *<sup>k</sup>*þ<sup>1</sup> *F x*ð Þ*<sup>k</sup>* ; by mean value theorem

*F*0

*F*0

*s T*

have the following solution also regarded as the optimal solution:

*<sup>Δ</sup><sup>k</sup>* <sup>¼</sup> *<sup>s</sup> T <sup>k</sup> yk* � *s T <sup>k</sup> Dk*þ<sup>1</sup>*sk*

some norms. The updated formula for *Dk* follows after the theorem below: **Theorem 3:** Suppose *Dk*þ<sup>1</sup> is the update of the diagonal matrix *Dk* and

> min <sup>1</sup> 2 k k *Δ<sup>k</sup>* 2

Since *Dk*þ<sup>1</sup> is the update of diagonal matrix *Dk*, let us assume *Dk*þ<sup>1</sup> satisfy the

*T*

*<sup>k</sup> Dk*þ<sup>1</sup>*sk* ¼ *s*

which would be used to minimize the deviation between *Dk*þ<sup>1</sup> and *Dk* under

such that Eq. (15) holds and k k*: <sup>F</sup>* denotes the Frobenius norm. From Eq. (16), we

*tr* Ω<sup>2</sup> *k*

Substituting Eq. (12) in Eq. (13), we have

*Δ<sup>k</sup>* ¼ *Dk*þ<sup>1</sup> � *Dk*, *sk* 6¼ 0. Consider the problem

[15] for the proof of the convergence theorem described below.

*A Shamanskii-Like Accelerated Scheme for Nonlinear Systems of Equations*

*DOI: http://dx.doi.org/10.5772/intechopen.87246*

Eq. (10), and this method possesses at least cubic order of convergence.

**3. Diagonal updating scheme for solving nonlinear systems**

*F*0

*F*0

**49**

(MVT), we have

weak secant equation:

which is described as follows.

It would be worthwhile to construct a derivative-free approach and analyze with existing techniques [20–22]. The aim of this work is to derive a diagonal matrix for the approximate Jacobian of Shamanskii method by means of variational techniques. The expectation would be to reduce computational cost, storage, and CPU time of evaluating any problem. The proposed method works efficiently by combining the good convergence rate of Shamanskii method and the derivate free approach employed, and the results are very encouraging. The next section presents the Shamanskii method for nonlinear systems of equations.

#### **2. Shamanskii method**

It is known that the Newton method defined in Eq. (2) converges quadratically to *x*<sup>∗</sup> when the initial guess is sufficiently close to the root [7, 10, 19]. The major concern about this method is the evaluation and storage of the Jacobian matrix at every iteration [23]. A scheme that almost completely overcomes this is the chord method. This method factored the Jacobian matrix only once in the case of a finite dimensional problem, thereby reducing the evaluation cost of each iteration as in Eq. (3) and thereby degrading the convergence rate to linear [10].

Motivated by this, a method due originally to Shamanskii [11] was developed and analyzed by [7, 13, 14, 16, 24]. Starting with an initial approximation *x*0, this method uses the multiple pseudo-Newton approach as described below:

$$\boldsymbol{\infty}\_{k+\frac{1}{2}} = \boldsymbol{\infty}\_{k} - F'(\boldsymbol{\infty}\_{k})^{-1} F(\boldsymbol{\infty}\_{k}) \tag{8}$$

$$\varkappa\_{k+1} = \varkappa\_{k+\frac{1}{2}} - F'(\varkappa\_k)^{-1} F\left(\varkappa\_{k+\frac{1}{2}}\right) \tag{9}$$

after little simplification, we have

$$\boldsymbol{\omega}\_{k+1} = \boldsymbol{\omega}\_k - F'(\boldsymbol{\omega}\_k)^{-1} \left[ \boldsymbol{F}(\boldsymbol{\omega}\_k) + \boldsymbol{F} \left( \boldsymbol{\omega}\_k - \boldsymbol{F}'(\boldsymbol{\omega}\_k)^{-1} \boldsymbol{F}(\boldsymbol{\omega}\_k) \right) \right] \tag{10}$$

This method converges superlinearly with *q*-order of at least *t* þ 1 when the initial approximation *<sup>x</sup>*<sup>0</sup> is sufficiently chosen near the solution point *<sup>x</sup>*<sup>∗</sup> and *<sup>F</sup>*<sup>0</sup> *<sup>x</sup>*<sup>∗</sup> ð Þ is nonsingular. This implies that there exists *Ks* >0, such that

$$||\mathfrak{x}\_{n+1} - \mathfrak{x}^\*|| \le K\_s ||\mathfrak{x}\_n - \mathfrak{x}^\*||^{t+1} \tag{11}$$

Combining Eq. (7) and the quadratic convergence of Newton method produces the convergence rate of the Shamanskii method as in Eq. (8). Thus, the balance is between the reduced evaluation cost of Fréchet derivative and Jacobian computation for Shamanskii method and Newton method rapid convergence. This low-cost derivative evaluation as well as the rapid convergence rate of several methods including the Shamanskii method has been studied and analyzed in [13, 15]. From the analysis, the researchers conclude that that Shamanskii method has shown

*A Shamanskii-Like Accelerated Scheme for Nonlinear Systems of Equations DOI: http://dx.doi.org/10.5772/intechopen.87246*

superior performance compared to Newton method in terms of efficiency whenever work is measured in terms of function evaluations [9]. Also, if the value of *t* is sufficiently chosen, then, as the dimension increases, the performance of the Shamanskii method improves and thus reduces the limit of complexity of factoring the approximate Jacobian for two pseudo-Newton iterations [14]. Please refer to [15] for the proof of the convergence theorem described below.

**Theorem 2** [15]: Let *<sup>F</sup>* : *<sup>D</sup>* <sup>⊂</sup> *Rn* ! *<sup>R</sup><sup>n</sup>* conform hypotheses H1 2ð Þ, H2, and H3. Then the solution point *x*<sup>∗</sup> is a point of attraction of the Shamanskii iteration, i.e., Eq. (10), and this method possesses at least cubic order of convergence.

#### **3. Diagonal updating scheme for solving nonlinear systems**

Evaluation or inversion of the Jacobian matrix at every iteration or after few iterations does not seem relevant even though the computational cost has generally been reduced as in Shamanskii method [14, 25–28]. As a matter of fact, it can be easily shown that by adding a diagonal updating scheme to a method, we would have a new low memory iterative approach which would approximate the Jacobian *F*0 ð Þ *xk* into nonsingular diagonal matrix that can be updated in every iteration [29–31]. Indeed, using the Shamanskii procedure, the proposed method avoids the main complexity of the Newton-type methods by reusing the evaluated Jacobian during the iteration process. This is the basic idea of the Shamanskii-like method which is described as follows.

