**Abstract**

In this chapter, we develop an efficient numerical scheme for the solution of boundary value problems of fractional order differential equations as well as their coupled systems by using Bernstein polynomials. On using the mentioned polynomial, we construct operational matrices for both fractional order derivatives and integrations. Also we construct a new matrix for the boundary condition. Based on the suggested method, we convert the considered problem to algebraic equation, which can be easily solved by using Matlab. In the last section, numerical examples are provided to illustrate our main results.

**Keywords:** Bernstein polynomials, coupled systems, fractional order differential equations, operational matrices of integration, approximate solutions **2010 MSC:** 34L05, 65L05, 65T99, 34G10

## **1. Introduction**

Generalization of classical calculus is known as fractional calculus, which is one of the fastest growing area of research, especially the theory of fractional order differential equations because this area has wide range of applications in real-life problems. Differential equations of fractional order provide an excellent tool for the description of many physical biological phenomena. The said equations play important roles for the description of hereditary characteristics of various materials and genetical problems in biological models as compared with integer order differential equations in the form of mathematical models. Nowadays, most of its applications are found in bio-medical engineering as well as in other scientific and engineering disciplines such as mechanics, chemistry, viscoelasticity, control theory, signal and image processing phenomenon, economics, optimization theory, etc.; for details, we refer the reader to study [1–9] and the references there in. Due to these important applications of fractional order differential equations, mathematicians are taking interest in the study of these equations because their models are more realistic and practical. In the last decade, many researchers have studied the existence and uniqueness of solutions to boundary value problems and their

coupled systems for fractional order differential equations (see [10–17]). Hence the area devoted to existence theory has been very well explored. However, every fractional differential equation cannot be solved for its analytical solutions easily due to the complex nature of fractional derivative; so, in such a situation, approximate solutions to such a problem is most efficient and helpful. Recently, many methods such as finite difference method, Fourier series method, Adomian decomposition method (ADM), inverse Laplace technique (ILT), variational iteration methods (VIM), fractional transform method (FTM), differential transform method (DTM), homotopy analysis method (HAM), method of radial base function (MRBM), wavelet techniques (WT), spectral methods and many more (for more details, see [9, 18–38]) have been developed for obtaining numerical solutions of such differential equations. These methods have their own merits and demerits. Some of them provide a very good approximation. However, in the last few years, some operational matrices were constructed to achieve good approximation as in [39]. After this, a variety of operational matrices were developed for different wavelet methods. This method uses operational matrices, where every operation, for example differentiation and integration, involved in these equations is performed with the help of a matrix. A large variety of operational matrices are available in the literature for different orthogonal polynomials like Legendre, Laguerre, Jacobi and Bernstein polynomials [40–48]. Motivated by the above applications and uses of fractional differential equations, in this chapter, we developed a numerical scheme based on operational matrices via Bernstein polynomials. Our proof is more generalized and there is no need to convert the Bernstein polynomial function vector to another basis like block pulse function or Legendre polynomials. To the best of our knowledge, the method we consider provides a very good approximation to the solution. By the use of these operational matrices, we apply our scheme to a single fractional order differential equation with given boundary conditions as

some interesting practical examples and their images. Section 5 describes the

*Approximate Solutions of Some Boundary Value Problems by Using Operational Matrices…*

In this section, we recall some fundamental definitions and results from the

ð*t* 0

ð*t* 0

*y t*ðÞ¼ 0, *n* � 1<*γ* ≤*n*,

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

ð Þ <sup>1</sup> � *<sup>t</sup> <sup>m</sup>*�*<sup>i</sup>*

Θð Þ *<sup>i</sup>*,*k*,*<sup>m</sup> t*

*k*þ*i*

**Lemma 2.2.** *The following result holds for fractional differential equations*

*y t*ðÞ¼ *y t*ð Þþ *d*<sup>0</sup> þ *c*1*t* þ *d*2*t*

The Bernstein polynomials *Bi*,*<sup>m</sup>*ð Þ*t* on 0, 1 ½ � can be defined as

*m i* � �*<sup>t</sup> i*

*Bi*,*<sup>m</sup>*ðÞ¼ *<sup>t</sup>* <sup>X</sup>*<sup>m</sup>*�*<sup>i</sup>*

*k*¼0

**Definition 2.2.** *The Caputo fractional order derivative of a function y on the interval*

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

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

*y*ð Þ*τ dτ:*

*<sup>y</sup>*ð Þ *<sup>n</sup>* ð Þ*<sup>τ</sup> <sup>d</sup>τ*,

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

*<sup>k</sup>*þ*<sup>γ</sup>* and *D<sup>γ</sup>*

*t*

, for *i* ¼ 0, 1, 2 … *m*,

, *i* ¼ 0, 1, 2 … *m*, (3)

ð Þ *<sup>m</sup>*�*<sup>i</sup>* !*i*! , which on further simplification can be written in the

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

*n*�1 ,

½ �¼ *constant* 0*:*

ð Þ ½ � 0, 1 ,

*<sup>n</sup>*�1, where

**Definition 2.1.** *The fractional integral of order <sup>γ</sup>* <sup>∈</sup> <sup>þ</sup> *of a function y*∈*L*<sup>1</sup>

Γð Þ*γ*

Γð Þ *n* � *γ*

conclusion of the chapter.

*is defined as*

½ � 0, 1 *is defined by*

**2. Basic definitions and results**

literature, which can be found in [10–16].

*DOI: http://dx.doi.org/10.5772/intechopen.90302*

*Dγ*

*dk* ∈*R* and *k* ¼ 0, 1, 2, 3, *:* … , *n* � 1*:*.

*I γ Dγ*

Hence it follows that

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

**2.1 The Bernstein polynomials**

*m i* � �

most simplified form as

*Dγ t*

where

**167**

for arbitrary *dk* ∈ *R*, *k* ¼ 0, 1, 2, … , *n* � 1.

*Bi*,*<sup>m</sup>*ðÞ¼ *t*

<sup>¼</sup> *<sup>m</sup>*!

*t k*�*γ* , *I γ t*

*I γ*

<sup>0</sup>þ*y t*ðÞ¼ <sup>1</sup>

<sup>0</sup>þ*y t*ðÞ¼ <sup>1</sup>

where *n* ¼ ½ �þ *γ* 1 and ½ � *γ* represents the integer part of *γ*. **Lemma 2.1.** *The fractional differential equation of order γ* >0

*Dγ*

has a unique solution of the form *y t*ðÞ¼ *d*<sup>0</sup> þ *d*1*t* þ *d*2*t*

$$\begin{cases} D^a \boldsymbol{\chi}(t) + A D^\mu \boldsymbol{\chi}(t) + B \boldsymbol{\chi}(t) = \boldsymbol{f}(t), \quad 1 < a \le 2, \quad 0 < \mu \le 1, \\ \boldsymbol{\chi}(\mathbf{0}) = a, \quad \boldsymbol{\chi}(\mathbf{1}) = b, \end{cases} \tag{1}$$

where *f t*ð Þ is the source term, *A*, *B* are any real numbers; then we extend our method to solve a boundary value problem of coupled system of fractional order differential equations of the form

$$\begin{cases} D^{\mathfrak{a}}\mathbf{x}(t) + A\_{1}D^{\mu\_{1}}\mathbf{x}(t) + B\_{1}D^{\nu\_{1}}\mathbf{y}(t) + C\_{1}\mathbf{x}(t) + D\_{1}\mathbf{y}(t) = \mathbf{f}(t), \quad 1 < a \le 2, \quad 0 < \mu\_{1}, \nu\_{1} \le 1, \\\\ D^{\mathfrak{b}}\mathbf{y}(t) + A\_{2}D^{\mu\_{2}}\mathbf{x}(t) + B\_{2}D^{\nu\_{2}}\mathbf{y}(t) + C\_{2}\mathbf{x}(t) + D\_{2}\mathbf{y}(t) = \mathbf{g}(t), \quad 1 < \boldsymbol{\rho} \le 2, \quad 0 < \mu\_{2}, \nu\_{2} \le 1, \\\\ \mathbf{y}(0) = a, \ \mathbf{y}(1) = b, \ \mathbf{y}(0) = c, \ \mathbf{y}(1) = d, \end{cases} \tag{2}$$

where *f t*ð Þ, *g t*ð Þ are source terms of the system, *Ai*, *Bi*,*Ci*, *Di*ð Þ *i* ¼ 1, 2 are any real constants. Also we compare our approximations to exact values and approximations of other methods like Jacobi polynomial approximations and Haar wavelets methods to evaluate the efficiency of the proposed method. We also provide some examples for the illustration of our main results.

This chapter is designed in five sections. In the first section of the chapter, we have cited some basic works related to the numerical and analytical solutions of differential equations of arbitrary order by various methods. The necessary definitions and results related to fractional calculus and Bernstein polynomials along with the construction of some operational matrices are given in Section 2. In Section 3, we have discussed the main theory for the numerical procedure. Section 4 contains

*Approximate Solutions of Some Boundary Value Problems by Using Operational Matrices… DOI: http://dx.doi.org/10.5772/intechopen.90302*

some interesting practical examples and their images. Section 5 describes the conclusion of the chapter.

## **2. Basic definitions and results**

coupled systems for fractional order differential equations (see [10–17]). Hence the area devoted to existence theory has been very well explored. However, every fractional differential equation cannot be solved for its analytical solutions easily due to the complex nature of fractional derivative; so, in such a situation, approximate solutions to such a problem is most efficient and helpful. Recently, many methods such as finite difference method, Fourier series method, Adomian decomposition method (ADM), inverse Laplace technique (ILT), variational iteration methods (VIM), fractional transform method (FTM), differential transform method (DTM), homotopy analysis method (HAM), method of radial base function (MRBM), wavelet techniques (WT), spectral methods and many more (for more details, see [9, 18–38]) have been developed for obtaining numerical solutions of such differential equations. These methods have their own merits and demerits. Some of them provide a very good approximation. However, in the last few years, some operational matrices were constructed to achieve good approximation as in [39]. After this, a variety of operational matrices were developed for different wavelet methods. This method uses operational matrices, where every operation, for example differentiation and integration, involved in these equations is performed with the help of a matrix. A large variety of operational matrices are available in the literature for different orthogonal polynomials like Legendre, Laguerre, Jacobi and Bernstein polynomials [40–48]. Motivated by the above applications and uses of fractional differential equations, in this chapter, we developed a numerical scheme based on operational matrices via Bernstein polynomials. Our proof is more generalized and there is no need to convert the Bernstein polynomial function vector to another basis like block pulse function or Legendre polynomials. To the best of our knowledge, the method we consider provides a very good approximation to the solution. By the use of these operational matrices, we apply our scheme to a single fractional order differential equation with given

*y t*ðÞþ *By t*ðÞ¼ *f t*ð Þ, 1<*α* ≤2, 0<*μ*≤1,

where *f t*ð Þ is the source term, *A*, *B* are any real numbers; then we extend our method to solve a boundary value problem of coupled system of fractional order

*<sup>D</sup>αx t*ðÞþ *<sup>A</sup>*1*Dμ*1*x t*ðÞþ *<sup>B</sup>*1*Dν*<sup>1</sup> *y t*ðÞþ *<sup>C</sup>*1*x t*ðÞþ *<sup>D</sup>*1*y t*ðÞ¼ *f t*ð Þ, 1<*<sup>α</sup>* <sup>≤</sup>2, 0<sup>&</sup>lt; *<sup>μ</sup>*1, *<sup>ν</sup>*<sup>1</sup> <sup>≤</sup> 1, *<sup>D</sup>βy t*ðÞþ *<sup>A</sup>*2*Dμ*2*x t*ðÞþ *<sup>B</sup>*2*Dν*<sup>2</sup> *y t*ðÞþ *<sup>C</sup>*2*x t*ðÞþ *<sup>D</sup>*2*y t*ðÞ¼ *g t*ð Þ, 1 <sup>&</sup>lt;*<sup>β</sup>* <sup>≤</sup>2, 0<sup>&</sup>lt; *<sup>μ</sup>*2, *<sup>ν</sup>*<sup>2</sup> <sup>≤</sup> 1,

where *f t*ð Þ, *g t*ð Þ are source terms of the system, *Ai*, *Bi*,*Ci*, *Di*ð Þ *i* ¼ 1, 2 are any real constants. Also we compare our approximations to exact values and approximations

This chapter is designed in five sections. In the first section of the chapter, we have cited some basic works related to the numerical and analytical solutions of differential equations of arbitrary order by various methods. The necessary definitions and results related to fractional calculus and Bernstein polynomials along with the construction of some operational matrices are given in Section 2. In Section 3, we have discussed the main theory for the numerical procedure. Section 4 contains

of other methods like Jacobi polynomial approximations and Haar wavelets methods to evaluate the efficiency of the proposed method. We also provide some

(1)

(2)

boundary conditions as

*Functional Calculus*

8 ><

>:

**166**

�

differential equations of the form

*<sup>D</sup>αy t*ðÞþ *AD<sup>μ</sup>*

*y*ð Þ¼ 0 *a*, *y*ð Þ¼ 1 *b*, *y*ð Þ¼ 0 *c*, *y*ð Þ¼ 1 *d*,

examples for the illustration of our main results.

*y*ð Þ¼ 0 *a*, *y*ð Þ¼ 1 *b*,

In this section, we recall some fundamental definitions and results from the literature, which can be found in [10–16].

**Definition 2.1.** *The fractional integral of order <sup>γ</sup>* <sup>∈</sup> <sup>þ</sup> *of a function y*∈*L*<sup>1</sup> ð Þ ½ � 0, 1 , *is defined as*

$$I\_{0+}^{\prime}\boldsymbol{\jmath}(t) = \frac{1}{\boldsymbol{\Gamma}(\boldsymbol{\gamma})} \int\_{0}^{t} (t-\boldsymbol{\tau})^{\boldsymbol{\gamma}-1} \boldsymbol{\jmath}(\boldsymbol{\tau}) d\boldsymbol{\tau}.$$

**Definition 2.2.** *The Caputo fractional order derivative of a function y on the interval* ½ � 0, 1 *is defined by*

$$D\_{0+}^{\gamma}\jmath(t) = \frac{1}{\Gamma(n-\gamma)}\int\_{0}^{t} (t-\tau)^{n-\gamma-1}\jmath^{(n)}(\tau)d\tau,$$

where *n* ¼ ½ �þ *γ* 1 and ½ � *γ* represents the integer part of *γ*. **Lemma 2.1.** *The fractional differential equation of order γ* >0

$$D^\circ y(t) = 0, n - 1 < \chi \le n,$$

has a unique solution of the form *y t*ðÞ¼ *d*<sup>0</sup> þ *d*1*t* þ *d*2*t* <sup>2</sup> <sup>þ</sup> … <sup>þ</sup> *dn*�<sup>1</sup>*<sup>t</sup> <sup>n</sup>*�1, where *dk* ∈*R* and *k* ¼ 0, 1, 2, 3, *:* … , *n* � 1*:*.