Given an initial approximation *x*0, we compute Eq. (2) to obtain the Jacobian *F*0 ð Þ *xk* and present a diagonal approximation to the Jacobian say *Dk* as follows:

$$F'(\mathbf{x}\_{k+1}) \approx D\_{k+1} \tag{12}$$

Suppose *sk* ¼ *xk*þ<sup>1</sup> � *xk* and *yk* ¼ *F x*ð Þ� *<sup>k</sup>*þ<sup>1</sup> *F x*ð Þ*<sup>k</sup>* ; by mean value theorem (MVT), we have

$$D\_{k+1} \varsigma\_k \approx \jmath\_k \tag{13}$$

Substituting Eq. (12) in Eq. (13), we have

$$F'(\mathcal{X}\_k)\mathbf{s}\_k \approx \mathcal{y}\_k \tag{14}$$

Since *Dk*þ<sup>1</sup> is the update of diagonal matrix *Dk*, let us assume *Dk*þ<sup>1</sup> satisfy the weak secant equation:

$$s\_k^T D\_{k+1} s\_k = s\_k^T y\_k \tag{15}$$

which would be used to minimize the deviation between *Dk*þ<sup>1</sup> and *Dk* under some norms. The updated formula for *Dk* follows after the theorem below:

**Theorem 3:** Suppose *Dk*þ<sup>1</sup> is the update of the diagonal matrix *Dk* and *Δ<sup>k</sup>* ¼ *Dk*þ<sup>1</sup> � *Dk*, *sk* 6¼ 0. Consider the problem

$$\min \frac{1}{2} \|\Delta\_k\|\_F^2 \tag{16}$$

such that Eq. (15) holds and k k*: <sup>F</sup>* denotes the Frobenius norm. From Eq. (16), we have the following solution also regarded as the optimal solution:

$$
\Delta\_k = \frac{s\_k^T \mathcal{Y}\_k - s\_k^T D\_{k+1} s\_k}{tr\left(\Omega\_k^2\right)} \Omega\_k \tag{17}
$$

The convergence theorems and proof of Eq. (7) can be referred to [9, 10]. Motivated by the excellent convergence of Newton method and low cost of Jacobian evaluation of chord method, a method due originally to Shamanskii [11, 12] that lies between Newton method and chord method was proposed and has been analyzed in Kelly [9, 13–15]. Other variants of Newton methods with different Jacobian approximation schemes include [9, 14, 16–18]. However, most of these methods require the computation and storage of the full or approximate Jacobian, which become very difficult and time-consuming as the dimension of systems increases [10, 19].

It would be worthwhile to construct a derivative-free approach and analyze with existing techniques [20–22]. The aim of this work is to derive a diagonal matrix for the approximate Jacobian of Shamanskii method by means of variational techniques. The expectation would be to reduce computational cost, storage, and CPU time of evaluating any problem. The proposed method works efficiently by combining the good convergence rate of Shamanskii method and the derivate free approach employed, and the results are very encouraging. The next section presents

It is known that the Newton method defined in Eq. (2) converges quadratically to *x*<sup>∗</sup> when the initial guess is sufficiently close to the root [7, 10, 19]. The major concern about this method is the evaluation and storage of the Jacobian matrix at every iteration [23]. A scheme that almost completely overcomes this is the chord method. This method factored the Jacobian matrix only once in the case of a finite dimensional problem, thereby reducing the evaluation cost of each iteration as in

Motivated by this, a method due originally to Shamanskii [11] was developed and analyzed by [7, 13, 14, 16, 24]. Starting with an initial approximation *x*0, this

ð Þ *xk* �<sup>1</sup>

ð Þ *xk* �<sup>1</sup>

*F xk*þ<sup>1</sup> 2 � �

h i � �

*xn*þ<sup>1</sup> � *<sup>x</sup>*<sup>∗</sup> k k<sup>≤</sup> *Ks xn* � *<sup>x</sup>*<sup>∗</sup> k k*<sup>t</sup>*þ<sup>1</sup> (11)

ð Þ *xk* �<sup>1</sup>

*F x*ð Þ*<sup>k</sup>* (8)

*F x*ð Þ*<sup>k</sup>*

(9)

(10)

the Shamanskii method for nonlinear systems of equations.

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

Eq. (3) and thereby degrading the convergence rate to linear [10].

*xk*þ<sup>1</sup>

is nonsingular. This implies that there exists *Ks* >0, such that

after little simplification, we have

**48**

*xk*þ<sup>1</sup> ¼ *xk* � *F*<sup>0</sup>

*xk*þ<sup>1</sup> ¼ *xk*þ<sup>1</sup>

method uses the multiple pseudo-Newton approach as described below:

<sup>2</sup> ¼ *xk* � *F*<sup>0</sup>

<sup>2</sup> � *F*<sup>0</sup>

ð Þ *xk* �<sup>1</sup> *F x*ð Þþ *<sup>k</sup> F xk* � *<sup>F</sup>*<sup>0</sup>

This method converges superlinearly with *q*-order of at least *t* þ 1 when the initial approximation *<sup>x</sup>*<sup>0</sup> is sufficiently chosen near the solution point *<sup>x</sup>*<sup>∗</sup> and *<sup>F</sup>*<sup>0</sup> *<sup>x</sup>*<sup>∗</sup> ð Þ

Combining Eq. (7) and the quadratic convergence of Newton method produces the convergence rate of the Shamanskii method as in Eq. (8). Thus, the balance is between the reduced evaluation cost of Fréchet derivative and Jacobian computation for Shamanskii method and Newton method rapid convergence. This low-cost derivative evaluation as well as the rapid convergence rate of several methods including the Shamanskii method has been studied and analyzed in [13, 15]. From the analysis, the researchers conclude that that Shamanskii method has shown

**2. Shamanskii method**

$$\begin{aligned} \text{where } \Omega\_k = \text{diag}\left(\left(\mathbf{s}\_k^{(1)}\right)^2, \left(\mathbf{s}\_k^{(2)}\right)^2, \dots, \left(\mathbf{s}\_k^{(n)}\right)^2\right), \sum\_{i=1}^n \left(\mathbf{s}\_k^{(i)}\right)^4 = tr\left(\Omega\_k^2\right), \text{ and } Tr \text{ is} \\ \text{a two-connection.} \end{aligned}$$

the trace operation.