**Lemma 2.2.** *The following result holds for fractional differential equations*

$$I^\prime D^\gamma \chi(t) = \chi(t) + d\_0 + c\_1 t + d\_2 t^2 + \dots + d\_{n-1} t^{n-1},$$

for arbitrary *dk* ∈ *R*, *k* ¼ 0, 1, 2, … , *n* � 1. Hence it follows that

$$D^\prime t^k = \frac{\Gamma(k+1)}{\Gamma(k-\gamma+1)} t^{k-\gamma}, \quad I^\prime t^k = \frac{\Gamma(k+1)}{\Gamma(k+\gamma+1)} t^{k+\gamma} \text{ and } \; D^\prime[constant] = \mathbf{0}.$$

#### **2.1 The Bernstein polynomials**

The Bernstein polynomials *Bi*,*<sup>m</sup>*ð Þ*t* on 0, 1 ½ � can be defined as

$$B\_{i,m}(t) = \binom{m}{i} t^i (1-t)^{m-i}, \text{for } i = 0, 1, 2 \dots m,$$

where *m i* � � <sup>¼</sup> *<sup>m</sup>*! ð Þ *<sup>m</sup>*�*<sup>i</sup>* !*i*! , which on further simplification can be written in the most simplified form as

$$B\_{i,m}(t) = \sum\_{k=0}^{m-i} \Theta\_{(i,k,m)} t^{k+i}, \; i = 0, 1, 2\dots m,\tag{3}$$

where

$$
\Theta\_{(i,k,m)} = (-1)^k \binom{m}{i} \binom{m-i}{k} .
$$

*Y* ¼ *span B*f g 0,*m*, *B*1,*m*, … *Bm*,*<sup>m</sup>*

*Approximate Solutions of Some Boundary Value Problems by Using Operational Matrices…*

is a finite dimensional and closed subspace. So if *f* ∈ *H* is an arbitrary element then its best approximation is unique in *Y:* This terminology can be achieved by using *y*<sup>0</sup> ∈*Y* and for all *y*∈*Y*, we have ∥ *f* � *y*0∥≤∥ *f* � *y*∥*:* Thus any function can be

*f t*ðÞ¼ <sup>X</sup>*<sup>m</sup>*

where coefficient *ci* can easily be calculated by multiplying (4) by *B*ð Þ *<sup>j</sup>*,*<sup>m</sup>* ð Þ*t* , *j* ¼ 0, 1, 2, … *m* and integrating over 0, 1 ½ � by using inner product and

> *i*¼0 *ci* ð1 0

ð1 0 X*m i*¼0

*f t*ð Þ*B*ð Þ *<sup>j</sup>*,*<sup>m</sup>* ð Þ*<sup>t</sup> dt* <sup>¼</sup> <sup>X</sup>*<sup>m</sup>*

*i*¼0

*ciB*ð Þ *<sup>i</sup>*,*<sup>m</sup>* , (4)

<sup>0</sup>*B*ð Þ *<sup>i</sup>*,*<sup>m</sup>* ð Þ*t B*ð Þ *<sup>j</sup>*,*<sup>m</sup>* ð Þ*t dt*, *i*, *j* ¼ 0, 1, 2*:* … *m*, we have

*B*ð Þ *<sup>i</sup>*,*<sup>m</sup>* ð Þ*t :B*ð Þ *<sup>j</sup>*,*<sup>m</sup>* ð Þ*t dt*, *j* ¼ 0, 1, 2 … *m*

*θ*ð Þ 0,0 *θ*ð Þ 0,1 ⋯ *θ*ð Þ 0,*<sup>r</sup>* ⋯ *θ*ð Þ 0,*<sup>m</sup> θ*ð Þ 1,0 *θ*ð Þ 1,1 ⋯ *θ*ð Þ 1,*<sup>r</sup>* ⋯ *θ*ð Þ 1,*<sup>m</sup>* ⋮ ⋮⋮⋮⋮ ⋮ *θ*ð Þ *<sup>r</sup>*,0 *θ*ð Þ *<sup>r</sup>*,1 ⋯ *θ*ð Þ *<sup>r</sup>*,*<sup>r</sup>* ⋯ *θ*ð Þ *<sup>r</sup>*,*<sup>m</sup>* ⋮ ⋮⋮⋮⋮ ⋮ *θ*ð Þ *<sup>m</sup>*,0 *θ*ð Þ *<sup>m</sup>*,1 ⋯ *θ*ð Þ *<sup>m</sup>*,*<sup>r</sup>* ⋯ *θ*ð Þ *<sup>m</sup>*,*<sup>m</sup>*

, so

*<sup>M</sup>*�*<sup>M</sup>:* (6)

*:*

(5)

*ciB*ð Þ *<sup>i</sup>*,*<sup>m</sup>* ð Þ*t :B*ð Þ *<sup>j</sup>*,*<sup>m</sup>* ð Þ*t dt*, *j* ¼ 0, 1, 2 … *m*,

Let *XM* ¼ ½ � *d*<sup>0</sup> *d*<sup>1</sup> *:* … *dm* , *CM* ¼ ½ � *c*<sup>0</sup> *c*1*:* … *cm* , where *M* ¼ *m* þ 1 where *M* is the

*XM* <sup>¼</sup> *CM*Φ*<sup>M</sup>*�*<sup>M</sup>* ) *CM* <sup>¼</sup> *XM*Φ�<sup>1</sup>

where Φ*<sup>m</sup>*�*<sup>M</sup>* is called the dual matrix of the Bernstein polynomials. After

*BM*ðÞ¼ *t B*ð Þ 0,*<sup>m</sup>* , *B*ð Þ 1,*<sup>m</sup>* , *:* … *B*ð Þ *<sup>m</sup>*,*<sup>m</sup>*

*<sup>M</sup>*ð Þ*t* , *CM* is coefficient matrix

� �*:* (7)

*<sup>M</sup>*ð Þ*t be the function vector defined in* (3), *then the fractional order*

*θ*ð Þ 0,0 *θ*ð Þ 0,1 ⋯ *θ*ð Þ 0,*<sup>r</sup>* ⋯ *θ*ð Þ 0,*<sup>m</sup> θ*ð Þ 1,0 *θ*ð Þ 1,1 ⋯ *θ*ð Þ 1,*<sup>r</sup>* ⋯ *θ*ð Þ 1,*<sup>m</sup>* ⋮ ⋮⋮⋮⋮ ⋮ *θ*ð Þ *<sup>r</sup>*,0 *θ*ð Þ *<sup>r</sup>*,1 ⋯ *θ*ð Þ *<sup>r</sup>*,*<sup>r</sup>* ⋯ *θ*ð Þ *<sup>r</sup>*,*<sup>m</sup>* ⋮ ⋮⋮⋮⋮ ⋮ *θ*ð Þ *<sup>m</sup>*,0 *θ*ð Þ *<sup>m</sup>*,1 ⋯ *θ*ð Þ *<sup>m</sup>*,*<sup>r</sup>* ⋯ *θ*ð Þ *<sup>m</sup>*,*<sup>m</sup>*

approximated in terms of Bernstein polynomials as

*DOI: http://dx.doi.org/10.5772/intechopen.90302*

<sup>0</sup>*B*ð Þ *<sup>i</sup>*,*<sup>m</sup>* ð Þ*<sup>t</sup> f t*ð Þ*dt*, *<sup>θ</sup>*ð Þ *<sup>i</sup>*,*<sup>j</sup>* <sup>¼</sup> <sup>Ð</sup> <sup>1</sup>

*f t*ð Þ*B*ð Þ *<sup>j</sup>*,*<sup>m</sup>* ð Þ*t dt* ¼

0

which implies that ½*d*<sup>0</sup> *d*<sup>1</sup> *:* … *dm*� ¼ ½ � *c*<sup>0</sup> *c*1*:* … *cm*

*f t*ðÞ¼ *CMBT*

*<sup>M</sup>*ð Þ*t is given by*

ð1 0

which implies that <sup>ð</sup><sup>1</sup>

scale level and Φ*<sup>M</sup>*�*<sup>M</sup>* ¼

where

*integration of BT*

**169**

**Lemma 2.4.** *Let BT*

calculating *ci*, (4) can be written as

*di* <sup>¼</sup> <sup>Ð</sup> <sup>1</sup>

Note that the sum of the Bernstein polynomials converges to 1.

**Lemma 2.3.** *Convergence Analysis: Assume that the function g* ∈*Cm*þ<sup>1</sup> ½ � 0, 1 *that is m* þ 1 *times continuously differentiable function and let X* ¼ h i *B*0,*m*, *B*1, *<sup>m</sup>*, … , *Bm*, *<sup>m</sup> . If <sup>C</sup>T*Ψð Þ *<sup>x</sup> is the best approximation of g out of X, then the error bound is presented as*

$$\|\|\mathbf{g} - \mathbf{C}^T \Psi\|\|\_2 \le \frac{\sqrt{2} \text{MS}^{\frac{2m+3}{2}}}{\Gamma(m+2)\sqrt{2m+3}},$$

where *<sup>M</sup>* <sup>¼</sup> max *<sup>x</sup>*∈½ � 0,1 <sup>∣</sup> *<sup>g</sup>*ð Þ *<sup>m</sup>*þ<sup>1</sup> ð Þ *<sup>x</sup>* <sup>∣</sup>, *<sup>S</sup>* <sup>¼</sup> max 1f g � *<sup>x</sup>*0, *<sup>x</sup>*<sup>0</sup> *: Proof.* In view of Taylor polynomials, we have

$$F(\boldsymbol{\mathfrak{x}}) = \mathbf{g}(\boldsymbol{\mathfrak{x}}\_0) + (\boldsymbol{\mathfrak{x}} - \boldsymbol{\mathfrak{x}}\_0)\mathbf{g}^{(1)}(\boldsymbol{\mathfrak{x}}\_0) + \frac{(\boldsymbol{\mathfrak{x}} - \boldsymbol{\mathfrak{x}}\_0)^2}{\Gamma \mathbf{3}}\mathbf{g}^{(2)} + \dots + \frac{(\boldsymbol{\mathfrak{x}} - \boldsymbol{\mathfrak{x}}\_0)^m}{\Gamma (m+1)}\mathbf{g}^{(m)},$$

from which we know that

$$|\mathbf{g} - F(\mathbf{x})| = |\mathbf{g}^{(m+1)}(\eta)| \frac{(\mathbf{x} - \mathbf{x}\_0)^{m+1}}{\Gamma(m+2)}, \text{ there exist } \eta \in (\mathbf{0}, \mathbf{1}).$$

Due the best approximation *<sup>C</sup><sup>T</sup>*Ψð Þ *<sup>x</sup>* of *<sup>g</sup>*, we have

$$\begin{aligned} \|\|g - \mathbf{C}^T \Psi(\mathbf{x})\|\|\_{2}^{2} &\le \|g - F\|\|\_{2}^{2} \\ &= \int\_{0}^{1} (g(\mathbf{x}) - F(\mathbf{x}))^{2} d\mathbf{x} \\ &= \int\_{0}^{1} \left[ |g^{(m+1)}(\eta)| \frac{(\mathbf{x} - \mathbf{x}\_{0})^{m+1}}{\Gamma(m+2)} \right]^{2} d\mathbf{x} \\ &\le \frac{M^{2}}{\Gamma^{2}(m+2)} \int\_{0}^{1} (\mathbf{x} - \mathbf{x}\_{0})^{2m+2} d\mathbf{x} \\ &\le \frac{2M^{2} S^{2m+3}}{\Gamma^{2}(m+2)(2m+3)}. \end{aligned}$$

Hence we have

$$\|\|g - C^T \Psi(\varkappa)\|\|\_2 \le \frac{\sqrt{2} \text{MS}^{\frac{2m+3}{2}}}{\Gamma(m+2)\sqrt{(2m+3)}} \cdot \|\square\|$$

Let *<sup>H</sup>* <sup>¼</sup> *<sup>L</sup>*<sup>2</sup> ½ � 0, 1 be a Hilbert space, then the inner product can be defined as

$$ = \int\_0^1 f(\mathbf{x}) \, \mathbf{g}(\mathbf{x}) d\mathbf{x}$$

and

*Approximate Solutions of Some Boundary Value Problems by Using Operational Matrices… DOI: http://dx.doi.org/10.5772/intechopen.90302*

$$Y = \operatorname{span}\left\{B\_{0,m}, B\_{1,m}, \dots B\_{m,m}\right\},$$

is a finite dimensional and closed subspace. So if *f* ∈ *H* is an arbitrary element then its best approximation is unique in *Y:* This terminology can be achieved by using *y*<sup>0</sup> ∈*Y* and for all *y*∈*Y*, we have ∥ *f* � *y*0∥≤∥ *f* � *y*∥*:* Thus any function can be approximated in terms of Bernstein polynomials as

$$f(t) = \sum\_{i=0}^{m} c\_i B\_{(i,m)},\tag{4}$$

where coefficient *ci* can easily be calculated by multiplying (4) by *B*ð Þ *<sup>j</sup>*,*<sup>m</sup>* ð Þ*t* , *j* ¼ 0, 1, 2, … *m* and integrating over 0, 1 ½ � by using inner product and *di* <sup>¼</sup> <sup>Ð</sup> <sup>1</sup> <sup>0</sup>*B*ð Þ *<sup>i</sup>*,*<sup>m</sup>* ð Þ*<sup>t</sup> f t*ð Þ*dt*, *<sup>θ</sup>*ð Þ *<sup>i</sup>*,*<sup>j</sup>* <sup>¼</sup> <sup>Ð</sup> <sup>1</sup> <sup>0</sup>*B*ð Þ *<sup>i</sup>*,*<sup>m</sup>* ð Þ*t B*ð Þ *<sup>j</sup>*,*<sup>m</sup>* ð Þ*t dt*, *i*, *j* ¼ 0, 1, 2*:* … *m*, we have

$$\begin{aligned} \label{def:} \int\_{0}^{1} f(t) B\_{(j,m)}(t) dt &= \int\_{0}^{1} \sum\_{i=0}^{m} c\_{i} B\_{(j,m)}(t) B\_{(j,m)}(t) dt, \quad j = 0, 1, 2 \dots m, \\ \text{which implies that } \int\_{0}^{1} f(t) B\_{(j,m)}(t) dt &= \sum\_{i=0}^{m} c\_{i} \Big|\_{0}^{1} B\_{(i,m)}(t) B\_{(j,m)}(t) dt, \quad j = 0, 1, 2 \dots m \\ \text{which implies that } [d\_{0} \ \operatorname{d}\_{1} \ \dots \ \operatorname{d}\_{m}] &= [c\_{0} \ \operatorname{c}\_{1} \dots \operatorname{c}\_{m}] \begin{bmatrix} \theta\_{(0,0)} & \theta\_{(0,1)} & \cdots & \theta\_{(0,r)} & \cdots & \theta\_{(0,m)} \\ \theta\_{(1,0)} & \theta\_{(1,1)} & \cdots & \theta\_{(1,r)} & \cdots & \theta\_{(1,m)} \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ \theta\_{(r,0)} & \theta\_{(r,1)} & \cdots & \theta\_{(r,r)} & \cdots & \theta\_{(r,m)} \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ \theta\_{(m,0)} & \theta\_{(m,1)} & \cdots & \theta\_{(m,r)} & \cdots & \theta\_{(m,m)} \end{bmatrix}. \end{aligned} \tag{5}$$

$$\text{Let } \mathbf{X}\_M = [d\_0 \ \mathbf{d}\_1 \ \dots \ \mathbf{d}\_m], \ \mathbf{C}\_M = [c\_0 \ \mathbf{c}\_1 \ \dots \ \mathbf{c}\_m], \text{ where } \\ \mathbf{M} = m + 1 \\ \text{where } \\ \mathbf{M} = \begin{bmatrix} \theta\_{(0,0)} & \theta\_{(0,1)} & \cdots & \theta\_{(0,r)} & \cdots & \theta\_{(0,m)} \\ \theta\_{(1,0)} & \theta\_{(1,1)} & \cdots & \theta\_{(1,r)} & \cdots & \theta\_{(1,m)} \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ \theta\_{(r,0)} & \theta\_{(r,1)} & \cdots & \theta\_{(r,r)} & \cdots & \theta\_{(r,m)} \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ \theta\_{(m,0)} & \theta\_{(m,1)} & \cdots & \theta\_{(m,r)} & \cdots & \theta\_{(m,m)} \end{bmatrix}, \text{ so}$$

$$\mathbf{X}\_M = \mathbf{C}\_M \boldsymbol{\Phi}\_{M \times M} \to \mathbf{C}\_M = X\_M \boldsymbol{\Phi}\_{M \times M}^{-1}. \tag{6}$$

where Φ*<sup>m</sup>*�*<sup>M</sup>* is called the dual matrix of the Bernstein polynomials. After calculating *ci*, (4) can be written as

$$f(t) = \mathbf{C}\_{M}\mathbf{B}\_{M}^{T}(t), \ \mathbf{C}\_{M} \text{ is coefficient matrix}$$

where

$$B\_M(t) = \begin{bmatrix} B\_{(0,m)}, B\_{(1,m)}, \dots, B\_{(m,m)} \end{bmatrix}. \tag{7}$$

**Lemma 2.4.** *Let BT <sup>M</sup>*ð Þ*t be the function vector defined in* (3), *then the fractional order integration of BT <sup>M</sup>*ð Þ*t is given by*

where

*Functional Calculus*

<sup>Θ</sup>ð Þ *<sup>i</sup>*,*k*,*<sup>m</sup>* ¼ �ð Þ<sup>1</sup> *<sup>k</sup> <sup>m</sup>*

Note that the sum of the Bernstein polynomials converges to 1. **Lemma 2.3.** *Convergence Analysis: Assume that the function g* ∈*Cm*þ<sup>1</sup>

<sup>∥</sup> *<sup>g</sup>* � *<sup>C</sup>T*Ψ∥<sup>2</sup> <sup>≤</sup>

where *<sup>M</sup>* <sup>¼</sup> max *<sup>x</sup>*∈½ � 0,1 <sup>∣</sup> *<sup>g</sup>*ð Þ *<sup>m</sup>*þ<sup>1</sup> ð Þ *<sup>x</sup>* <sup>∣</sup>, *<sup>S</sup>* <sup>¼</sup> max 1f g � *<sup>x</sup>*0, *<sup>x</sup>*<sup>0</sup> *:*

*Proof.* In view of Taylor polynomials, we have

*F x*ð Þ¼ *g x*ð Þþ <sup>0</sup> ð Þ *<sup>x</sup>* � *<sup>x</sup>*<sup>0</sup> *<sup>g</sup>*ð Þ<sup>1</sup> ð Þþ *<sup>x</sup>*<sup>0</sup>

<sup>∣</sup> *<sup>g</sup>* � *F x*ð Þ<sup>∣</sup> <sup>¼</sup> <sup>∣</sup> *<sup>g</sup>*ð Þ *<sup>m</sup>*þ<sup>1</sup> ð Þ*<sup>η</sup>* <sup>∣</sup>

<sup>∥</sup> *<sup>g</sup>* � *<sup>C</sup><sup>T</sup>*Ψð Þ *<sup>x</sup>* <sup>∥</sup><sup>2</sup>

Due the best approximation *<sup>C</sup><sup>T</sup>*Ψð Þ *<sup>x</sup>* of *<sup>g</sup>*, we have

from which we know that

Hence we have

Let *<sup>H</sup>* <sup>¼</sup> *<sup>L</sup>*<sup>2</sup>

and

**168**

*i*

ffiffi 2 <sup>p</sup> *MS*<sup>2</sup>*m*þ<sup>3</sup> 2

ð Þ *x* � *x*<sup>0</sup> 2

ð Þ *<sup>x</sup>* � *<sup>x</sup>*<sup>0</sup> *<sup>m</sup>*þ<sup>1</sup>

2

ð Þ *g x*ð Þ� *F x*ð Þ <sup>2</sup>

ð1 0

*S*<sup>2</sup>*m*þ<sup>3</sup>

ð Þ *<sup>m</sup>* <sup>þ</sup> <sup>2</sup> ð Þ <sup>2</sup>*<sup>m</sup>* <sup>þ</sup> <sup>3</sup> *:*

ffiffi 2 <sup>p</sup> *MS*<sup>2</sup>*m*þ<sup>3</sup> 2

<sup>Γ</sup>ð Þ *<sup>m</sup>* <sup>þ</sup> <sup>2</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

½ � 0, 1 be a Hilbert space, then the inner product can be defined as

*f x*ð Þ*:g x*ð Þ*dx*

<sup>2</sup> ≤ ∥ *<sup>g</sup>* � *<sup>F</sup>*∥<sup>2</sup>

¼ ð1 0

¼ ð1 0

≤

≤

<sup>∥</sup> *<sup>g</sup>* � *<sup>C</sup><sup>T</sup>*Ψð Þ *<sup>x</sup>* <sup>∥</sup><sup>2</sup> <sup>≤</sup>

Γ2

Γ2

<*f*, *g* > ¼

*M*<sup>2</sup>

ð Þ *m* þ 2

2*M*<sup>2</sup>

ð1 0

<sup>Γ</sup>ð Þ *<sup>m</sup>* <sup>þ</sup> <sup>2</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>2</sup>*<sup>m</sup>* <sup>þ</sup> <sup>3</sup> <sup>p</sup> ,

<sup>Γ</sup><sup>3</sup> *<sup>g</sup>*ð Þ<sup>2</sup> <sup>þ</sup> … <sup>þ</sup> ð Þ *<sup>x</sup>* � *<sup>x</sup>*<sup>0</sup> *<sup>m</sup>*

<sup>Γ</sup>ð Þ *<sup>m</sup>* <sup>þ</sup> <sup>2</sup> , there exist *<sup>η</sup>*∈ð Þ 0, 1 *:*

*dx*

Γð Þ *m* þ 2

ð Þ <sup>2</sup>*<sup>m</sup>* <sup>þ</sup> <sup>3</sup> <sup>p</sup> *:* □

2*m*þ2 *dx* *dx*

<sup>j</sup> *<sup>g</sup>*ð Þ *<sup>m</sup>*þ<sup>1</sup> ð Þj *<sup>η</sup>* ð Þ *<sup>x</sup>* � *<sup>x</sup>*<sup>0</sup> *<sup>m</sup>*þ<sup>1</sup>

" #<sup>2</sup>

ð Þ *x* � *x*<sup>0</sup>

<sup>Γ</sup>ð Þ *<sup>m</sup>* <sup>þ</sup> <sup>1</sup> *<sup>g</sup>*ð Þ *<sup>m</sup>* ,

*m* þ 1 *times continuously differentiable function and let X* ¼ h i *B*0,*m*, *B*1, *<sup>m</sup>*, … , *Bm*, *<sup>m</sup> . If <sup>C</sup>T*Ψð Þ *<sup>x</sup> is the best approximation of g out of X, then the error bound is presented as*

� � *m* � *i*

*k* � �

*:*

½ � 0, 1 *that is*

$$I^a B\_M^T(t) = P\_{M \times M}^a B\_M^T(t),\tag{8}$$

evaluating this result for i = 0,1,2....m, we have

*I <sup>α</sup>B*0,*m*ð Þ*<sup>t</sup>*

*X*ð Þ <sup>0</sup> *<sup>M</sup>* Φ�<sup>1</sup> *<sup>M</sup>*�*MBT <sup>M</sup>*ð Þ*t*

*Approximate Solutions of Some Boundary Value Problems by Using Operational Matrices…*

*X*ð Þ<sup>1</sup> *<sup>M</sup>* Φ�<sup>1</sup> *<sup>M</sup>*�*MB<sup>T</sup> <sup>M</sup>*ð Þ*t*

*X*ð Þ *<sup>m</sup> <sup>M</sup>* Φ�<sup>1</sup> *<sup>M</sup>*�*MBT*

Γð Þ *k* þ *i* þ *α* Γð Þ *k* þ *i* þ *α* þ 1

*<sup>M</sup>*�*<sup>M</sup>:*Φ�<sup>1</sup>

*<sup>M</sup>*�*<sup>M</sup>* <sup>¼</sup> *<sup>P</sup><sup>α</sup>*

*<sup>M</sup>*�*<sup>M</sup>:BT*

*<sup>M</sup>*�*<sup>M</sup>:BT*

*<sup>M</sup>*�*<sup>M</sup>* is the operational matrix of fractional order derivative given by

*<sup>M</sup>*�*<sup>M</sup>*Φ�<sup>1</sup>

Ψð Þ 0,0 Ψð Þ 0,1 ⋯ Ψð Þ 0,*<sup>r</sup>* ⋯ Ψð Þ 0,*<sup>m</sup>* Ψð Þ 1,0 Ψð Þ 1,1 ⋯ Ψð Þ 1,*<sup>r</sup>* ⋯ Ψð Þ 1,*<sup>m</sup>* ⋮ ⋮⋮⋮⋮ ⋮

Ψð Þ *<sup>r</sup>*,0 Ψð Þ *<sup>r</sup>*,1 ⋯ Ψð Þ *<sup>r</sup>*,*<sup>r</sup>* ⋯ Ψð Þ *<sup>r</sup>*,*<sup>m</sup>* ⋮ ⋮⋮⋮⋮ ⋮

Ψð Þ *<sup>m</sup>*,0 Ψð Þ *<sup>m</sup>*,1 ⋯ Ψð Þ *<sup>m</sup>*,*<sup>r</sup>* ⋯ Ψð Þ *<sup>m</sup>*,*<sup>m</sup>*

*<sup>M</sup>*�*<sup>M</sup>:BT*

*M*�*M*

*<sup>M</sup>*ð Þ*t be the function vector as defined in* (3), *then fractional order*

⋮ ⋮ ⋮

*<sup>M</sup>*ð ÞÞ *t*

1 ð Þ *k* þ *l* þ *j* þ *i* þ *α* þ 1

*<sup>M</sup>*ð Þ*t :* (17)

*<sup>M</sup>*ð Þ*t :* (18)

*<sup>M</sup>*ð Þ*t* (19)

*<sup>M</sup>*�*<sup>M</sup>*, (20)

, (21)

(16)

□

¼

*DOI: http://dx.doi.org/10.5772/intechopen.90302*

*I <sup>α</sup>B*1,*m*ð Þ*<sup>t</sup>* ⋮ ⋮ ⋮

*I <sup>α</sup>Bm*,*m*ð Þ*<sup>t</sup>*

*θ*ð Þ *<sup>i</sup>*,*k*,*<sup>m</sup> :θ*ð Þ *<sup>j</sup>*,*l*,*<sup>m</sup>*

*I <sup>α</sup>BT*

*<sup>M</sup>*ðÞ¼ *<sup>t</sup> <sup>P</sup>*^*<sup>α</sup>*

*<sup>M</sup>*�*<sup>M</sup>:*Φ�<sup>1</sup>

*<sup>M</sup>*ðÞ¼ *<sup>t</sup> <sup>P</sup><sup>α</sup>*

*<sup>M</sup>*ðÞ¼ *<sup>t</sup> <sup>G</sup><sup>α</sup>*

*<sup>M</sup>*�*<sup>M</sup>* <sup>¼</sup> *<sup>G</sup>*^ *<sup>α</sup>*

*P*^*α*

*I <sup>α</sup>BT*

*D<sup>α</sup>BT*

*Gα*

*where* Φ*<sup>M</sup>*�*<sup>M</sup> is the dual matrix given in* (3) *and*

further writing

Let us represent

**Lemma 2.5.** *Let BT*

*G*^ *α <sup>M</sup>*�*<sup>M</sup>* ¼

*derivative is defined as*

where *G<sup>α</sup>*

**171**

we get

thus

<sup>Ψ</sup>*<sup>i</sup>*,*<sup>j</sup>* <sup>¼</sup> <sup>X</sup>*<sup>m</sup>*�*<sup>i</sup>*

*k*¼0

X*m*�*j*

*l*¼0

where *P<sup>α</sup> <sup>M</sup>*�*<sup>M</sup>* is the fractional integration's operational matrix defined as

$$P^{a}\_{M \times M} = \hat{P}^{a}\_{M \times M} \Phi^{-1}\_{M \times M}$$

*and* Φ�<sup>1</sup> *<sup>M</sup>*�*<sup>M</sup> is given in* (3) *and P<sup>α</sup> <sup>M</sup>*�*<sup>M</sup> is given by*

$$
\hat{P}\_{M \times M}^{a} = \begin{bmatrix}
\Psi\_{(0,0)} & \Psi\_{(0,1)} & \dots & \Psi\_{(0,r)} & \dots & \Psi\_{(0,m)} \\
\Psi\_{(1,0)} & \Psi\_{(1,1)} & \dots & \Psi\_{(1,r)} & \dots & \Psi\_{(1,m)} \\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\
\Psi\_{(r,0)} & \Psi\_{(r,1)} & \dots & \Psi\_{(r,r)} & \dots & \Psi\_{(r,m)} \\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\
\Psi\_{(m,0)} & \Psi\_{(m,1)} & \dots & \Psi\_{(m,r)} & \dots & \Psi\_{(m,m)} \\
\end{bmatrix},\tag{9}
$$

where

$$\Psi\_{ij} = \sum\_{k=0}^{m-i} \sum\_{l=0}^{m-j} \theta\_{(i,k,m)} \theta\_{(j,l,m)} \frac{\Gamma(k+i+1)}{(i+j+k+l+a+1)\Gamma(k+i+a+1)}.\tag{10}$$

*Proof.* Consider

$$B\_{i,m}(t) = \sum\_{k=0}^{m-i} \theta\_{(i,k,m)} t^{k+i} \tag{11}$$

taking fractional integration of both sides, we have

$$I^{a}B\_{i,m}(t) = \sum\_{k=0}^{m-i} \theta\_{(i,k,m)} I^{a}t^{k+i} = \sum\_{k=0}^{m-i} \theta\_{(i,k,m)} \frac{\Gamma(k+i+a)}{\Gamma(k+i+a+1)} t^{k+i+a}.\tag{12}$$

Now to approximate right-hand sides of above

$$\sum\_{k=0}^{m-i} \theta\_{(i,k,m)} \frac{\Gamma(k+i+a)}{\Gamma(k+i+a+1)} t^{k+i+a} = \mathbf{C}\_{M}^{(i)} B\_{M}^{T}(t) \tag{13}$$

where *C*ð Þ*<sup>i</sup> <sup>M</sup>* can be approximated by using (3) as

$$\mathbf{C}\_{M}^{(i)} = \mathbf{X}\_{M}^{(i)} \boldsymbol{\Phi}\_{M \times M}^{-1},\tag{14}$$

where entries of the vector *X*ð Þ*<sup>i</sup> <sup>M</sup>* can be calculated in generalized form as

$$\begin{split} &X\_{M}^{(j)} = \int\_{0}^{1} \sum\_{k=0}^{m-i} \theta\_{(i,k,m)} \frac{\Gamma(k+i+a)}{\Gamma(k+i+a+1)} t^{k+i+a} B\_{j,m}(t) dt, \quad j = 0, 1, 2 \dots m \\ &\Rightarrow X\_{M}^{(j)} = \int\_{0}^{1} \sum\_{k=0}^{m-i} \theta\_{(i,k,m)} \frac{\Gamma(k+i+a)}{\Gamma(k+i+a+1)} \sum\_{l=0}^{m-j} \theta\_{(j,l,m)} t^{k+i+a} x^{l+j} dt, \; j = 0, 1, 2 \dots m \\ &= \sum\_{k=0}^{m-i} \theta\_{(i,k,m)} \sum\_{l=0}^{m-j} \theta\_{(j,l,m)} \frac{\Gamma(k+i+a)}{\Gamma(k+i+a+1)} \frac{1}{(k+l+j+i+a+1)}, \quad j = 0, 1, 2, \dots m \end{split} \tag{15}$$

*Approximate Solutions of Some Boundary Value Problems by Using Operational Matrices… DOI: http://dx.doi.org/10.5772/intechopen.90302*

evaluating this result for i = 0,1,2....m, we have

$$\begin{bmatrix} I^{a}B\_{0,m}(t) \\ I^{a}B\_{1,m}(t) \\ \vdots \\ \vdots \\ \vdots \\ \vdots \\ \vdots \\ I^{a}B\_{m,m}(t) \end{bmatrix} = \begin{bmatrix} X\_{M}^{(0)}\Phi\_{M\times M}^{-1}B\_{M}^{T}(t) \\ X\_{M}^{(1)}\Phi\_{M\times M}^{-1}B\_{M}^{T}(t) \\ \vdots \\ \vdots \\ \vdots \\ \vdots \\ X\_{M}^{(m)}\Phi\_{M\times M}^{-1}B\_{M}^{T}(t) \end{bmatrix} \tag{16}$$

further writing

$$\Psi\_{i,j} = \sum\_{k=0}^{m-i} \sum\_{l=0}^{m-j} \theta\_{(i,k,m)} \cdot \theta\_{(j,l,m)} \frac{\Gamma(k+i+a)}{\Gamma(k+i+a+1)} \frac{1}{(k+l+j+i+a+1)}$$

we get

*I <sup>α</sup>BT*

*Pα*

*<sup>M</sup>*�*<sup>M</sup> is given in* (3) *and P<sup>α</sup>*

*θ*ð Þ *<sup>i</sup>*,*k*,*<sup>m</sup> θ*ð Þ *<sup>j</sup>*,*l*,*<sup>m</sup>*

taking fractional integration of both sides, we have

*θ*ð Þ *<sup>i</sup>*,*k*,*<sup>m</sup> I αt*

Now to approximate right-hand sides of above

*θ*ð Þ *<sup>i</sup>*,*k*,*<sup>m</sup>*

*<sup>M</sup>* can be approximated by using (3) as

Γð Þ *k* þ *i* þ *α* Γð Þ *k* þ *i* þ *α* þ 1

> Γð Þ *k* þ *i* þ *α* Γð Þ *k* þ *i* þ *α* þ 1

> Γð Þ *k* þ *i* þ *α* Γð Þ *k* þ *i* þ *α* þ 1

*C*ð Þ*<sup>i</sup> <sup>M</sup>* <sup>¼</sup> *<sup>X</sup>*ð Þ*<sup>i</sup>*

*P*^*α <sup>M</sup>*�*<sup>M</sup>* ¼

where *P<sup>α</sup>*

*Functional Calculus*

*and* Φ�<sup>1</sup>

where

<sup>Ψ</sup>*<sup>i</sup>*,*<sup>j</sup>* <sup>¼</sup> <sup>X</sup>*<sup>m</sup>*�*<sup>i</sup>*