**Proof:** It is known that the objective function and the constraint of Eq. (16) are convex; thus, we intend to use its Lagrangian function to obtain the unique solution as follows:

$$\mathcal{L}(\Delta\_k, \mu) = \frac{1}{2} ||\Delta\_k||\_F^2 + \mu \left( s\_k^T \Delta\_k s\_k - s\_k^T \mathcal{y}\_k - s\_k^T D\_{k+1} s\_k \right) \tag{18}$$

where *μ* is the corresponding Lagrangian multiplier. Simplifying Eq. (18), we have

$$\mu = \frac{s\_k^T y\_k - s\_k^T D\_{k+1} s\_k}{\sum\_{i=1}^n \left(s\_k^{(i)}\right)^4} \tag{19}$$

and

$$\Delta\_k^{(i)} = \frac{s\_k^T y\_k - s\_k^T D\_{k+1} s\_k}{\sum\_{i=1}^n \left(s\_k^{(i)}\right)^4} \left(s\_k^{(i)}\right)^2 \quad \forall \, i = 1, 2, \dots, n \tag{20}$$

Also, for diagonal matrix *Dk*, the element of the diagonal component is given as *D*ð Þ*<sup>i</sup> <sup>k</sup>* , and the *i th* component of the vector *sk* is *s* ð Þ*i <sup>k</sup>* . Then Ω*<sup>k</sup>* ¼ diag *s* ð Þ1 *k* � �<sup>2</sup> *; ; s* ð Þ2 *k* � �<sup>2</sup> *;* …*; ; s* ð Þ *n k* � �<sup>2</sup> � �*,* and∑*<sup>n</sup> <sup>i</sup>*¼<sup>1</sup> *<sup>s</sup>* ð Þ*i k* � �<sup>4</sup> <sup>¼</sup> *tr* <sup>Ω</sup><sup>2</sup> *k* � �. To complete the proof, we rewrite Eq. (20) as follows:

$$
\Delta\_k = \frac{\left(s\_k^T \mathcal{Y}\_k - s\_k^T D\_{k+1} \mathfrak{s}\_k\right)}{tr\left(\Omega\_k^2\right)} \Omega\_k. \tag{21}
$$

This completes the proof. ∎

Now, from the above description of the theorem, we deduce that the best possible diagonal update D*<sup>k</sup>*þ<sup>1</sup> is as follows:

$$\mathbf{D}\_{k+1} = \mathbf{D}\_k + \frac{\left(\mathbf{s}\_k^T \mathbf{y}\_k - \mathbf{s}\_k^T \mathbf{D}\_{k+1} \mathbf{s}\_k\right)}{tr\left(\boldsymbol{\Omega}\_k^2\right)} \boldsymbol{\Omega}\_k \tag{22}$$

carried out using a designed computer code for its algorithm. The problems could be artificial or real-life problems. The artificial problems check the performance of any algorithm in situation such as point of singularity, function with many solutions,

While the real-life problems emerge from fields such as chemistry, engineering,

This section demonstrates the proposed method and illustrates its advantages on some benchmark problems with dimensions ranging from 25 to 1,000 variables.

management, etc., the real-life problems often contain large data or complex

and null space effect as presented in **Figures 1–3** [7, 32].

algebraic expression which makes it difficult to solve.

**4. Numerical results**

*Essentially unimodal function.*

*Functions with significant null space.*

**Figure 3.**

**51**

**Figure 2.**

**Figure 1.**

*Functions with a huge number of significant local optima.*

*DOI: http://dx.doi.org/10.5772/intechopen.87246*

*A Shamanskii-Like Accelerated Scheme for Nonlinear Systems of Equations*

However, for possibly small k k *sk* and *trΩk*, we need to define a condition that would be applied for such cases. To this end, we require that k k *sk* ≥*s*<sup>1</sup> for some chosen small *s*<sup>1</sup> > 0. Otherwise, we set the updated diagonal D*<sup>k</sup>*þ<sup>1</sup> ¼ D*<sup>k</sup>* where D*<sup>k</sup>*þ<sup>1</sup> is defined as

$$\mathbf{D}\_{k+1} = \begin{cases} \mathbf{D}\_k + \frac{\left(s\_k^T \mathbf{y}\_k - s\_k^T D\_{k+1} \mathbf{s}\_k\right)}{tr\left(\mathfrak{Q}\_k^2\right)} \mathbf{\Omega}\_k; ||\mathbf{s}\_k|| \ge c\_1 \\\\ \mathbf{D}\_k; \text{Otherwise} \end{cases} \tag{23}$$

Thus, the proposed accelerated method is described as follows:

$$\mathbf{x}\_{k+1} = \mathbf{x}\_k - D\_k^{-1} \left[ F(\mathbf{x}\_k) + F(\mathbf{x}\_k - D\_k^{-1} F(\mathbf{x}\_k)) \right] \tag{24}$$

The performance of this proposed method would be tested on well-known benchmark problems employed by researchers on existing methods. This would be *A Shamanskii-Like Accelerated Scheme for Nonlinear Systems of Equations DOI: http://dx.doi.org/10.5772/intechopen.87246*

**Figure 1.** *Functions with a huge number of significant local optima.*

where Ω*<sup>k</sup>* ¼ diag *s*

the trace operation.

as follows:

and

given as *D*ð Þ*<sup>i</sup>*

ð Þ1 *k* � �<sup>2</sup>

diag *s*

**50**

ð Þ1 *k* � �<sup>2</sup>

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

Lð Þ¼ *Δk; μ*

*Δ*ð Þ*<sup>i</sup> <sup>k</sup>* <sup>¼</sup> *<sup>s</sup> T <sup>k</sup> yk* � *s T <sup>k</sup> Dk*þ<sup>1</sup>*sk*

*;* …*; ; s* ð Þ *n k*

*<sup>k</sup>* , and the *i*

� �<sup>2</sup> � �

proof, we rewrite Eq. (20) as follows:

possible diagonal update D*<sup>k</sup>*þ<sup>1</sup> is as follows:

*; ; s* ð Þ2 *k* � �<sup>2</sup> *; ; s* ð Þ2 *k* � �<sup>2</sup>

1 2 k k *Δ<sup>k</sup>* 2 *<sup>F</sup>* þ *μ s T <sup>k</sup> Δksk* � *s*

∑*<sup>n</sup> <sup>i</sup>*¼<sup>1</sup> *<sup>s</sup>* ð Þ*i k* � �<sup>4</sup> *<sup>s</sup>*

*<sup>Δ</sup><sup>k</sup>* <sup>¼</sup> *<sup>s</sup> T <sup>k</sup> yk* � *s T <sup>k</sup> Dk*þ<sup>1</sup>*sk* � � *tr* Ω<sup>2</sup> *k*