*Proof.* Consider

*I*

where *C*ð Þ*<sup>i</sup>*

ð1 0 X*m*�*i k*¼0

> ð1 0 X*m*�*i k*¼0

*θ*ð Þ *<sup>i</sup>*,*k*,*<sup>m</sup>*

*X*ð Þ*<sup>j</sup> <sup>M</sup>* ¼

) *<sup>X</sup>*ð Þ*<sup>j</sup> <sup>M</sup>* ¼

<sup>¼</sup> <sup>X</sup>*<sup>m</sup>*�*<sup>i</sup> k*¼0

**170**

*k*¼0

*<sup>α</sup>Bi*,*<sup>m</sup>*ðÞ¼ *<sup>t</sup>* <sup>X</sup>*<sup>m</sup>*�*<sup>i</sup>*

*k*¼0

X*m*�*i k*¼0

where entries of the vector *X*ð Þ*<sup>i</sup>*

*θ*ð Þ *<sup>i</sup>*,*k*,*<sup>m</sup>*

X*m*�*j*

*l*¼0

*θ*ð Þ *<sup>i</sup>*,*k*,*<sup>m</sup>*

*θ*ð Þ *<sup>j</sup>*,*l*,*<sup>m</sup>*

X*m*�*j*

*l*¼0

*<sup>M</sup>*ðÞ¼ *<sup>t</sup> <sup>P</sup><sup>α</sup>*

*<sup>M</sup>*�*<sup>M</sup>* <sup>¼</sup> *<sup>P</sup>*^*<sup>α</sup>*

*Bi*,*<sup>m</sup>*ðÞ¼ *<sup>t</sup>* <sup>X</sup>*<sup>m</sup>*�*<sup>i</sup>*

*<sup>k</sup>*þ*<sup>i</sup>* <sup>¼</sup> <sup>X</sup>*<sup>m</sup>*�*<sup>i</sup>*

*k*¼0

Γð Þ *k* þ *i* þ *α* Γð Þ *k* þ *i* þ *α* þ 1

*t*

*k*¼0

*θ*ð Þ *<sup>i</sup>*,*k*,*<sup>m</sup>*

*t*

*<sup>M</sup>* Φ�<sup>1</sup>

X*m*�*j*

*θ*ð Þ *<sup>j</sup>*,*l*,*<sup>m</sup> t*

*l*¼0

*θ*ð Þ *<sup>i</sup>*,*k*,*<sup>m</sup> t*

*<sup>M</sup>*�*MBT*

*<sup>M</sup>*�*M*Φ�<sup>1</sup> *M*�*M*

Ψð Þ 0,0 Ψð Þ 0,1 ⋯ Ψð Þ 0,*<sup>r</sup>* ⋯ Ψð Þ 0,*<sup>m</sup>* Ψð Þ 1,0 Ψð Þ 1,1 ⋯ Ψð Þ 1,*<sup>r</sup>* ⋯ Ψð Þ 1,*<sup>m</sup>* ⋮ ⋮⋮⋮⋮ ⋮ Ψð Þ *<sup>r</sup>*,0 Ψð Þ *<sup>r</sup>*,1 ⋯ Ψð Þ *<sup>r</sup>*,*<sup>r</sup>* ⋯ Ψð Þ *<sup>r</sup>*,*<sup>m</sup>* ⋮ ⋮⋮⋮⋮ ⋮ Ψð Þ *<sup>m</sup>*,0 Ψð Þ *<sup>m</sup>*,1 ⋯ Ψð Þ *<sup>m</sup>*,*<sup>r</sup>* ⋯ Ψð Þ *<sup>m</sup>*,*<sup>m</sup>*

Γð Þ *k* þ *i* þ 1

ð Þ *<sup>i</sup>* <sup>þ</sup> *<sup>j</sup>* <sup>þ</sup> *<sup>k</sup>* <sup>þ</sup> *<sup>l</sup>* <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> <sup>Γ</sup>ð Þ *<sup>k</sup>* <sup>þ</sup> *<sup>i</sup>* <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> *:* (10)

Γð Þ *k* þ *i* þ *α* Γð Þ *k* þ *i* þ *α* þ 1

*<sup>k</sup>*þ*i*þ*<sup>α</sup>* <sup>¼</sup> *<sup>C</sup>*ð Þ*<sup>i</sup>*

*<sup>M</sup>* can be calculated in generalized form as

*<sup>k</sup>*þ*i*þ*<sup>α</sup><sup>B</sup> <sup>j</sup>*,*<sup>m</sup>*ð Þ*<sup>t</sup> dt*, *<sup>j</sup>* <sup>¼</sup> 0, 1, 2*:* … *<sup>m</sup>*

*<sup>k</sup>*þ*i*þ*<sup>α</sup>:t l*þ*j*

1

*<sup>M</sup> BT*

*<sup>M</sup>*�*<sup>M</sup>* is the fractional integration's operational matrix defined as

*<sup>M</sup>*�*<sup>M</sup> is given by*

*<sup>M</sup>*ð Þ*t* , (8)

*<sup>k</sup>*þ*<sup>i</sup>* (11)

*t*

*<sup>M</sup>*�*<sup>M</sup>*, (14)

ð Þ *<sup>k</sup>* <sup>þ</sup> *<sup>l</sup>* <sup>þ</sup> *<sup>j</sup>* <sup>þ</sup> *<sup>i</sup>* <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> , *<sup>j</sup>* <sup>¼</sup> 0, 1, 2, *:* … *<sup>m</sup>*

*<sup>k</sup>*þ*i*þ*α:* (12)

*<sup>M</sup>*ð Þ*t* (13)

*dt*, *j* ¼ 0, 1, 2*:* … *m*

(15)

, (9)

$$I^a B\_M^T(t) = \hat{P}\_{M \times M}^a \Phi\_{M \times M}^{-1} B\_M^T(t). \tag{17}$$

Let us represent

$$
\hat{P}^{a}\_{\mathcal{M}\times\mathcal{M}} \Phi^{-1}\_{\mathcal{M}\times\mathcal{M}} = P^{a}\_{\mathcal{M}\times\mathcal{M}}
$$

thus

$$
\boldsymbol{P}^{a}\boldsymbol{B}\_{\mathcal{M}}^{T}(\boldsymbol{t}) = \boldsymbol{P}\_{\mathcal{M}\times\mathcal{M}}^{a}\boldsymbol{B}\_{\mathcal{M}}^{T}(\boldsymbol{t}).\tag{18}
$$

□ **Lemma 2.5.** *Let BT <sup>M</sup>*ð Þ*t be the function vector as defined in* (3), *then fractional order derivative is defined as*

$$D^a B\_M^T(\mathbf{t}) = G\_{M \times M}^a B\_M^T(\mathbf{t}) \tag{19}$$

where *G<sup>α</sup> <sup>M</sup>*�*<sup>M</sup>* is the operational matrix of fractional order derivative given by

$$G^{a}\_{\mathcal{M}\times\mathcal{M}} = \hat{G}^{a}\_{\mathcal{M}\times\mathcal{M}} \Phi^{-1}\_{\mathcal{M}\times\mathcal{M}}.\tag{20}$$

*where* Φ*<sup>M</sup>*�*<sup>M</sup> is the dual matrix given in* (3) *and*

$$
\hat{\mathbf{G}}\_{M\times M}^{a} = \begin{bmatrix}
\Psi\_{(0,0)} & \Psi\_{(0,1)} & \cdots & \Psi\_{(0,r)} & \cdots & \Psi\_{(0,m)} \\
\Psi\_{(1,0)} & \Psi\_{(1,1)} & \cdots & \Psi\_{(1,r)} & \cdots & \Psi\_{(1,m)} \\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\
\Psi\_{(r,0)} & \Psi\_{(r,1)} & \cdots & \Psi\_{(r,r)} & \cdots & \Psi\_{(r,m)} \\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\
\Psi\_{(m,0)} & \Psi\_{(m,1)} & \cdots & \Psi\_{(m,r)} & \cdots & \Psi\_{(m,m)} \\
\end{bmatrix}, \tag{21}
$$

where Ψð Þ *<sup>i</sup>*,*<sup>j</sup>* is defined for two different cases as **Case I:** ð Þ *i*<½ � *α*

$$\Psi\_{i,j} = \sum\_{k=\left[a\right]}^{m-i} \sum\_{l=0}^{m-j} \theta\_{(i,k,m)} \cdot \theta\_{(j,l,m)} \frac{\Gamma(k+i-a)}{\Gamma(k+i-a+1)} \frac{1}{(k+l+j+i-a+1)}\tag{22}$$

*<sup>D</sup><sup>α</sup>B*0,*m*ð Þ*<sup>t</sup> <sup>D</sup><sup>α</sup>B*1,*m*ð Þ*<sup>t</sup>* ⋮ ⋮ ⋮ *<sup>D</sup><sup>α</sup>Bm*,*m*ð Þ*<sup>t</sup>*

¼

*X*ð Þ <sup>0</sup> *<sup>M</sup>* Φ�<sup>1</sup> *<sup>M</sup>*�*MBT <sup>M</sup>*ð Þ*t*

*Approximate Solutions of Some Boundary Value Problems by Using Operational Matrices…*

*X*ð Þ<sup>1</sup> *<sup>M</sup>* Φ�<sup>1</sup> *<sup>M</sup>*�*MB<sup>T</sup> <sup>M</sup>*ð Þ*t*

*X*ð Þ *<sup>m</sup> <sup>M</sup>* Φ�<sup>1</sup> *<sup>M</sup>*�*MBT*

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

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

*<sup>M</sup>*ð Þ*t :*

*<sup>M</sup>*ðÞ¼ *<sup>t</sup> <sup>G</sup><sup>α</sup>*

*<sup>M</sup>*�*<sup>M</sup>* <sup>¼</sup> *<sup>G</sup><sup>α</sup>*

which is the desired result. □ **Lemma 2.6.** *An operational matrix corresponding to the boundary condition by*

*u t*ðÞ¼ *KB t* ^ð Þ

*<sup>M</sup>*ð Þ*t is function vector and K is coefficient vector by taking the approximation*

Ωð Þ 0,0 Ωð Þ 0,1 ⋯ Ωð Þ 0,*<sup>r</sup>* ⋯ Ωð Þ 0,*<sup>m</sup>* Ωð Þ 1,0 Ωð Þ 1,1 ⋯ Ωð Þ 1,*<sup>r</sup>* ⋯ Ωð Þ 1,*<sup>m</sup>* ⋮ ⋮⋮⋮⋮ ⋮ Ωð Þ *<sup>r</sup>*,0 Ωð Þ *<sup>r</sup>*,1 ⋯ Ωð Þ *<sup>r</sup>*,*<sup>r</sup>* ⋯ Ωð Þ *<sup>r</sup>*,*<sup>m</sup>* ⋮ ⋮⋮⋮⋮ ⋮ Ωð Þ *<sup>m</sup>*,0 Ωð Þ *<sup>m</sup>*,1 ⋯ Ωð Þ *<sup>m</sup>*,*<sup>r</sup>* ⋯ Ωð Þ *<sup>m</sup>*,*<sup>m</sup>*

Δ*<sup>i</sup>*,*<sup>m</sup>ϕ*ð Þ*t Bj*ð Þ*t dt*, *i*, *j* ¼ 0, 1, 2*:* … *m:*

*<sup>M</sup>*�*MBT <sup>M</sup>*ð Þ*t*

*M*�*M*

*<sup>M</sup>*�*<sup>M</sup>:BT*

⋮ ⋮ ⋮

*<sup>M</sup>*ð ÞÞ *t*

1

1 ð Þ *k* þ *l* þ *j* þ *i* � *α* þ 1

ð Þ *<sup>k</sup>* <sup>þ</sup> *<sup>l</sup>* <sup>þ</sup> *<sup>j</sup>* <sup>þ</sup> *<sup>i</sup>* � *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> ð Þ *<sup>i</sup>*<½ � *<sup>α</sup>*

, (31)

*:* (29)

(30)

*DOI: http://dx.doi.org/10.5772/intechopen.90302*

On further simplification, we have

*θ*ð Þ *<sup>i</sup>*,*k*,*<sup>m</sup> :θ*ð Þ *<sup>j</sup>*,*l*,*<sup>m</sup>*

*θ*ð Þ *<sup>i</sup>*,*k*,*<sup>m</sup> :θ*ð Þ *<sup>j</sup>*,*l*,*<sup>m</sup>*

*<sup>M</sup>*�*<sup>M</sup>*Φ�<sup>1</sup>

*G*^ *α <sup>M</sup>*�*<sup>M</sup>*Φ�<sup>1</sup>

*D<sup>α</sup>BT*

*<sup>M</sup>*ðÞ¼ *<sup>t</sup> <sup>G</sup>*^ *<sup>α</sup>*

<sup>Ψ</sup>*i*,*<sup>j</sup>* <sup>¼</sup> <sup>X</sup>*m*�*<sup>i</sup>*

<sup>Ψ</sup>*<sup>i</sup>*,*<sup>j</sup>* <sup>¼</sup> <sup>X</sup>*<sup>m</sup>*�*<sup>i</sup>*

Let

so

*taking BT*

is given by

where

**173**

*Q<sup>α</sup>*,*<sup>ϕ</sup> <sup>M</sup>*�*<sup>M</sup>* ¼

Ω*<sup>i</sup>*,*<sup>j</sup>* ¼

*Proof.* Let us take *u t*ðÞ¼ *KB t* ^ð Þ, then

ð1 0

*k*¼½ � *α*

*k*¼0

we get *D<sup>α</sup>BT*

X*m*�*j*

*l*¼0

X*m*�*j*

*l*¼0

**Case II:** ð Þ *i*≥ ½ � *α*

$$\Psi\_{i,j} = \sum\_{k=0}^{m-i} \sum\_{l=0}^{m-j} \theta\_{(i,k,m)} \cdot \theta\_{(j,l,m)} \frac{\Gamma(k+i-a)}{\Gamma(k+i-a+1)} \frac{1}{(k+l+j+i-a+1)}.\tag{23}$$

*Proof.* Consider the general element as

$$D^a B\_{i,m}(t) = D^a \left( \sum\_{k=0}^{m-i} \theta\_{(i,k,m)}, t^{k+i} \right) = \sum\_{k=0}^{m-i} \theta\_{(i,k,m)} D^a t^{k+i}.\tag{24}$$

It is to be noted in the polynomial function *Bi*,*<sup>m</sup>* the power of the variable '*t*' is an ascending order and the lowest power is '*i*' therefore the first ½ � *α* � 1 terms becomes zero when we take derivative of order *α:*

**Case I**: ð Þ *i*< ½ � *α* By the use of definition of fractional derivative

$$D^a B\_{i,m}(t) = \sum\_{k=[a]}^{m-i} \theta\_{(i,k,m)} \frac{\Gamma(k+i+1)}{\Gamma(k+i-a+1)} t^{k+i-a}.\tag{25}$$

Now approximating RHS of (25) as

$$\sum\_{k=[a]}^{m-i} \theta\_{(i,k,m)} \frac{\Gamma(k+i+1)}{\Gamma(k+i-a+1)} t^{k+i-a} = \mathbf{C}\_{M}^{(i)} B\_{M}^{T}(t) \tag{26}$$

further implies that

$$X\_M^{(j)} = \int\_0^1 \sum\_{k=[a]}^{m-i} \theta\_{(i,k,m)} \frac{\Gamma(k+i+1)}{\Gamma(k+i-a+1)} t^{k+i-a} B\_{j,m}(t) dt, \; j = 0, 1, 2...m$$

$$\Rightarrow X\_M^{(j)} = \sum\_{k=[a]}^{m-i} \theta\_{(i,k,m)} \sum\_{l=0}^{m-j} \theta\_{(j,l,m)} \frac{\Gamma(k+i+1)}{\Gamma(k+i-a+1)(k+i+l+j-a+1)}, \; j = 0, 1, 2...m \tag{27}$$