D*<sup>k</sup>*þ<sup>1</sup> ¼ D*<sup>k</sup>* þ

<sup>D</sup>*<sup>k</sup>*þ<sup>1</sup> <sup>¼</sup> <sup>D</sup>*<sup>k</sup>* <sup>þ</sup>

8 ><

>:

*xk*þ<sup>1</sup> <sup>¼</sup> *xk* � *<sup>D</sup>*�<sup>1</sup>

� �<sup>2</sup> � �

*<sup>μ</sup>* <sup>¼</sup> *<sup>s</sup> T <sup>k</sup> yk* � *s T <sup>k</sup> Dk*þ<sup>1</sup>*sk*

*;* …*; ; s* ð Þ *n k*

**Proof:** It is known that the objective function and the constraint of Eq. (16) are convex; thus, we intend to use its Lagrangian function to obtain the unique solution

where *μ* is the corresponding Lagrangian multiplier. Simplifying Eq. (18), we have

ð Þ*i k* � �<sup>2</sup>

*<sup>i</sup>*¼<sup>1</sup> *<sup>s</sup>* ð Þ*i k* � �<sup>4</sup>

This completes the proof. ∎ Now, from the above description of the theorem, we deduce that the best

However, for possibly small k k *sk* and *trΩk*, we need to define a condition that would be applied for such cases. To this end, we require that k k *sk* ≥*s*<sup>1</sup> for some chosen small *s*<sup>1</sup> > 0. Otherwise, we set the updated diagonal D*<sup>k</sup>*þ<sup>1</sup> ¼ D*<sup>k</sup>* where D*<sup>k</sup>*þ<sup>1</sup> is defined as

*<sup>k</sup> F x*ð Þþ *<sup>k</sup> F xk* � *<sup>D</sup>*�<sup>1</sup>

The performance of this proposed method would be tested on well-known benchmark problems employed by researchers on existing methods. This would be

� � <sup>Ω</sup>*k; s* k k*<sup>k</sup>* <sup>≥</sup>*ϵ*<sup>1</sup>

*<sup>k</sup> F x*ð Þ*<sup>k</sup>* � � � � (24)

*s T <sup>k</sup> yk* � *s T <sup>k</sup> Dk*þ<sup>1</sup>*sk* � � *tr* Ω<sup>2</sup> *k*

*s T <sup>k</sup> yk* � *s T <sup>k</sup> Dk*þ<sup>1</sup>*sk* � � *tr* Ω<sup>2</sup> *k*

*Dk; Otherwise*

Thus, the proposed accelerated method is described as follows:

∑*<sup>n</sup> <sup>i</sup>*¼<sup>1</sup> *<sup>s</sup>* ð Þ*i k*

Also, for diagonal matrix *Dk*, the element of the diagonal component is

*th* component of the vector *sk* is *s*

*,* and∑*<sup>n</sup>*

, ∑*<sup>n</sup> <sup>i</sup>*¼<sup>1</sup> *<sup>s</sup>* ð Þ*i k* � �<sup>4</sup>

*T <sup>k</sup> yk* � *s T <sup>k</sup> Dk*þ<sup>1</sup>*sk* � � (18)

<sup>¼</sup> *tr* <sup>Ω</sup><sup>2</sup> *k* � �, and *Tr* is

� �<sup>4</sup> (19)

*<sup>k</sup>* . Then Ω*<sup>k</sup>* ¼

� � <sup>Ω</sup>*k:* (21)

� � <sup>Ω</sup>*<sup>k</sup>* (22)

(23)

ð Þ*i*

<sup>¼</sup> *tr* <sup>Ω</sup><sup>2</sup> *k*

∀ *i* ¼ 1*,* 2*, ::, n* (20)

� �. To complete the

**Figure 2.** *Functions with significant null space.*

**Figure 3.** *Essentially unimodal function.*

carried out using a designed computer code for its algorithm. The problems could be artificial or real-life problems. The artificial problems check the performance of any algorithm in situation such as point of singularity, function with many solutions, and null space effect as presented in **Figures 1–3** [7, 32].

While the real-life problems emerge from fields such as chemistry, engineering, management, etc., the real-life problems often contain large data or complex algebraic expression which makes it difficult to solve.

#### **4. Numerical results**

This section demonstrates the proposed method and illustrates its advantages on some benchmark problems with dimensions ranging from 25 to 1,000 variables.

These include problems with restrictions such as singular Jacobian or problems with only one point of singularity. To evaluate the performance of the proposed diagonal updating Shamanskii method (DUSM), we employ some tools by Dolan and Moré [33] and compare the performance with two classical Newton-type methods based on the number of iterations and CPU time in seconds. The methods include:

At the point of failure due to any of the above conditions as in the tabulated results, it is assumed the number of iteration and CPU time is zero and thus that point has been denoted by " ∗ ." The following are the details of the standard test problems, the initial points used, and the exact solutions for systems of nonlinear equations.

<sup>þ</sup> *xi*ð<sup>1</sup> <sup>þ</sup> *xixn*�<sup>2</sup>*xn*�<sup>1</sup>*xn*Þ � <sup>2</sup>

*<sup>i</sup>* � cosð Þ *xi* � 1

*xi* � <sup>1</sup>

*n*

**NI CPU NI CPU NI CPU**

*i* ¼ 1*,* 2*,* 3*,* …*, n, x*<sup>0</sup> ¼ ð Þ 0*:*3*;* 0*;* 3*;* …*;* 0*:*3

*i* ¼ 1*,* 2*,* 3*,* …*, n, x*<sup>0</sup> ¼ ð Þ 0*:*2*;* 0*:*2*;* …*;* 0*:*2

*Fn*ð Þ¼ *<sup>x</sup> xn* � <sup>0</sup>*:*1*x*<sup>2</sup>

*i* ¼ 1*,* 2*,* 3*,* …*, n, x*<sup>0</sup> ¼ ð Þ 0*:*05*;* 0*:*05*;* …*;* 0*:*05

 25 13 0.016102 8 0.034777 13 0.015999 25 6 0.009522 7 0.028231 7 0.010412 25 \* \* 4 0.023766 \* \* 25 16 0.019679 17 0.077072 22 0.022889 25 4 0.006605 16 0.061750 4 0.005761 50 13 0.032998 8 0.090310 13 0.032271 50 10 0.022134 7 0.089785 7 0.017036 50 4 0.010350 4 0.052899 4 0.010238 50 30 0.054640 17 0.228077 23 0.041569 50 4 0.012361 16 0.201262 4 0.010735 100 13 0.073565 8 0.339333 13 0.066363 100 10 0.054075 7 0.292001 7 0.044512 100 \* \* 4 0.175300 \* \* 100 15 0.075073 18 0.770165 25 0.118170 100 4 0.029221 17 0.755556 4 0.023154 1000 13 1.868606 8 27.171776 13 2.042222 1000 10 1.444533 7 24.295632 7 1.045329 1000 \* \* 4 27.1250 \* \* 1000 52 6.757533 19 63.981376 39 5.138997 1000 4 0.610145 18 62.364143 4 0.612590

*Fi*ð Þ¼ *x e*

**Problem Dim NM DUSM SM**

**Problem 1** [31]: System of *n* nonlinear equations

*DOI: http://dx.doi.org/10.5772/intechopen.87246*

*Fi*ð Þ¼ *<sup>x</sup>* <sup>1</sup> � *<sup>x</sup>*<sup>2</sup>

**Problem 2** [34]: Systems of *n* nonlinear equations

**Problem 3** [31]: Structured exponential function

**Table 1.**

**53**

*Numerical comparison of NM, DUSM, and SM.*

*i*

*A Shamanskii-Like Accelerated Scheme for Nonlinear Systems of Equations*

*Fi*ð Þ¼ *<sup>x</sup> <sup>x</sup>*<sup>2</sup>


These tools are used to represent the efficiency, robustness, and numerical comparisons of different algorithms. Suppose there exist *ns* solvers and *np* problems; for each problem *p* and solver *s,*they define:

*tp,s* ¼ computing time needed to solve a problem by solver the number of iteration or CPU time ð Þ

Requiring a baseline for comparisons, they compared the performance on problem *p* by solver *s* with the best performance by any solver for this problem using the performance ratio:

$$r\_{p,s} = \frac{t\_{p,s}}{\min\{t\_{p,s} : s \in \mathcal{S}\}}$$

We suppose that parameter *rm* ≥*rp,s* for all *p, s* is chosen and *rp,s* ¼ *rM* if and only if solver *s* does not solve problem *p*. The performance of solvers *s* on any given problem might be of interest, but because we would prefer obtaining the overall assessment of the performance of the solver, then it was defined as

$$p\_s(t) = \frac{1}{n\_p} size\{p \in P : r\_{p,s} \le t\}.$$

Thus, *ps* ð Þ*t* was the probability for solver *s*∈ *S* that a performance ratio *rp,s* was within a factor *t*∈*R* of the best possible ratio. Then, function *ps* was the cumulative distribution function for the performance ratio. The performance profile *ps* : *R* ! ½ � 0*;* 1 for a solver was nondecreasing, piecewise, and continuous from right. The value of *ps* ð Þ1 is the probability that the solver will win over the rest of the solvers. In general, a solver with high value of *p*ð Þ*τ* or at the top right of the figure is preferable or represents the best solver.

All problems considered in this research are solved using MATLAB (R 2015a) subroutine programming [37]. This was run on an Intel® Core™ i5-2410M CPU @ 2.30 GHz processor, 4GB for RAM memory and Windows 7 Professional operating system. The termination condition is set as

$$||s\_k|| + ||F(\mathbf{x}\_k)|| \le 10^{-6}$$

and the program has been designed to terminate whenever:


*A Shamanskii-Like Accelerated Scheme for Nonlinear Systems of Equations DOI: http://dx.doi.org/10.5772/intechopen.87246*

At the point of failure due to any of the above conditions as in the tabulated results, it is assumed the number of iteration and CPU time is zero and thus that point has been denoted by " ∗ ." The following are the details of the standard test problems, the initial points used, and the exact solutions for systems of nonlinear equations.

**Problem 1** [31]: System of *n* nonlinear equations

These include problems with restrictions such as singular Jacobian or problems with only one point of singularity. To evaluate the performance of the proposed diagonal updating Shamanskii method (DUSM), we employ some tools by Dolan and Moré [33] and compare the performance with two classical Newton-type methods based on the number of iterations and CPU time in seconds. The methods include:

These tools are used to represent the efficiency, robustness, and numerical comparisons of different algorithms. Suppose there exist *ns* solvers and *np* problems;

*tp,s* ¼ computing time needed to solve a problem by solver the number of iteration or CPU time ð Þ

*rp,s* <sup>¼</sup> *tp,s*

if solver *s* does not solve problem *p*. The performance of solvers *s* on any given problem might be of interest, but because we would prefer obtaining the overall

assessment of the performance of the solver, then it was defined as

1 *np*

distribution function for the performance ratio. The performance profile

*ps* ðÞ¼ *t*

preferable or represents the best solver.

system. The termination condition is set as

• The CPU time in seconds reaches 500.

• Insufficient memory to initiate the run.

termination condition.

Requiring a baseline for comparisons, they compared the performance on problem *p* by solver *s* with the best performance by any solver for this problem using the

min *tp,s* : *s*∈ *S*

*size p* ∈*P* : *rp,s* ≤ *t :*

ð Þ*t* was the probability for solver *s*∈ *S* that a performance ratio *rp,s* was

ð Þ1 is the probability that the solver will win over the rest of the

within a factor *t*∈*R* of the best possible ratio. Then, function *ps* was the cumulative

*ps* : *R* ! ½ � 0*;* 1 for a solver was nondecreasing, piecewise, and continuous from right.

solvers. In general, a solver with high value of *p*ð Þ*τ* or at the top right of the figure is

All problems considered in this research are solved using MATLAB (R 2015a) subroutine programming [37]. This was run on an Intel® Core™ i5-2410M CPU @ 2.30 GHz processor, 4GB for RAM memory and Windows 7 Professional operating

k k *sk* <sup>þ</sup> k k *F x*ð Þ*<sup>k</sup>* <sup>≤</sup>10�<sup>6</sup>

• The number of iterations exceeds 500, and no point of *xk* satisfies the

and the program has been designed to terminate whenever:

We suppose that parameter *rm* ≥*rp,s* for all *p, s* is chosen and *rp,s* ¼ *rM* if and only

1. The Newton method (NM)

performance ratio:

Thus, *ps*

The value of *ps*

**52**

2. The Shamanskii method (SM)

for each problem *p* and solver *s,*they define:

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

$$F\_i(\mathbf{x}) = \left(\mathbf{1} - \mathbf{x}\_i^2\right) + \mathbf{x}\_i(\mathbf{1} + \mathbf{x}\_i \mathbf{x}\_{n-2} \mathbf{x}\_{n-1} \mathbf{x}\_n) - \mathbf{2}$$

$$i = \mathbf{1}, \mathbf{2}, \mathbf{3}, \dots, n, \qquad \mathbf{x}\_0 = (\mathbf{0}.\mathbf{3}, \mathbf{0}, \mathbf{3}, \dots, \mathbf{0}.\mathbf{3})$$