**Case II**: ð Þ *i*≥ ½ � *α* if *i*≤½ � *α* then

$$X\_M^{(j)} = \sum\_{k=0}^{m-i} \theta\_{(i,k,m)} \sum\_{l=0}^{m-j} \theta\_{(j,l,m)} \frac{\Gamma(k+i+1)}{\Gamma(k+i-a+1)(k+i+l+j-a+1)}, \quad j = 0, 1, 2, \dots \\ m.s.t. \tag{28}$$

After careful simplification, we get

*Approximate Solutions of Some Boundary Value Problems by Using Operational Matrices… DOI: http://dx.doi.org/10.5772/intechopen.90302*

$$\begin{bmatrix} D^a B\_{0,m}(t) \\ D^a B\_{1,m}(t) \\ \vdots \\ \vdots \\ \vdots \\ \vdots \\ D^a B\_{m,m}(t) \end{bmatrix} = \begin{bmatrix} X\_M^{(0)} \Phi\_{M \times M}^{-1} B\_M^T(t) \\ X\_M^{(1)} \Phi\_{M \times M}^{-1} B\_M^T(t) \\ \vdots \\ \vdots \\ \vdots \\ \vdots \\ X\_M^{(m)} \Phi\_{M \times M}^{-1} B\_M^T(t) \end{bmatrix}. \tag{29}$$

On further simplification, we have

$$\begin{split} \Psi\_{ij} &= \sum\_{k=[a]}^{m-i} \sum\_{l=0}^{m-j} \theta\_{(i,k,m)} \, \theta\_{(j,l,m)} \frac{\Gamma(k+i+1)}{\Gamma(k+i-a+1)} \frac{1}{(k+l+j+i-a+1)} \quad (i < [a]) \\ \Psi\_{ij} &= \sum\_{k=0}^{m-i} \sum\_{l=0}^{m-j} \theta\_{(i,k,m)} \, \theta\_{(j,l,m)} \frac{\Gamma(k+i+1)}{\Gamma(k+i-a+1)} \frac{1}{(k+l+j+i-a+1)} \\ \text{we get } \, D^a \mathbf{B}\_M^T(t) &= \hat{\mathbf{G}}\_{M \times M}^a \Phi\_{M \times M}^{-1} \mathbf{B}\_M^T(t). \end{split} \tag{30}$$

Let

where Ψð Þ *<sup>i</sup>*,*<sup>j</sup>* is defined for two different cases as

*θ*ð Þ *<sup>i</sup>*,*k*,*<sup>m</sup> :θ*ð Þ *<sup>j</sup>*,*l*,*<sup>m</sup>*

*θ*ð Þ *<sup>i</sup>*,*k*,*<sup>m</sup> :θ*ð Þ *<sup>j</sup>*,*l*,*<sup>m</sup>*

*<sup>D</sup><sup>α</sup>Bi*,*<sup>m</sup>*ðÞ¼ *<sup>t</sup> <sup>D</sup><sup>α</sup>* <sup>X</sup>*<sup>m</sup>*�*<sup>i</sup>*

*<sup>D</sup><sup>α</sup>Bi*,*<sup>m</sup>*ðÞ¼ *<sup>t</sup>* <sup>X</sup>*<sup>m</sup>*�*<sup>i</sup>*

*θ*ð Þ *<sup>i</sup>*,*k*,*<sup>m</sup>*

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

*θ*ð Þ *<sup>j</sup>*,*l*,*<sup>m</sup>*

*k*¼0

**Case I**: ð Þ *i*< ½ � *α* By the use of definition of fractional derivative

*θ*ð Þ *<sup>i</sup>*,*k*,*<sup>m</sup>*

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

*t*

*k*¼½ � *α*

*Proof.* Consider the general element as

zero when we take derivative of order *α:*

Now approximating RHS of (25) as

X*m*�*i k*¼½ � *α*

further implies that

*θ*ð Þ *<sup>i</sup>*,*k*,*<sup>m</sup>*

*θ*ð Þ *<sup>i</sup>*,*k*,*<sup>m</sup>*

**Case II**: ð Þ *i*≥ ½ � *α* if *i*≤½ � *α* then

X*m*�*j*

*θ*ð Þ *<sup>j</sup>*,*l*,*<sup>m</sup>*

*l*¼0

After careful simplification, we get

*θ*ð Þ *<sup>i</sup>*,*k*,*<sup>m</sup>*

X*m*�*j l*¼0

X*m*�*i k*¼½ � *α*

*<sup>M</sup>* <sup>¼</sup> <sup>X</sup>*<sup>m</sup>*�*<sup>i</sup> k*¼½ � *α*

*X*ð Þ*<sup>j</sup> <sup>M</sup>* ¼ ð1 0

) *<sup>X</sup>*ð Þ*<sup>j</sup>*

*X*ð Þ*<sup>j</sup> <sup>M</sup>* <sup>¼</sup> <sup>X</sup>*<sup>m</sup>*�*<sup>i</sup> k*¼0

**172**

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

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

*θ*ð Þ *<sup>i</sup>*,*k*,*<sup>m</sup> :t*

!

*k*þ*i*

It is to be noted in the polynomial function *Bi*,*<sup>m</sup>* the power of the variable '*t*' is an ascending order and the lowest power is '*i*' therefore the first ½ � *α* � 1 terms becomes

<sup>¼</sup> <sup>X</sup>*<sup>m</sup>*�*<sup>i</sup> k*¼0

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

*t*

*<sup>k</sup>*þ*i*�*<sup>α</sup>* <sup>¼</sup> *<sup>C</sup>*ð Þ*<sup>i</sup>*

*<sup>k</sup>*þ*i*�*<sup>α</sup>Bj*,*<sup>m</sup>*ð Þ*<sup>t</sup> dt*, *<sup>j</sup>* <sup>¼</sup> 0, 1, 2*:* … *<sup>m</sup>*

Γð Þ *k* þ *i* þ 1

Γð Þ *k* þ *i* þ 1

1

1

*<sup>θ</sup>*ð Þ *<sup>i</sup>*,*k*,*<sup>m</sup> <sup>D</sup>α<sup>t</sup>*

*t*

*<sup>M</sup> BT*

<sup>Γ</sup>ð Þ *<sup>k</sup>* <sup>þ</sup> *<sup>i</sup>* � *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> ð Þ *<sup>k</sup>* <sup>þ</sup> *<sup>i</sup>* <sup>þ</sup> *<sup>l</sup>* <sup>þ</sup> *<sup>j</sup>* � *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> , *<sup>j</sup>* <sup>¼</sup> 0, 1, 2*:* … *<sup>m</sup>*

<sup>Γ</sup>ð Þ *<sup>k</sup>* <sup>þ</sup> *<sup>i</sup>* � *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> ð Þ *<sup>k</sup>* <sup>þ</sup> *<sup>i</sup>* <sup>þ</sup> *<sup>l</sup>* <sup>þ</sup> *<sup>j</sup>* � *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> , *<sup>j</sup>* <sup>¼</sup> 0, 1, 2*:* … *<sup>m</sup>:*

ð Þ *<sup>k</sup>* <sup>þ</sup> *<sup>l</sup>* <sup>þ</sup> *<sup>j</sup>* <sup>þ</sup> *<sup>i</sup>* � *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> (22)

ð Þ *<sup>k</sup>* <sup>þ</sup> *<sup>l</sup>* <sup>þ</sup> *<sup>j</sup>* <sup>þ</sup> *<sup>i</sup>* � *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> *:* (23)

*k*þ*i*

*:* (24)

*<sup>k</sup>*þ*i*�*α:* (25)

*<sup>M</sup>*ð Þ*t* (26)

(27)

(28)

**Case I:** ð Þ *i*<½ � *α*

*Functional Calculus*

<sup>Ψ</sup>*i*,*<sup>j</sup>* <sup>¼</sup> <sup>X</sup>*m*�*<sup>i</sup>*

**Case II:** ð Þ *i*≥ ½ � *α*

<sup>Ψ</sup>*i*,*<sup>j</sup>* <sup>¼</sup> <sup>X</sup>*m*�*<sup>i</sup>*

*k*¼½ � *α*

*k*¼0

X*m*�*j*

*l*¼0

X*m*�*j*

*l*¼0

$$
\hat{G}^{a}\_{M \times M} \Phi^{-1}\_{M \times M} = G^{a}\_{M \times M}
$$

so

$$D^a B\_M^T(t) = G^a\_{M \times M} B\_M^T(t)$$

which is the desired result. □ **Lemma 2.6.** *An operational matrix corresponding to the boundary condition by taking BT <sup>M</sup>*ð Þ*t is function vector and K is coefficient vector by taking the approximation*

$$u(t) = K\hat{B}(t)$$

is given by

$$\mathbf{Q}\_{M\times M}^{a,\phi} = \begin{bmatrix} \boldsymbol{\Omega}\_{(0,0)} & \boldsymbol{\Omega}\_{(0,1)} & \cdots & \boldsymbol{\Omega}\_{(0,r)} & \cdots & \boldsymbol{\Omega}\_{(0,m)} \\ \boldsymbol{\Omega}\_{(1,0)} & \boldsymbol{\Omega}\_{(1,1)} & \cdots & \boldsymbol{\Omega}\_{(1,r)} & \cdots & \boldsymbol{\Omega}\_{(1,m)} \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ \boldsymbol{\Omega}\_{(r,0)} & \boldsymbol{\Omega}\_{(r,1)} & \cdots & \boldsymbol{\Omega}\_{(r,r)} & \cdots & \boldsymbol{\Omega}\_{(r,m)} \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ \boldsymbol{\Omega}\_{(m,0)} & \boldsymbol{\Omega}\_{(m,1)} & \cdots & \boldsymbol{\Omega}\_{(m,r)} & \cdots & \boldsymbol{\Omega}\_{(m,m)} \end{bmatrix},\tag{31}$$

where

$$\mathfrak{Q}\_{i\dot{j}} = \int\_0^1 \Delta\_{i,m} \phi(t) B\_{\dot{j}}(t) dt, \quad i, j = 0, 1, 2 \dots m.$$

*Proof.* Let us take *u t*ðÞ¼ *KB t* ^ð Þ, then

$$\,\_0I\_1^\alpha K\hat{B}(t) = K\_0I\_1^\alpha \hat{B}(t) = K \begin{bmatrix} \,\_0I\_1^\alpha B\_0(t) \\ \,\_0I\_1^\alpha B\_1(t) \\ \vdots \\ \vdots \\ \vdots \\ \,\_0I\_1^\alpha B\_m(t) \end{bmatrix}.$$

Let us evaluate the general terms

$$\begin{split} \, \_0I\_1^a B\_i(t)dt &= \frac{1}{\Gamma \alpha} \Big|\_0^1 (1-s)^{a-1} B\_{i,m}(s) ds \\ &= \frac{1}{\Gamma \alpha} \sum\_{k=0}^{m-i} \Theta\_{i,k,m} \Big|\_0^1 (1-s)^{a-1} s^{k+i} ds. \end{split} \tag{32}$$

On further simplification, we get

*DOI: http://dx.doi.org/10.5772/intechopen.90302*

> *ϕ*ð Þ*t* <sup>0</sup>*I α*

Ω*<sup>i</sup>*,*<sup>j</sup>* ¼

**3. Applications of operational matrices**

in the following procedure.

ential equation

**175**

**3.1 Fractional differential equations**

*<sup>D</sup><sup>α</sup>y t*ð Þþ *AD<sup>μ</sup>*

ð1 0

Ωð Þ 0,0 Ωð Þ 0,1 ⋯ Ωð Þ 0,*<sup>r</sup>* ⋯ Ωð Þ 0,*<sup>m</sup>* Ωð Þ 1,0 Ωð Þ 1,1 ⋯ Ωð Þ 1,*<sup>r</sup>* ⋯ Ωð Þ 1,*<sup>m</sup>* ⋮ ⋮⋮⋮⋮ ⋮ Ωð Þ *<sup>r</sup>*,0 Ωð Þ *<sup>r</sup>*,1 ⋯ Ωð Þ *<sup>r</sup>*,*<sup>r</sup>* ⋯ Ωð Þ *<sup>r</sup>*,*<sup>m</sup>* ⋮ ⋮⋮⋮⋮ ⋮ Ωð Þ *<sup>m</sup>*,0 Ωð Þ *<sup>m</sup>*,1 ⋯ Ωð Þ *<sup>m</sup>*,*<sup>r</sup>* ⋯ Ωð Þ *<sup>m</sup>*,*<sup>m</sup>*

*Approximate Solutions of Some Boundary Value Problems by Using Operational Matrices…*

<sup>1</sup>*u t*ðÞ¼ *KQ<sup>α</sup>*,*<sup>ϕ</sup>*

which is the required result. □

In this section, we are going to approximate a boundary value problem of fractional order differential equation as well as a coupled system of fractional order boundary value problem. The application of obtained operational matrices is shown

Consider the following problem in generalized form of fractional order differ-

*y t*ðÞþ *By t*ðÞ¼ *f t*ð Þ, 1< *α*≤2, 0 <*μ*≤1, subject to the boundary conditions *<sup>y</sup>*ð Þ¼ <sup>0</sup> *<sup>a</sup>*, *<sup>y</sup>*ð Þ¼ <sup>1</sup> *<sup>b</sup>*, (37)

where *f t*ð Þ is a source term; *A*, *B* are any real constants and *y t*ð Þ is an unknown solution which we want to determine. To obtain a numerical solution of the above

Let *<sup>D</sup><sup>α</sup>y t*ðÞ¼ *KMB<sup>T</sup>*

*<sup>M</sup>*�*MBT*

*<sup>M</sup>*ð Þ�*t c*<sup>0</sup> þ *c*1*t*

*<sup>M</sup>*�*MBT <sup>M</sup>*ð Þ*t* � � *t*¼1 *:*

problem in terms of Bernstein polynomials, we proceed as

Applying fractional integral of order *α* we have

using boundary conditions, we have

*y t*ðÞ¼ *KMP<sup>α</sup>*

*<sup>c</sup>*<sup>0</sup> <sup>¼</sup> *<sup>a</sup>*, *<sup>c</sup>*<sup>1</sup> <sup>¼</sup> *<sup>b</sup>* � *<sup>a</sup>* � *KMP<sup>α</sup>*

*<sup>M</sup>*�*<sup>M</sup>BT*

*<sup>M</sup>*ð Þ*t* ,

Δ*<sup>i</sup>*,*<sup>m</sup>ϕ*ð Þ*t Bj*ð Þ*t dt*, *i*, *j* ¼ 0, 1, 2*:* … *m:* (36)

*<sup>M</sup>*ð Þ*t :* (38)

*B*0ð Þ*t B*1ð Þ*t* ⋮ ⋮ ⋮ *Bm*ð Þ*t*

*:* (35)

*<sup>ϕ</sup>*ð Þ*<sup>t</sup> KI<sup>α</sup>B t* ^ðÞ¼ *<sup>K</sup>*

So

and

Since by

$$L\left(\int\_0^1 (\mathbf{1} - s)^{a-1} s^{k+i} ds\right) = \frac{\Gamma a \Gamma(k + i + \mathbf{1})}{\tau^{k+a+i}}$$

taking inverse Laplace of both sides, we get

$$\int\_0^1 (\mathbf{1} - \mathbf{s})^{a-1} \mathbf{s}^{k+i} d\mathbf{s} = L^{-1} \left[ \frac{\Gamma a.\Gamma(k+i+\mathbf{1})}{\pi^{k+a+i}} \right] = \frac{\Gamma a.\Gamma(k+i+\mathbf{1})}{\Gamma(k+i+a+\mathbf{1})}$$

now Eq. (32) implies that

$${}\_{0}I\_{1}^{a}B\_{i}(t)dt = \sum\_{k=0}^{m-i} \Theta\_{i,k,m} \frac{\Gamma(k+i+1)}{\Gamma(k+i+a+1)} = \Delta\_{i,m}.\tag{33}$$

Now using the approximation <sup>Δ</sup>*<sup>i</sup>*,*<sup>m</sup>ϕ*ðÞ¼ *<sup>t</sup>* <sup>P</sup>*<sup>m</sup> <sup>i</sup>*¼<sup>0</sup>^*ciBi*ðÞ¼ *<sup>t</sup> <sup>C</sup><sup>i</sup> MBT <sup>M</sup>*, and using Eq. (3) we get *C<sup>i</sup> <sup>M</sup>* <sup>¼</sup> *<sup>K</sup><sup>i</sup> M*Φ�<sup>1</sup> *<sup>M</sup>*�*MBT <sup>M</sup>* and using *cj* <sup>¼</sup> <sup>Ð</sup> <sup>1</sup> <sup>0</sup>*ϕ*ð Þ*t Bj*ð Þ*t dt*,