**Problem 2** [34]: Systems of *n* nonlinear equations

$$F\_i(\mathbf{x}) = \mathbf{x}\_i^2 - \cos\left(\mathbf{x}\_i - \mathbf{1}\right)$$
 
$$i = \mathbf{1}, \mathbf{2}, \mathbf{3}, \dots, n, \qquad \mathbf{x}\_0 = (\mathbf{0}.\mathbf{2}, \mathbf{0}.\mathbf{2}, \dots, \mathbf{0}.\mathbf{2})$$

**Problem 3** [31]: Structured exponential function

$$F\_i(\mathbf{x}) = e^{\mathbf{x}\_i} - \mathbf{1}$$

$$F\_n(\mathbf{x}) = \mathbf{x}\_n - \mathbf{0.1x}\_n^2$$

$$i = \mathbf{1}, \mathbf{2}, \mathbf{3}, \dots, n, \quad \mathbf{x}\_0 = (\mathbf{0.05}, \mathbf{0.05}, \dots, \mathbf{0.05})$$


#### **Table 1.**

*Numerical comparison of NM, DUSM, and SM.*

**Problem 4** [35]: Extended trigonometric of Byeong-Chun

$$F\_i(\mathbf{x}) = \cos\left(\mathbf{x}\_i^2 - \mathbf{1}\right) - \mathbf{1}$$

$$i = \mathbf{1}, \mathbf{2}, \mathbf{3}, \dots, n, \qquad \mathbf{x}\_0 = (\mathbf{0}.\mathbf{0}\mathbf{6}, \mathbf{0}.\mathbf{0}\mathbf{6}, \dots, \mathbf{0}.\mathbf{0}\mathbf{6})$$

**References**

LLC; 1998

Abidin; 2018

Springer; 1999

[1] Rainer K. Numerical Analysis. New York: Springer Science+Business Media,

*DOI: http://dx.doi.org/10.5772/intechopen.87246*

*A Shamanskii-Like Accelerated Scheme for Nonlinear Systems of Equations*

computers. Ukrains'kyi Matematychnyi

[13] Traub JF. Iterative Methods for the Solution of Equations. Englewood Cliffs:

Zhurnal. 1966;**18**(6):135-140

[14] Kchouk B, Dussault J. The Chebyshev–Shamanskii method for solving systems of nonlinear equations. Journal of Optimization Theory and Applications. 2013;**157**:148-167

[15] Ortega JM, Rheinboldt WC. Iterative Solution of Nonlinear Equations in Several Variables. New

for solving systems of nonlinear equations. Journal of Nucleic Acids.

DOI: 10.5829/idosi.wasj.2013.21.

solving nonlinear simultaneous equations. Mathematics of

[19] Chong EKP, Zak SH. An

and Sons; 2013

Computation. 1965;**19**(92):577-593

Introduction to Optimization, Wiley Series in Discrete Mathematics and Optimization. New York: John Wiley

[20] Jose LH, Eulalia M, Juan RM. Modified Newton's method for systems of nonlinear equations with singular Jacobian. Journal of Computational and Applied Mathematics. 2009;**224**:77-83

[21] Leong WJ, Hassan MA, Waziri MY. A matrix-free quasi-Newton method for solving large-scale nonlinear systems.

[16] Brent RP. Some efficient algorithms

[17] Waziri MY, Leong WJ, Mamat M, Moyi AU. Two-step derivative-free diagonally Newton's method for largescale nonlinear equations. World Applied Sciences Journal. 2013;**21**:86-94.

[18] Broyden CG. A class of methods for

York: Academic Press; 1970

1973;**10**(2):327-344

am.2045

Prentice-Hall; 1964

[2] Sulaiman IM. New iterative methods

[3] Wenyu S, Ya-Xiang Y. Optimization

[4] John RH. Numerical Methods for Nonlinear Engineering Models. Netherlands: Springer; 2009

[5] Burden RL, Faires JD. Numerical Analysis. 8th ed. USA: Thomson; 2005

[6] Wright SJ, Nocedal J. Numerical Optimization. 2nd ed. Berlin, Germany:

Method for Unconstrained

Houston, Texas: SIAM; 1996

Nonlinear Optimization. 2nd ed. Philadelphia: SIAM; 2009

[9] Kelley CT. A Shamanskii-like acceleration scheme for nonlinear equations at singular roots. Mathematics of Computation. 1986;**47**:609-623

[10] Kelley CT. Iterative Methods for Linear and Nonlinear Equations. Philadelphia, PA: SIAM; 1995

[11] Shamanskii VE. A modification of Newton's method. Ukrains'kyi

Matematychnyi Zhurnal. 1967;**19**:133-138

[12] Shamanskii VE. On a realization of

Newton's method on electronic

**55**

[7] Dennis JE Jr, Schnabel RB. Numerical

Optimization and Nonlinear Equations.

[8] Griva I, Nash SG, Sofer A. Linear and

for solving fuzzy and dual fuzzy nonlinear equations [PhD thesis]. Malaysia: Faculty of Informatics and Computing, Universiti Sultan Zainal

Theory and Methods, Springer Optimization and Its Applications. Boston, MA: Springer; 2006

**Problem 5** [36]: Extended spare system of Byeong

$$F\_i(\mathbf{x}) = \mathbf{x}\_i^2 - \mathbf{x}\_i - \mathbf{2}$$

$$i = \mathbf{1, 2, 3, ..., n}, \qquad \mathbf{x}\_0 = (\mathbf{1.1, 11.1, ..., 1.1)}$$

**Table 1** shows the number of iterations (NI) and CPU time for Newton method (NM), Shamanskii method (SM), and the proposed diagonal updating method (DUSM), respectively. The performance of these methods was analyzed via storage locations and execution time. It can be observed that the proposed DUSM was able to solve the test problems perfectly, while NM and SM fail at some points due to the matrix being singular to working precision. This shows that the diagonal scheme employed has provided an option in the case of singularity, thereby reducing the computational cost of the classical Newton-type methods.

#### **5. Conclusion**

This chapter proposes a diagonal updating formula for systems of nonlinear equations which attributes to reduction in Jacobian evaluation cost. By computational experiments, we reach the conclusion that the proposed scheme is reliable and efficient and reduces Jacobian computational cost during the iteration process. Meanwhile, the proposed scheme is superior compared to the result of the classical and existing numerical methods for solving systems of equations.

#### **Author details**

Ibrahim Mohammed Sulaiman\*, Mustafa Mamat and Umar Audu Omesa Universiti Sultan Zainal Abidin, Kuala Terengganu, Malaysia

\*Address all correspondence to: sulaimanib@unisza.edu.my

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*A Shamanskii-Like Accelerated Scheme for Nonlinear Systems of Equations DOI: http://dx.doi.org/10.5772/intechopen.87246*

#### **References**

**Problem 4** [35]: Extended trigonometric of Byeong-Chun

**Problem 5** [36]: Extended spare system of Byeong

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

computational cost of the classical Newton-type methods.

and existing numerical methods for solving systems of equations.

Ibrahim Mohammed Sulaiman\*, Mustafa Mamat and Umar Audu Omesa

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

Universiti Sultan Zainal Abidin, Kuala Terengganu, Malaysia

\*Address all correspondence to: sulaimanib@unisza.edu.my

provided the original work is properly cited.

**5. Conclusion**

**Author details**

**54**

*Fi*ð Þ¼ *<sup>x</sup>* cos *<sup>x</sup>*<sup>2</sup>

*Fi*ð Þ¼ *<sup>x</sup> <sup>x</sup>*<sup>2</sup>

*i* ¼ 1*,* 2*,* 3*,* …*, n, x*<sup>0</sup> ¼ ð Þ 0*:*06*;* 0*:*06*;* …*;* 0*:*06

*i* ¼ 1*,* 2*,* 3*,* …*, n, x*<sup>0</sup> ¼ ð Þ 1*:*1*;* 11*:*1*;* …*;* 1*:*1

(NM), Shamanskii method (SM), and the proposed diagonal updating method (DUSM), respectively. The performance of these methods was analyzed via storage locations and execution time. It can be observed that the proposed DUSM was able to solve the test problems perfectly, while NM and SM fail at some points due to the matrix being singular to working precision. This shows that the diagonal scheme employed has provided an option in the case of singularity, thereby reducing the

This chapter proposes a diagonal updating formula for systems of nonlinear equations which attributes to reduction in Jacobian evaluation cost. By computational experiments, we reach the conclusion that the proposed scheme is reliable and efficient and reduces Jacobian computational cost during the iteration process. Meanwhile, the proposed scheme is superior compared to the result of the classical

**Table 1** shows the number of iterations (NI) and CPU time for Newton method

*<sup>i</sup>* � <sup>1</sup> � <sup>1</sup>

*<sup>i</sup>* � *xi* � 2

[1] Rainer K. Numerical Analysis. New York: Springer Science+Business Media, LLC; 1998

[2] Sulaiman IM. New iterative methods for solving fuzzy and dual fuzzy nonlinear equations [PhD thesis]. Malaysia: Faculty of Informatics and Computing, Universiti Sultan Zainal Abidin; 2018

[3] Wenyu S, Ya-Xiang Y. Optimization Theory and Methods, Springer Optimization and Its Applications. Boston, MA: Springer; 2006

[4] John RH. Numerical Methods for Nonlinear Engineering Models. Netherlands: Springer; 2009

[5] Burden RL, Faires JD. Numerical Analysis. 8th ed. USA: Thomson; 2005

[6] Wright SJ, Nocedal J. Numerical Optimization. 2nd ed. Berlin, Germany: Springer; 1999

[7] Dennis JE Jr, Schnabel RB. Numerical Method for Unconstrained Optimization and Nonlinear Equations. Houston, Texas: SIAM; 1996

[8] Griva I, Nash SG, Sofer A. Linear and Nonlinear Optimization. 2nd ed. Philadelphia: SIAM; 2009

[9] Kelley CT. A Shamanskii-like acceleration scheme for nonlinear equations at singular roots. Mathematics of Computation. 1986;**47**:609-623

[10] Kelley CT. Iterative Methods for Linear and Nonlinear Equations. Philadelphia, PA: SIAM; 1995

[11] Shamanskii VE. A modification of Newton's method. Ukrains'kyi Matematychnyi Zhurnal. 1967;**19**:133-138

[12] Shamanskii VE. On a realization of Newton's method on electronic

computers. Ukrains'kyi Matematychnyi Zhurnal. 1966;**18**(6):135-140

[13] Traub JF. Iterative Methods for the Solution of Equations. Englewood Cliffs: Prentice-Hall; 1964

[14] Kchouk B, Dussault J. The Chebyshev–Shamanskii method for solving systems of nonlinear equations. Journal of Optimization Theory and Applications. 2013;**157**:148-167

[15] Ortega JM, Rheinboldt WC. Iterative Solution of Nonlinear Equations in Several Variables. New York: Academic Press; 1970

[16] Brent RP. Some efficient algorithms for solving systems of nonlinear equations. Journal of Nucleic Acids. 1973;**10**(2):327-344

[17] Waziri MY, Leong WJ, Mamat M, Moyi AU. Two-step derivative-free diagonally Newton's method for largescale nonlinear equations. World Applied Sciences Journal. 2013;**21**:86-94. DOI: 10.5829/idosi.wasj.2013.21. am.2045

[18] Broyden CG. A class of methods for solving nonlinear simultaneous equations. Mathematics of Computation. 1965;**19**(92):577-593

[19] Chong EKP, Zak SH. An Introduction to Optimization, Wiley Series in Discrete Mathematics and Optimization. New York: John Wiley and Sons; 2013

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Computational and Applied Mathematics. 2011;**625**:2354-2363

[22] Natasa K, Zorana L. Newton-like methods with modification of the righthand side vector. Mathematics of Computation. 2002;**71**:237-250

[23] Waziri MY, Leong WJ, Hassan MA, Monsi M. A new Newton's method with diagonal Jacobian approximation for systems of nonlinear equations. Journal of Mathematics and Statistics. 2010;**6** (3):246-252

[24] Lampariello F, Sciandrone M. Global convergence technique for the Newton method with periodic Hessian evaluation. Journal of Optimization Theory and Applications. 2001;**111**(2): 341-358

[25] Sulaiman IM, Mamat M, Nurnadiah Z, Puspa LG. Solving dual fuzzy nonlinear equations via Shamanskii method. International Journal of Engineering & Technology. 2018;**7** (3.28):89-91

[26] Ypma TJ. Historical development of the Newton-Raphson method. SIAM Review. 1995;**37**(4):531-551

[27] Hao L, Qin N. Incomplete Jacobian Newton method for nonlinear equation. Computers and Mathematics with Applications. 2008;**56**(1):218-227

[28] Chency E, Kincaid D. Numerical Mathematics and Computing. Asia: Nelson Education, Cengage Learning; 2012

[29] Sulaiman IM, Mamat M, Afendee MM, Waziri MY. Diagonal updating Shamanskii-like method for solving singular fuzzy nonlinear equation. Far East Journal of Mathematical Sciences. 2017;**103**(10):1619-1629

[30] Waziri MY, Leong WJ, Hassan MA, Monsi M. Jacobian-free Newton's method for systems of nonlinear

equations. Journal of Numerical Mathematics and Stochastics. 2010;**2**(1): 54-63

**Chapter 4**

**Abstract**

classical MLS.