*<sup>ϕ</sup>*ð Þ*<sup>t</sup> KI<sup>α</sup>B t* ^ð Þ¼ *<sup>K</sup>* Δ0,*<sup>m</sup>ϕ*ð Þ*t* Δ1,*<sup>m</sup>ϕ*ð Þ*t* ⋮ ⋮ ⋮ Δ*<sup>m</sup>*,*<sup>m</sup>ϕ*ð Þ*t* 2 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 5 ¼ *K C*0 *M*Φ�<sup>1</sup> *<sup>M</sup>*�*MBT <sup>M</sup>*ð Þ*t C*1 *M*Φ�<sup>1</sup> *<sup>M</sup>*�*MBT <sup>M</sup>*ð Þ*t* ⋮ ⋮ ⋮ *Cm M*Φ�<sup>1</sup> *<sup>M</sup>*�*MBT <sup>M</sup>*ð Þ*t* 2 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 5 ¼ *K c*0 <sup>0</sup> *c*<sup>0</sup> <sup>1</sup> ⋯ *c*<sup>0</sup> *<sup>r</sup>* ⋯ *c*<sup>0</sup> *m c*1 <sup>0</sup> *c*<sup>1</sup> <sup>1</sup> ⋯ *c*<sup>1</sup> *<sup>r</sup>* ⋯ *c*<sup>1</sup> *m* ⋮⋮⋮⋮⋮⋮ *cr* <sup>0</sup> *c<sup>r</sup>* <sup>1</sup> ⋯ *c<sup>r</sup> <sup>r</sup>* ⋯ *c<sup>r</sup> m* ⋮⋮⋮⋮⋮⋮ *cm* <sup>0</sup> *cm* <sup>1</sup> ⋯ *c<sup>m</sup> <sup>r</sup>* ⋯ *c<sup>m</sup> m* 2 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 5 Φ�<sup>1</sup> *<sup>M</sup>*�*MBT <sup>M</sup>*ð Þ*t* Φ�<sup>1</sup> *<sup>M</sup>*�*MBT <sup>M</sup>*ð Þ*t* ⋮ ⋮ ⋮ Φ�<sup>1</sup> *<sup>M</sup>*�*MBT <sup>M</sup>*ð Þ*t* 2 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 5 *:* (34)

**174**

*Approximate Solutions of Some Boundary Value Problems by Using Operational Matrices… DOI: http://dx.doi.org/10.5772/intechopen.90302*

On further simplification, we get

$$
\phi(t)K\Omega^{a}\hat{B}(t) = K \begin{bmatrix}
\Omega\_{(0,0)} & \Omega\_{(0,1)} & \cdots & \Omega\_{(0,r)} & \cdots & \Omega\_{(0,m)} \\
\Omega\_{(1,0)} & \Omega\_{(1,1)} & \cdots & \Omega\_{(1,r)} & \cdots & \Omega\_{(1,m)} \\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\
\Omega\_{(r,0)} & \Omega\_{(r,1)} & \cdots & \Omega\_{(r,r)} & \cdots & \Omega\_{(r,m)} \\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\
\Omega\_{(m,0)} & \Omega\_{(m,1)} & \cdots & \Omega\_{(m,r)} & \cdots & \Omega\_{(m,m)} \\
\end{bmatrix} \begin{bmatrix}
B\_{0}(t) \\
B\_{1}(t) \\
\vdots \\
\vdots \\
\vdots \\
B\_{m}(t)
\end{bmatrix}.\tag{35}$$

So

0*I α* <sup>1</sup>*KB t*

Let us evaluate the general terms

Since by

*Functional Calculus*

ð1 0

Eq. (3) we get *C<sup>i</sup>*

**174**

*<sup>ϕ</sup>*ð Þ*<sup>t</sup> KI<sup>α</sup>B t*

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

0*I α*

*<sup>M</sup>* <sup>¼</sup> *<sup>K</sup><sup>i</sup>*

^ð Þ¼ *<sup>K</sup>*

¼ *K*

now Eq. (32) implies that

0*I α*

> *L* ð1 0

taking inverse Laplace of both sides, we get

*:s k*þ*i*

<sup>1</sup>*Bi*ð Þ*<sup>t</sup> dt* <sup>¼</sup> <sup>X</sup>*<sup>m</sup>*�*<sup>i</sup>*

Now using the approximation <sup>Δ</sup>*<sup>i</sup>*,*<sup>m</sup>ϕ*ðÞ¼ *<sup>t</sup>* <sup>P</sup>*<sup>m</sup>*

*M*Φ�<sup>1</sup> *<sup>M</sup>*�*MBT*

Δ0,*<sup>m</sup>ϕ*ð Þ*t* Δ1,*<sup>m</sup>ϕ*ð Þ*t* ⋮ ⋮ ⋮

Δ*<sup>m</sup>*,*<sup>m</sup>ϕ*ð Þ*t*

*c*0 <sup>0</sup> *c*<sup>0</sup>

*c*1 <sup>0</sup> *c*<sup>1</sup>

*cr* <sup>0</sup> *c<sup>r</sup>*

*cm* <sup>0</sup> *cm*

*k*¼0

¼ *K*

<sup>1</sup> ⋯ *c*<sup>0</sup>

<sup>1</sup> ⋯ *c*<sup>1</sup>

<sup>1</sup> ⋯ *c<sup>r</sup>*

<sup>1</sup> ⋯ *c<sup>m</sup>*

⋮⋮⋮⋮⋮⋮

⋮⋮⋮⋮⋮⋮

<sup>1</sup>*Bi*ð Þ*<sup>t</sup> dt* <sup>¼</sup> <sup>1</sup>

Γ*α* ð1 0

¼ 1 Γ*α* X*m*�*i k*¼0

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

� �

*s k*þ*i ds*

*ds* <sup>¼</sup> *<sup>L</sup>*�<sup>1</sup> <sup>Γ</sup>*α:*Γð Þ *<sup>k</sup>* <sup>þ</sup> *<sup>i</sup>* <sup>þ</sup> <sup>1</sup>

Θ*<sup>i</sup>*,*k*,*<sup>m</sup>*

*<sup>M</sup>* and using *cj* <sup>¼</sup> <sup>Ð</sup> <sup>1</sup>

*C*0 *M*Φ�<sup>1</sup> *<sup>M</sup>*�*MBT <sup>M</sup>*ð Þ*t*

*C*1 *M*Φ�<sup>1</sup> *<sup>M</sup>*�*MBT <sup>M</sup>*ð Þ*t*

*Cm M*Φ�<sup>1</sup> *<sup>M</sup>*�*MBT <sup>M</sup>*ð Þ*t*

*<sup>r</sup>* ⋯ *c*<sup>0</sup>

*<sup>r</sup>* ⋯ *c*<sup>1</sup>

*<sup>r</sup>* ⋯ *c<sup>r</sup>*

*<sup>r</sup>* ⋯ *c<sup>m</sup> m*

*τ<sup>k</sup>*þ*α*þ*<sup>i</sup>* � �

Γð Þ *k* þ *i* þ 1

⋮ ⋮ ⋮

*m*

Φ�<sup>1</sup> *<sup>M</sup>*�*MBT <sup>M</sup>*ð Þ*t*

Φ�<sup>1</sup> *<sup>M</sup>*�*MBT <sup>M</sup>*ð Þ*t*

Φ�<sup>1</sup> *<sup>M</sup>*�*MBT <sup>M</sup>*ð Þ*t*

⋮ ⋮ ⋮

(34)

*:*

*m*

*m*

*<sup>i</sup>*¼<sup>0</sup>^*ciBi*ðÞ¼ *<sup>t</sup> <sup>C</sup><sup>i</sup>*

<sup>0</sup>*ϕ*ð Þ*t Bj*ð Þ*t dt*,

^ðÞ¼ *<sup>K</sup>*0*<sup>I</sup>*

*α* <sup>1</sup>*B t* ^ðÞ¼ *<sup>K</sup>*

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

Θ*i*,*k*,*<sup>m</sup>* ð1 0 0*I α* <sup>1</sup>*B*0ð Þ*t*

*:*

0*I α* <sup>1</sup>*B*1ð Þ*t* ⋮ ⋮ ⋮

0*I α* <sup>1</sup>*Bm*ð Þ*t*

*Bi*,*m*ð Þ*s ds*

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

<sup>¼</sup> <sup>Γ</sup>*α*Γð Þ *<sup>k</sup>* <sup>þ</sup> *<sup>i</sup>* <sup>þ</sup> <sup>1</sup> *τ<sup>k</sup>*þ*α*þ*<sup>i</sup>*

*s k*þ*i ds:*

<sup>¼</sup> <sup>Γ</sup>*α:*Γð Þ *<sup>k</sup>* <sup>þ</sup> *<sup>i</sup>* <sup>þ</sup> <sup>1</sup> Γð Þ *k* þ *i* þ *α* þ 1

<sup>Γ</sup>ð Þ *<sup>k</sup>* <sup>þ</sup> *<sup>i</sup>* <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> <sup>¼</sup> <sup>Δ</sup>*<sup>i</sup>*,*<sup>m</sup>:* (33)

*MBT*

*<sup>M</sup>*, and using

(32)

$$
\phi(t)\_0 I\_1^a \mu(t) = K Q\_{M \times M}^{a, \phi} B\_M^T(t),
$$

and

$$\Omega\_{i,j} = \int\_0^1 \Delta\_{i,m} \phi(t) B\_j(t) dt, \quad i, j = 0, 1, 2 \dots m. \tag{36}$$

which is the required result. □

#### **3. Applications of operational matrices**

In this section, we are going to approximate a boundary value problem of fractional order differential equation as well as a coupled system of fractional order boundary value problem. The application of obtained operational matrices is shown in the following procedure.

#### **3.1 Fractional differential equations**

Consider the following problem in generalized form of fractional order differential equation

$$\begin{aligned} D^a \jmath(t) + A D^a \jmath(t) + B \jmath(t) &= f(t), & 1 &< a \le 2, \ 0 < \mu \le 1, \\ \text{subject to the boundary conditions } \jmath(0) &= a, & \jmath(1) &= b, \end{aligned} \tag{37}$$

where *f t*ð Þ is a source term; *A*, *B* are any real constants and *y t*ð Þ is an unknown solution which we want to determine. To obtain a numerical solution of the above problem in terms of Bernstein polynomials, we proceed as

$$\text{Let } D^a \boldsymbol{\jmath}(\mathbf{t}) = \boldsymbol{K}\_M \boldsymbol{B}\_M^T(\mathbf{t}). \tag{38}$$

Applying fractional integral of order *α* we have

$$y(t) = K\_M P\_{M \times M}^a B\_M^T(t) - c\_0 + c\_1 t$$

using boundary conditions, we have

$$\mathcal{c}\_0 = \mathfrak{a}, \quad \mathcal{c}\_1 = b - \mathfrak{a} - K\_M P\_{M \times M}^{\mathfrak{a}} B\_M^T(t)|\_{t=1}.$$

**175**

Using the approximation and Lemma 2.2

$$\mathfrak{a} + \mathfrak{t}(\mathfrak{b} - \mathfrak{a}) = \mathbf{F}\_{\mathcal{M}}^{(1)} \mathbf{B}\_{\mathcal{M}}^{T}(\mathfrak{t}), \quad \mathfrak{t} \mathbf{P}\_{\mathcal{M} \times \mathcal{M}}^{\mu} \mathbf{B}\_{\mathcal{M}}^{T}(\mathfrak{t}) \Big|\_{\mathfrak{t} = \mathbf{1}} = \mathbf{Q}\_{\mathcal{M} \times \mathcal{M}}^{a, \phi} \mathbf{B}\_{\mathcal{M}}^{T}(\mathfrak{t}). \mathbf{1}$$

Hence

$$\boldsymbol{y}(t) = \boldsymbol{K}\_{M}\boldsymbol{P}\_{M\times M}^{\boldsymbol{a}}\boldsymbol{B}\_{M}^{T}(t) + \boldsymbol{a} + \boldsymbol{t}(\boldsymbol{b} - \boldsymbol{a}) - \boldsymbol{t}\boldsymbol{K}\_{M}\boldsymbol{P}\_{M\times M}^{\boldsymbol{a}}\boldsymbol{B}\_{M}^{T}(t)\big|\_{t=1},$$

$$\text{which gives } \boldsymbol{y}(t) = \boldsymbol{K}\_{M}\boldsymbol{P}\_{M\times M}^{\boldsymbol{a}}\boldsymbol{B}\_{M}^{T}(t) + \boldsymbol{F}\_{M}^{(1)}\boldsymbol{B}\_{M}^{T}(t) - \boldsymbol{Q}\_{M\times M}^{\boldsymbol{a},\phi}\boldsymbol{B}\_{M}^{T}(t) \tag{39}$$

$$= \boldsymbol{K}\_{M}\left(\boldsymbol{P}\_{M\times M}^{\boldsymbol{a}} - \boldsymbol{Q}\_{M\times M}^{\boldsymbol{a},\phi}\right)\boldsymbol{B}\_{M}^{T}(t) + \boldsymbol{F}\_{M}^{(1)}\boldsymbol{B}\_{M}^{T}(t).$$

Now

$$\begin{split} D^{\mu}\boldsymbol{\chi}(t) &= \boldsymbol{D}^{\mu} \left[ \boldsymbol{K}\_{M} \left( \boldsymbol{P}\_{M \times M}^{a} - \boldsymbol{Q}\_{M \times M}^{a, \phi} \right) \boldsymbol{B}\_{M}^{T}(t) + \boldsymbol{F}\_{M}^{(1)} \boldsymbol{B}\_{M}^{T}(t) \right] \\ &= \boldsymbol{K}\_{M} \left( \boldsymbol{P}\_{M \times M}^{a} - \boldsymbol{Q}\_{M \times M}^{a, \phi} \right) \boldsymbol{G}\_{M \times M}^{\mu} \boldsymbol{B}\_{M}^{T}(t) + \boldsymbol{F}\_{M}^{(1)} \boldsymbol{G}\_{M \times M}^{\mu} \boldsymbol{B}\_{M}^{T}(t) \end{split} \tag{40}$$

and

$$f(\mathbf{t}) = F\_M^{(2)} B\_M^T(\mathbf{t}). \tag{41}$$

subject to the boundary conditions

*DOI: http://dx.doi.org/10.5772/intechopen.90302*

*<sup>M</sup>*ð Þ*<sup>t</sup>* , *<sup>D</sup><sup>β</sup>*

*<sup>M</sup>*�*MBT*

*<sup>M</sup>*�*<sup>M</sup>BT*

*MBT*

*<sup>M</sup>*�*MB<sup>T</sup>*

*<sup>M</sup>*�*<sup>M</sup>BT*

*<sup>M</sup>*�*MBT*

*<sup>M</sup>*�*<sup>M</sup>* � *<sup>Q</sup>α*,*<sup>ϕ</sup>*

*<sup>M</sup>*�*<sup>M</sup>BT*

*<sup>M</sup>*�*<sup>M</sup>* � *<sup>Q</sup><sup>β</sup>*,*<sup>ϕ</sup>*

*<sup>M</sup>*�*MBT*

*<sup>M</sup>*�*<sup>M</sup>* � *<sup>Q</sup><sup>α</sup>*,*<sup>ϕ</sup>*

*<sup>M</sup>*�*<sup>M</sup>BT*

*<sup>M</sup>*�*<sup>M</sup>* � *<sup>Q</sup><sup>β</sup>*,*<sup>ϕ</sup>*

*<sup>M</sup>*�*<sup>M</sup>BT*

*<sup>M</sup>*ðÞþ*<sup>t</sup> <sup>F</sup>*ð Þ<sup>1</sup>

*<sup>M</sup>*ðÞþ*<sup>t</sup> <sup>F</sup>*ð Þ<sup>2</sup>

*<sup>M</sup>*ðÞþ*<sup>t</sup> <sup>F</sup>*ð Þ<sup>1</sup>

*M*�*M* � �*Gμ*<sup>1</sup>

*M*�*M* � �*G<sup>ν</sup>*<sup>1</sup>

*M*�*M* � �*G<sup>μ</sup>*<sup>2</sup>

*M*�*M* � �*G<sup>ν</sup>*<sup>2</sup>

*<sup>M</sup>*ð Þ*<sup>t</sup>* , *g t*ðÞ¼ *<sup>F</sup>*ð Þ <sup>4</sup> *BT*

*<sup>M</sup>*ðÞþ*<sup>t</sup> <sup>F</sup>*ð Þ<sup>2</sup>

*<sup>M</sup>*ðÞþ*<sup>t</sup> <sup>F</sup>*ð Þ<sup>1</sup>

*<sup>M</sup>*ðÞþ*<sup>t</sup> <sup>F</sup>*ð Þ<sup>2</sup>

*<sup>M</sup>*ð Þ*<sup>t</sup> <sup>t</sup>*¼<sup>1</sup> <sup>¼</sup> *<sup>Q</sup>α*,*<sup>ϕ</sup>*