**1. Introduction**

**57**

Modified Moving Least Squares

Linear and Nonlinear Systems of

*Massoumeh Poura'bd Rokn Saraei and Mashaallah Matinfar*

This work aims at focusing on modifying the moving least squares (MMLS) methods for solving two-dimensional linear and nonlinear systems of integral equations and system of differential equations. The modified shape function is our main aim, so for computing the shape function based on the moving least squares method (MLS), an efficient algorithm is presented. In this modification, additional terms is proposed to impose based on the coefficients of the polynomial base functions on the quadratic base functions (m = 2). So, the MMLS method is developed for solving the systems of two-dimensional linear and nonlinear integral equations at irregularly distributed nodes. This approach prevents the singular moment matrix in the context of MLS based on meshfree methods. Also, determining the best radius of the support domain of a field node is an open problem for MLS-based methods. Therefore, the next important thing is that the MMLS algorithm can automatically find the best neighborhood radius for each node. Then, numerical examples are presented that determine the main motivators for doing this so. These examples enable us to make comparisons of two methods: MMLS and

**Keywords:** moving least squares, modified moving least squares, systems of

In mathematics, there are many functional equations of the description of a real system in the natural sciences (such as physics, biology, Earth science, meteorology) and disciplines of engineering. For instance, we can point to some mathematical model from physics that describe heat as a partial differential equation and the inverse problem of it's as integro-differential equations. Also, another example in nature is Laplace's equation which corresponds to the construction of potential for a vector field whose effect is known at the boundary of Domain alone. Especially, the integral equations have wide applicability which has been cited in [1–4].

integral equations, algorithm of shape function, numerical solutions

**MSC 2010:** 45G15, 45F05,45F35, 65D15

Method for Two-Dimensional

Integral Equations

[31] Waziri MY, Abdulmajid Z. An improved diagonal Jacobian approximation via a quasi-Cauchy condition for solving large-scale systems of nonlinear equations. Journal of Applied Mathematics. 2013;**3**:1-6

[32] Andrei N. An unconstrained optimization test functions collection. Advanced Modeling and Optimization. 2008;**10**:147-161

[33] Dolan E, Moré JJ. Benchmarking optimization software with performance profiles. Mathematical Programming. 2002;**91**(2):201-213

[34] Hafiz MA, Muhammad SMB. An efficient two-step iterative method for solving system of nonlinear equation. Journal of Mathematics Research. 2012; **4**(4):28-34

[35] Shin BC, Darvishi M, Kim CH. A comparison of the Newton-Krylov method with high order newton-like methods to solve nonlinear systems. Applied Mathematics and Computation. 2010;**217**(7):3190-3198

[36] Mamat M, Muhammad K, Waziri MY. Trapezoidal Broyden's method for systems of nonlinear equations. Applied Mathematical Sciences. 2014;**8**(6):54-63

[37] Sulaiman I, Mamat M, Waziri MY, Umar AO, et al. Journal of Advanced Research in Modelling and Simulations. 2018;**1**(1):13-18

#### **Chapter 4**

Computational and Applied Mathematics. 2011;**625**:2354-2363

(3):246-252

341-358

(3.28):89-91

2012

**56**

[22] Natasa K, Zorana L. Newton-like methods with modification of the righthand side vector. Mathematics of Computation. 2002;**71**:237-250

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

equations. Journal of Numerical

[31] Waziri MY, Abdulmajid Z. An improved diagonal Jacobian approximation via a quasi-Cauchy condition for solving large-scale systems of nonlinear equations. Journal of Applied Mathematics. 2013;**3**:1-6

[32] Andrei N. An unconstrained optimization test functions collection. Advanced Modeling and Optimization.

[33] Dolan E, Moré JJ. Benchmarking

performance profiles. Mathematical Programming. 2002;**91**(2):201-213

[34] Hafiz MA, Muhammad SMB. An efficient two-step iterative method for solving system of nonlinear equation. Journal of Mathematics Research. 2012;

[35] Shin BC, Darvishi M, Kim CH. A comparison of the Newton-Krylov method with high order newton-like methods to solve nonlinear systems. Applied Mathematics and Computation.

[36] Mamat M, Muhammad K, Waziri MY. Trapezoidal Broyden's method for systems of nonlinear equations. Applied Mathematical Sciences. 2014;**8**(6):54-63

[37] Sulaiman I, Mamat M, Waziri MY, Umar AO, et al. Journal of Advanced Research in Modelling and Simulations.

2010;**217**(7):3190-3198

2018;**1**(1):13-18

optimization software with

2008;**10**:147-161

**4**(4):28-34

54-63

Mathematics and Stochastics. 2010;**2**(1):

[23] Waziri MY, Leong WJ, Hassan MA, Monsi M. A new Newton's method with diagonal Jacobian approximation for systems of nonlinear equations. Journal of Mathematics and Statistics. 2010;**6**

[24] Lampariello F, Sciandrone M. Global convergence technique for the Newton method with periodic Hessian evaluation. Journal of Optimization Theory and Applications. 2001;**111**(2):

[25] Sulaiman IM, Mamat M, Nurnadiah

[26] Ypma TJ. Historical development of the Newton-Raphson method. SIAM

[27] Hao L, Qin N. Incomplete Jacobian Newton method for nonlinear equation. Computers and Mathematics with Applications. 2008;**56**(1):218-227

[28] Chency E, Kincaid D. Numerical Mathematics and Computing. Asia: Nelson Education, Cengage Learning;

[29] Sulaiman IM, Mamat M, Afendee MM, Waziri MY. Diagonal updating Shamanskii-like method for solving singular fuzzy nonlinear equation. Far East Journal of Mathematical Sciences.

[30] Waziri MY, Leong WJ, Hassan MA, Monsi M. Jacobian-free Newton's method for systems of nonlinear

2017;**103**(10):1619-1629

Z, Puspa LG. Solving dual fuzzy nonlinear equations via Shamanskii method. International Journal of Engineering & Technology. 2018;**7**

Review. 1995;**37**(4):531-551