� � �

*x t*ðÞ¼ *KMP<sup>α</sup>*

*y t*ðÞ¼ *LMP<sup>β</sup>*

*<sup>D</sup>μ*1*x t*ðÞ¼ *KMP<sup>α</sup>*

<sup>¼</sup> *KM <sup>P</sup><sup>α</sup>*

<sup>¼</sup> *LM <sup>P</sup><sup>β</sup>*

<sup>¼</sup> *KM <sup>P</sup><sup>α</sup>*

*<sup>D</sup><sup>ν</sup>*<sup>1</sup> *y t*ðÞ¼ *<sup>D</sup><sup>ν</sup>*<sup>1</sup> *LMP<sup>β</sup>*

*<sup>D</sup><sup>μ</sup>*2*x t*ðÞ¼ *<sup>D</sup><sup>μ</sup>*<sup>2</sup> *KMP<sup>α</sup>*

*<sup>D</sup><sup>ν</sup>*<sup>2</sup> *y t*ðÞ¼ *<sup>D</sup><sup>ν</sup>*<sup>2</sup> *KMP<sup>β</sup>*

*f t*ðÞ¼ *<sup>F</sup>*ð Þ<sup>3</sup> *BT*

<sup>¼</sup> *LM <sup>P</sup><sup>β</sup>*

*<sup>M</sup>*�*MB<sup>T</sup>*

applying boundary conditions, we have

*x t*ðÞ¼ *KM*ð*P<sup>α</sup>*

*y t*ðÞ¼ *KM*ð*P<sup>β</sup>*

*<sup>a</sup>* <sup>þ</sup> *t b*ð Þ¼ � *<sup>a</sup> <sup>F</sup>*<sup>1</sup>

polynomials such as

*<sup>D</sup><sup>α</sup>x t*ðÞ¼ *KMBT*

*x t*ðÞ¼ *KMP<sup>α</sup>*

let us approximate

*tP<sup>α</sup> <sup>M</sup>*�*MBT*

then

and

**177**

*x*ð Þ¼ 0 *a*, *x*ð Þ¼ 1 *b y*ð Þ¼ 0 *c*, *y*ð Þ¼ 1 *d*, (45)

*<sup>M</sup>*�*<sup>M</sup>BT*

*<sup>M</sup>*�*MBT <sup>M</sup>*ð Þ*t* � � *t*¼1 ,

*<sup>M</sup>*�*<sup>M</sup>BT <sup>M</sup>*ð Þ*t* � � � *t*¼1 *:*

<sup>¼</sup> *<sup>Q</sup>β*,*<sup>ϕ</sup> <sup>M</sup>*�*<sup>M</sup>B<sup>T</sup> <sup>M</sup>*ð Þ*t*

*<sup>M</sup>*�*<sup>M</sup>BT <sup>M</sup>*ð Þ*t*

*<sup>M</sup>*�*<sup>M</sup>BT <sup>M</sup>*ð Þ*t*

> *<sup>M</sup>*�*<sup>M</sup>BT <sup>M</sup>*ð Þ*t*

*<sup>M</sup>*�*<sup>M</sup>BT <sup>M</sup>*ð Þ*t*

*<sup>M</sup>*�*<sup>M</sup>BT <sup>M</sup>*ð Þ*t*

*<sup>M</sup>*�*<sup>M</sup>BT <sup>M</sup>*ð Þ*t*

*<sup>M</sup>*�*<sup>M</sup>B<sup>T</sup> <sup>M</sup>*ð Þ*t*

*<sup>M</sup>*�*<sup>M</sup>BT <sup>M</sup>*ð Þ*t*

�

*MBT <sup>M</sup>*ð Þ*t*

*<sup>M</sup>*ðÞ�*<sup>t</sup> KMQ<sup>α</sup>*,*<sup>ϕ</sup>*

*<sup>M</sup>*ðÞ�*<sup>t</sup> LMQ<sup>β</sup>*,*<sup>ϕ</sup>*

*<sup>M</sup>*�*<sup>M</sup>* <sup>þ</sup> *<sup>F</sup>*ð Þ<sup>1</sup>

*<sup>M</sup> BT*

*<sup>M</sup> B<sup>T</sup>*

*<sup>M</sup> BT*

h i

*<sup>M</sup>*�*<sup>M</sup>* <sup>þ</sup> *<sup>F</sup>*ð Þ<sup>2</sup>

*<sup>M</sup>*ð Þ*t :*

h i

*<sup>M</sup>*�*<sup>M</sup>* <sup>þ</sup> *<sup>F</sup>*ð Þ<sup>1</sup>

h i

*<sup>M</sup>*�*<sup>M</sup>* <sup>þ</sup> *<sup>F</sup>*ð Þ<sup>2</sup>

*<sup>M</sup>*ð Þ�*<sup>t</sup> KMQ<sup>α</sup>*,*<sup>ϕ</sup>*

*<sup>M</sup> <sup>G</sup>μ*<sup>1</sup>

*<sup>M</sup>*ð Þ�*<sup>t</sup> LMQ<sup>β</sup>*,*<sup>ϕ</sup>*

*<sup>M</sup> <sup>G</sup><sup>ν</sup>*<sup>1</sup>

*<sup>M</sup>*ðÞ�*<sup>t</sup> KMQ<sup>α</sup>*,*<sup>ϕ</sup>*

*<sup>M</sup> <sup>G</sup><sup>μ</sup>*<sup>2</sup> *<sup>M</sup>*�*<sup>M</sup>BT <sup>M</sup>*ð Þ*t*

*<sup>M</sup>*ð Þ�*<sup>t</sup> KMQ<sup>β</sup>*,*<sup>ϕ</sup>*

*<sup>M</sup> <sup>G</sup><sup>ν</sup>*<sup>2</sup>

*<sup>M</sup>*�*<sup>M</sup>BT <sup>M</sup>*ð Þ*t*

� � � *t*¼1

*<sup>M</sup>*�*<sup>M</sup>BT <sup>M</sup>*ð Þ*t* *<sup>M</sup>*ðÞþ*t d*<sup>0</sup> þ *d*1*t*

where *Ai*, *Bi*,*Ci*, *Di*ð Þ *i* ¼ 1, 2 are any real constants, *f t*ð Þ, *g t*ð Þ are given source terms. We approximate the solution of the above system in terms of Bernstein

*Approximate Solutions of Some Boundary Value Problems by Using Operational Matrices…*

*<sup>M</sup>*ð Þ*t*

*<sup>M</sup>*ðÞþ*<sup>t</sup> <sup>a</sup>* <sup>þ</sup> *t b*ð Þ� � *<sup>a</sup> tKMP<sup>α</sup>*

*<sup>M</sup>*ðÞþ*<sup>t</sup> <sup>c</sup>* <sup>þ</sup> *t d*ð Þ� � *<sup>c</sup> tKMP<sup>β</sup>*

*<sup>M</sup>*ð Þ*<sup>t</sup>* ,*<sup>c</sup>* <sup>þ</sup> *t d*ð Þ¼ � *<sup>c</sup> <sup>F</sup>*<sup>2</sup>

*<sup>M</sup>*ð Þ*<sup>t</sup>* , *tP<sup>β</sup>*

*<sup>M</sup> BT*

*<sup>M</sup> B<sup>T</sup>*

*<sup>M</sup> BT*

h i

*y t*ðÞ¼ *LMBT*

*<sup>M</sup>*ðÞþ*<sup>t</sup> <sup>c</sup>*<sup>0</sup> <sup>þ</sup> *<sup>c</sup>*1*t*, *y t*ðÞ¼ *LM <sup>P</sup><sup>β</sup>*

Putting Eqs. (38)–(41) in Eq. (37), we get

$$\begin{split} \boldsymbol{K}\_{M}\boldsymbol{B}\_{M}^{T}(\boldsymbol{t}) + \boldsymbol{A}\boldsymbol{K}\_{M} \Big( \boldsymbol{P}\_{M \times M}^{a} - \boldsymbol{Q}\_{M \times M}^{a, \phi} \Big) \mathbf{G}\_{M \times M}^{\mu} \boldsymbol{B}\_{M}^{T}(\boldsymbol{t}) + \boldsymbol{A}\boldsymbol{F}\_{M}^{(1)} \boldsymbol{G}\_{M \times M}^{\mu} \boldsymbol{B}\_{M}^{T}(\boldsymbol{t}) \\ + \boldsymbol{B}\boldsymbol{K}\_{M} \Big( \boldsymbol{P}\_{M \times M}^{a} - \boldsymbol{Q}\_{M \times M}^{a, \phi} \Big) \boldsymbol{B}\_{M}^{T}(\boldsymbol{t}) + \boldsymbol{B}\boldsymbol{F}\_{M}^{(1)} \boldsymbol{B}\_{M}^{T}(\boldsymbol{t}) = \boldsymbol{F}\_{M}^{(2)} \boldsymbol{B}\_{M}^{T}(\boldsymbol{t}). \end{split} \tag{42}$$

In simple form, we can write (42) as

$$\begin{split} \boldsymbol{K}\_{M}\boldsymbol{B}\_{M}^{T}(\boldsymbol{t}) + \boldsymbol{A}\boldsymbol{K}\_{M} \Big(\boldsymbol{P}\_{M\times M}^{a} - \boldsymbol{Q}\_{M\times M}^{a,\phi}\Big) \boldsymbol{G}\_{M\times M}^{\mu}\boldsymbol{B}\_{M}^{T}(\boldsymbol{t}) + \boldsymbol{A}\boldsymbol{F}\_{M}^{(1)}\boldsymbol{G}\_{M\times M}^{\mu}\boldsymbol{B}\_{M}^{T}(\boldsymbol{t}) \\ + \boldsymbol{B}\boldsymbol{K}\_{M} \Big(\boldsymbol{P}\_{M\times M}^{\mu} - \boldsymbol{Q}\_{M\times M}^{a,\phi}\Big) \boldsymbol{B}\_{M}^{T}(\boldsymbol{t}) + \boldsymbol{B}\boldsymbol{F}\_{M}^{(1)}\boldsymbol{B}\_{M}^{T}(\boldsymbol{t}) - \boldsymbol{F}\_{M}^{(2)}\boldsymbol{B}\_{M}^{T}(\boldsymbol{t}) = \boldsymbol{0} \\ \boldsymbol{K}\_{M} + \boldsymbol{K}\_{M} \Big(\boldsymbol{A}\hat{\boldsymbol{P}}\_{M\times M}^{a}\boldsymbol{G}\_{M\times M}^{\mu} + \boldsymbol{B}\hat{\boldsymbol{P}}\_{M\times M}^{a}\Big) + \boldsymbol{A}\boldsymbol{F}\_{M}^{(1)}\boldsymbol{G}\_{M\times M}^{\mu} + \boldsymbol{B}\boldsymbol{F}\_{M}^{(1)} - \boldsymbol{F}\_{M}^{(2)}, \end{split} \tag{43}$$

where

$$
\hat{P}^{a}\_{\mathcal{M}\times\mathcal{M}} = P^{a}\_{\mathcal{M}\times\mathcal{M}} - Q^{a,\phi}\_{\mathcal{M}\times\mathcal{M}^\*},
$$

Eq. (43) is an algebraic equation of Lyapunov type, which can be easily solved for the unknown coefficient vector *KM*. When we find *KM*, then putting this in Eq. (39), we get the required approximate solution of the problem.

#### **3.2 Coupled system of boundary value problem of fractional order differential equations**

Consider a coupled system of a fractional order boundary value problem

$$\begin{aligned} D^a \mathbf{x}(t) + A\_1 D^{\mu\_1} \mathbf{x}(t) + B\_1 D^{\nu\_1} \mathbf{y}(t) + C\_1 \mathbf{x}(t) + D\_1 \mathbf{y}(t) &= f(t), \ \mathbf{1} < a \le 2, \ 0 < \mu\_1, \nu\_1 \le \mathbf{1}, \\ D^\beta \mathbf{y}(t) + A\_2 D^{\nu\_2} \mathbf{x}(t) + B\_2 D^{\nu\_2} \mathbf{y}(t) + C\_2 \mathbf{x}(t) + D\_2 \mathbf{y}(t) &= \mathbf{g}(t), \ \mathbf{1} < \beta \le 2, \ 0 < \mu\_2, \nu\_2 \le \mathbf{1}, \end{aligned} \tag{44}$$

*Approximate Solutions of Some Boundary Value Problems by Using Operational Matrices… DOI: http://dx.doi.org/10.5772/intechopen.90302*

subject to the boundary conditions

$$\varkappa(\mathbf{0}) = a, \quad \varkappa(\mathbf{1}) = b \quad \jmath(\mathbf{0}) = c, \quad \jmath(\mathbf{1}) = d,\tag{45}$$

where *Ai*, *Bi*,*Ci*, *Di*ð Þ *i* ¼ 1, 2 are any real constants, *f t*ð Þ, *g t*ð Þ are given source terms. We approximate the solution of the above system in terms of Bernstein polynomials such as

$$\begin{aligned} D^a \mathfrak{x}(t) &= K\_M B\_M^T(t), \quad D^\beta \mathfrak{y}(t) = L\_M B\_M^T(t) \\ \mathfrak{x}(t) &= K\_M P\_{M \times M}^x B\_M^T(t) + c\_0 + c\_1 t, \quad \mathfrak{y}(t) = L\_M \left( P\_{M \times M}^\beta B\_M^T(t) + d\_0 + d\_1 t \right) \end{aligned}$$

applying boundary conditions, we have

$$\begin{aligned} \boldsymbol{\omega}(t) &= \boldsymbol{K}\_{\boldsymbol{M}}(\boldsymbol{P}\_{\boldsymbol{M}\times\boldsymbol{M}}^{a}\boldsymbol{B}\_{\boldsymbol{M}}^{T}(t) + \boldsymbol{a} + \boldsymbol{t}(\boldsymbol{b} - \boldsymbol{a}) - \boldsymbol{t}\boldsymbol{K}\_{\boldsymbol{M}}\boldsymbol{P}\_{\boldsymbol{M}\times\boldsymbol{M}}^{a}\boldsymbol{B}\_{\boldsymbol{M}}^{T}(t))\big|\_{t=1}, \\ \boldsymbol{y}(t) &= \boldsymbol{K}\_{\boldsymbol{M}}(\boldsymbol{P}\_{\boldsymbol{M}\times\boldsymbol{M}}^{\boldsymbol{\beta}}\boldsymbol{B}\_{\boldsymbol{M}}^{T}(t) + \boldsymbol{c} + \boldsymbol{t}(\boldsymbol{d} - \boldsymbol{c}) - \boldsymbol{t}\boldsymbol{K}\_{\boldsymbol{M}}\boldsymbol{P}\_{\boldsymbol{M}\times\boldsymbol{M}}^{\boldsymbol{\beta}}\boldsymbol{B}\_{\boldsymbol{M}}^{T}(t)\Big|\_{t=1}. \end{aligned}$$

let us approximate

$$\begin{aligned} a + t(b - a) &= F\_M^1 B\_M^T(t), c + t(d - c) = F\_M^2 B\_M^T(t) \\ t P\_{M \times M}^a B\_M^T(t) \Big|\_{t=1} &= Q\_{M \times M}^{a, \phi} B\_M^T(t), t P\_{M \times M}^\beta B\_M^T(t) \Big|\_{t=1} = Q\_{M \times M}^{\beta, \phi} B\_M^T(t) \end{aligned}$$

then

Using the approximation and Lemma 2.2

*<sup>M</sup>*�*MBT*

which gives *y t*ðÞ¼ *KMP<sup>α</sup>*

*<sup>D</sup><sup>μ</sup>y t*ðÞ¼ *<sup>D</sup><sup>μ</sup> KM <sup>P</sup><sup>α</sup>*

*<sup>M</sup>*ðÞþ*<sup>t</sup> AKM <sup>P</sup><sup>α</sup>*

*<sup>M</sup>*ðÞþ*<sup>t</sup> AKM <sup>P</sup><sup>α</sup>*

<sup>þ</sup> *BKM <sup>P</sup><sup>α</sup>*

*<sup>M</sup>*�*MG<sup>μ</sup>*

<sup>þ</sup>*BKM <sup>P</sup><sup>α</sup>*

In simple form, we can write (42) as

<sup>¼</sup> *KM <sup>P</sup><sup>α</sup>*

Putting Eqs. (38)–(41) in Eq. (37), we get

*<sup>M</sup> BT*

<sup>¼</sup> *KM <sup>P</sup><sup>α</sup>*

*<sup>M</sup>*�*<sup>M</sup>* � *<sup>Q</sup>α*,*<sup>ϕ</sup>*

� �

*<sup>M</sup>*�*<sup>M</sup>* � *<sup>Q</sup>α*,*<sup>ϕ</sup>*

*<sup>M</sup>*�*<sup>M</sup>* � *<sup>Q</sup>α*,*<sup>ϕ</sup>*

*<sup>M</sup>*�*<sup>M</sup>* � *<sup>Q</sup>α*,*<sup>ϕ</sup>*

*<sup>M</sup>*�*<sup>M</sup>* � *<sup>Q</sup>α*,*<sup>ϕ</sup>*

� �

*<sup>M</sup>*�*<sup>M</sup>* <sup>þ</sup> *BP*^*<sup>α</sup>*

*P*^*α*

� �

� �

� �

� �

*<sup>M</sup>*�*<sup>M</sup>* � *<sup>Q</sup><sup>α</sup>*,*<sup>ϕ</sup>*

� �

*M*�*M*

*<sup>M</sup>*ð Þ*<sup>t</sup>* , *tP<sup>α</sup>*

*<sup>M</sup>*�*MBT*

*<sup>M</sup>*ðÞþ*<sup>t</sup> <sup>a</sup>* <sup>þ</sup> *t b*ð Þ� � *<sup>a</sup> tKMP<sup>α</sup>*

*<sup>M</sup>*�*<sup>M</sup>* � *<sup>Q</sup><sup>α</sup>*,*<sup>ϕ</sup>*

*M*�*M*

*f t*ðÞ¼ *<sup>F</sup>*ð Þ<sup>2</sup>

*Gμ <sup>M</sup>*�*<sup>M</sup>BT*

*BT*

*Gμ <sup>M</sup>*�*<sup>M</sup>BT*

*BT*

*<sup>M</sup>*ðÞþ*<sup>t</sup> BF*ð Þ<sup>1</sup>

*<sup>M</sup>*ð Þþ*<sup>t</sup> BF*ð Þ<sup>1</sup>

*<sup>M</sup>*�*<sup>M</sup>* � *<sup>Q</sup><sup>α</sup>*,*<sup>ϕ</sup>*

Eq. (43) is an algebraic equation of Lyapunov type, which can be easily solved for the unknown coefficient vector *KM*. When we find *KM*, then putting this in

**3.2 Coupled system of boundary value problem of fractional order differential**

*<sup>D</sup><sup>α</sup>x t*ðÞþ *<sup>A</sup>*1*D<sup>μ</sup>*1*x t*ðÞþ *<sup>B</sup>*1*D<sup>ν</sup>*<sup>1</sup> *y t*ðÞþ *<sup>C</sup>*1*x t*ð Þþ *<sup>D</sup>*1*y t*ðÞ¼ *f t*ð Þ, 1<sup>&</sup>lt; *<sup>α</sup>*≤2, 0<*μ*1, *<sup>ν</sup>*<sup>1</sup> <sup>≤</sup>1, *<sup>D</sup><sup>β</sup>y t*ð Þþ *<sup>A</sup>*2*D<sup>μ</sup>*2*x t*ðÞþ *<sup>B</sup>*2*D<sup>ν</sup>*<sup>2</sup> *y t*ðÞþ *<sup>C</sup>*2*x t*ðÞþ *<sup>D</sup>*2*y t*ðÞ¼ *g t*ð Þ, 1<*<sup>β</sup>* <sup>≤</sup> 2, 0 <sup>&</sup>lt;*μ*2, *<sup>ν</sup>*<sup>2</sup> <sup>≤</sup> 1,

Consider a coupled system of a fractional order boundary value problem

<sup>þ</sup> *AF*ð Þ<sup>1</sup> *<sup>M</sup> <sup>G</sup><sup>μ</sup>*

*M*�*M*

*M*�*M*

*M*�*M*

*M*�*M*

*M*�*M*

*<sup>M</sup>*�*<sup>M</sup>* <sup>¼</sup> *<sup>P</sup><sup>α</sup>*

Eq. (39), we get the required approximate solution of the problem.

*Gμ <sup>M</sup>*�*<sup>M</sup>BT*

h i

� �

*<sup>M</sup>*ð Þþ*<sup>t</sup> <sup>F</sup>*ð Þ<sup>1</sup>

*M*�*M*

*BT*

*<sup>M</sup> BT*

*<sup>M</sup> BT*

*BT*

*<sup>M</sup>*ð Þþ*<sup>t</sup> <sup>F</sup>*ð Þ<sup>1</sup>

*<sup>M</sup>*ðÞþ*<sup>t</sup> <sup>F</sup>*ð Þ<sup>1</sup>

*<sup>M</sup>*ðÞþ*<sup>t</sup> AF*ð Þ<sup>1</sup>

*<sup>M</sup> BT*

*<sup>M</sup>*ðÞþ*<sup>t</sup> AF*ð Þ<sup>1</sup>

*<sup>M</sup> B<sup>T</sup>*

*<sup>M</sup>*�*<sup>M</sup>:*

*<sup>M</sup>*�*MBT <sup>M</sup>*ð Þ*t* � � � *t*¼1

<sup>¼</sup> *<sup>Q</sup><sup>α</sup>*,*<sup>ϕ</sup> <sup>M</sup>*�*<sup>M</sup>BT*

> *<sup>M</sup>*�*MB<sup>T</sup> <sup>M</sup>*ð Þ*t*

*<sup>M</sup> BT <sup>M</sup>*ð Þ*t :*

*<sup>M</sup>*�*<sup>M</sup>BT <sup>M</sup>*ð Þ*t*

*<sup>M</sup>*ð Þ*t :* (41)

*<sup>M</sup>*�*<sup>M</sup>B<sup>T</sup> <sup>M</sup>*ð Þ*t*

> *<sup>M</sup> BT <sup>M</sup>*ð Þ*t :*

*<sup>M</sup>*�*<sup>M</sup>B<sup>T</sup> <sup>M</sup>*ð Þ*t*

*<sup>M</sup> BT*

*<sup>M</sup>* � *<sup>F</sup>*ð Þ<sup>2</sup> *<sup>M</sup>* ,

*<sup>M</sup>*ðÞ¼ *t* 0

*<sup>M</sup> <sup>G</sup><sup>μ</sup>*

*<sup>M</sup>*ðÞ¼ *<sup>t</sup> <sup>F</sup>*ð Þ<sup>2</sup>

*<sup>M</sup> <sup>G</sup><sup>μ</sup>*

*<sup>M</sup>*ðÞ�*<sup>t</sup> <sup>F</sup>*ð Þ<sup>2</sup>

*<sup>M</sup>*�*<sup>M</sup>* <sup>þ</sup> *BF*ð Þ<sup>1</sup>

*<sup>M</sup>*�*MBT <sup>M</sup>*ð Þ*t* � � *t*¼1 ,

*<sup>M</sup>*ðÞ�*<sup>t</sup> <sup>Q</sup><sup>α</sup>*,*<sup>ϕ</sup>*

*<sup>M</sup>*ðÞþ*<sup>t</sup> <sup>F</sup>*ð Þ<sup>1</sup>

*<sup>M</sup> BT <sup>M</sup>*ð Þ*t*

*<sup>M</sup> <sup>G</sup><sup>μ</sup>*

*<sup>M</sup>*ð Þ*t :*

(39)

(40)

(42)

(43)

(44)

*<sup>a</sup>* <sup>þ</sup> *t b*ð Þ¼ � *<sup>a</sup> <sup>F</sup>*ð Þ<sup>1</sup>

*y t*ðÞ¼ *KMP<sup>α</sup>*

Hence

*Functional Calculus*

Now

and

*KMBT*

*KMBT*

where

**equations**

**176**

*KM* <sup>þ</sup> *KM AP*^*<sup>α</sup>*

*x t*ðÞ¼ *KMP<sup>α</sup> <sup>M</sup>*�*MB<sup>T</sup> <sup>M</sup>*ðÞþ*<sup>t</sup> <sup>F</sup>*ð Þ<sup>1</sup> *<sup>M</sup> BT <sup>M</sup>*ðÞ�*<sup>t</sup> KMQ<sup>α</sup>*,*<sup>ϕ</sup> <sup>M</sup>*�*<sup>M</sup>BT <sup>M</sup>*ð Þ*t y t*ðÞ¼ *LMP<sup>β</sup> <sup>M</sup>*�*<sup>M</sup>BT <sup>M</sup>*ðÞþ*<sup>t</sup> <sup>F</sup>*ð Þ<sup>2</sup> *<sup>M</sup> B<sup>T</sup> <sup>M</sup>*ðÞ�*<sup>t</sup> LMQ<sup>β</sup>*,*<sup>ϕ</sup> <sup>M</sup>*�*<sup>M</sup>BT <sup>M</sup>*ð Þ*t <sup>D</sup>μ*1*x t*ðÞ¼ *KMP<sup>α</sup> <sup>M</sup>*�*MBT <sup>M</sup>*ðÞþ*<sup>t</sup> <sup>F</sup>*ð Þ<sup>1</sup> *<sup>M</sup> BT <sup>M</sup>*ð Þ�*<sup>t</sup> KMQ<sup>α</sup>*,*<sup>ϕ</sup> <sup>M</sup>*�*<sup>M</sup>BT <sup>M</sup>*ð Þ*t* h i <sup>¼</sup> *KM <sup>P</sup><sup>α</sup> <sup>M</sup>*�*<sup>M</sup>* � *<sup>Q</sup>α*,*<sup>ϕ</sup> M*�*M* � �*Gμ*<sup>1</sup> *<sup>M</sup>*�*<sup>M</sup>* <sup>þ</sup> *<sup>F</sup>*ð Þ<sup>1</sup> *<sup>M</sup> <sup>G</sup>μ*<sup>1</sup> *<sup>M</sup>*�*<sup>M</sup>BT <sup>M</sup>*ð Þ*t <sup>D</sup><sup>ν</sup>*<sup>1</sup> *y t*ðÞ¼ *<sup>D</sup><sup>ν</sup>*<sup>1</sup> *LMP<sup>β</sup> <sup>M</sup>*�*<sup>M</sup>BT <sup>M</sup>*ðÞþ*<sup>t</sup> <sup>F</sup>*ð Þ<sup>2</sup> *<sup>M</sup> BT <sup>M</sup>*ð Þ�*<sup>t</sup> LMQ<sup>β</sup>*,*<sup>ϕ</sup> <sup>M</sup>*�*<sup>M</sup>BT <sup>M</sup>*ð Þ*t* h i <sup>¼</sup> *LM <sup>P</sup><sup>β</sup> <sup>M</sup>*�*<sup>M</sup>* � *<sup>Q</sup><sup>β</sup>*,*<sup>ϕ</sup> M*�*M* � �*G<sup>ν</sup>*<sup>1</sup> *<sup>M</sup>*�*<sup>M</sup>* <sup>þ</sup> *<sup>F</sup>*ð Þ<sup>2</sup> *<sup>M</sup> <sup>G</sup><sup>ν</sup>*<sup>1</sup> *<sup>M</sup>*�*<sup>M</sup>BT <sup>M</sup>*ð Þ*t <sup>D</sup><sup>μ</sup>*2*x t*ðÞ¼ *<sup>D</sup><sup>μ</sup>*<sup>2</sup> *KMP<sup>α</sup> <sup>M</sup>*�*MBT <sup>M</sup>*ðÞþ*<sup>t</sup> <sup>F</sup>*ð Þ<sup>1</sup> *<sup>M</sup> B<sup>T</sup> <sup>M</sup>*ðÞ�*<sup>t</sup> KMQ<sup>α</sup>*,*<sup>ϕ</sup> <sup>M</sup>*�*<sup>M</sup>B<sup>T</sup> <sup>M</sup>*ð Þ*t* h i <sup>¼</sup> *KM <sup>P</sup><sup>α</sup> <sup>M</sup>*�*<sup>M</sup>* � *<sup>Q</sup><sup>α</sup>*,*<sup>ϕ</sup> M*�*M* � �*G<sup>μ</sup>*<sup>2</sup> *<sup>M</sup>*�*<sup>M</sup>* <sup>þ</sup> *<sup>F</sup>*ð Þ<sup>1</sup> *<sup>M</sup> <sup>G</sup><sup>μ</sup>*<sup>2</sup> *<sup>M</sup>*�*<sup>M</sup>BT <sup>M</sup>*ð Þ*t*

and

$$D^{\nu\_2}y(t) = D^{\nu\_2} \left[ K\_M \mathbf{P}\_{M \times M}^{\beta} \mathbf{B}\_M^T(t) + F\_M^{(2)} \mathbf{B}\_M^T(t) - K\_M \mathbf{Q}\_{M \times M}^{\beta, \phi} \mathbf{B}\_M^T(t) \right],$$

$$= L\_M \left( P\_{M \times M}^{\beta} - \mathbf{Q}\_{M \times M}^{\beta, \phi} \right) \mathbf{G}\_{M \times M}^{\nu\_2} + F\_M^{(2)} \mathbf{G}\_{M \times M}^{\omega\_2} \mathbf{B}\_M^T(t)$$

$$f(t) = F^{(3)} \mathbf{B}\_M^T(t), \quad \mathbf{g}(t) = F^{(4)} \mathbf{B}\_M^T(t).$$

Thus system (44) implies that

*KMBT <sup>M</sup>*ð Þþ*<sup>t</sup> <sup>A</sup>*1*KM <sup>P</sup><sup>α</sup> <sup>M</sup>*�*<sup>M</sup>* � *<sup>Q</sup><sup>α</sup>*,*<sup>ϕ</sup> M*�*M* � �*G<sup>μ</sup>*<sup>1</sup> *<sup>M</sup>*�*<sup>M</sup>* <sup>þ</sup> *<sup>A</sup>*1*F*ð Þ<sup>1</sup> *<sup>M</sup> <sup>G</sup><sup>μ</sup>*<sup>1</sup> *<sup>M</sup>*�*<sup>M</sup>BT <sup>M</sup>*ð Þ*t* <sup>þ</sup> *<sup>B</sup>*1*LM <sup>P</sup><sup>β</sup> <sup>M</sup>*�*<sup>M</sup>* � *<sup>Q</sup><sup>β</sup>*,*<sup>ϕ</sup> M*�*M* � �*G<sup>ν</sup>*<sup>1</sup> *<sup>M</sup>*�*<sup>M</sup>* <sup>þ</sup> *<sup>B</sup>*1*F*ð Þ<sup>2</sup> *<sup>M</sup> <sup>G</sup><sup>ν</sup>*<sup>1</sup> *<sup>M</sup>*�*<sup>M</sup>BT <sup>M</sup>*ðÞþ*<sup>t</sup> <sup>C</sup>*1*KMP<sup>α</sup> <sup>M</sup>*�*MBT <sup>M</sup>*ð Þ*t* <sup>þ</sup> *<sup>C</sup>*1*F*ð Þ<sup>1</sup> *<sup>M</sup> BT <sup>M</sup>*ðÞ�*<sup>t</sup> <sup>C</sup>*1*KMQ<sup>α</sup>*,*<sup>ϕ</sup> <sup>M</sup>*�*<sup>M</sup>BT <sup>M</sup>*ð Þþ*<sup>t</sup> <sup>D</sup>*1*LMP<sup>β</sup> <sup>M</sup>*�*<sup>M</sup>BT <sup>M</sup>*ð Þþ*<sup>t</sup> <sup>D</sup>*1*F*ð Þ<sup>2</sup> *<sup>M</sup> BT <sup>M</sup>*ð Þ*t* � *<sup>D</sup>*1*LMQ<sup>β</sup>*,*<sup>ϕ</sup> <sup>M</sup>*�*<sup>M</sup>BT <sup>M</sup>*ðÞ¼ *<sup>t</sup> <sup>F</sup>*ð Þ<sup>3</sup> *BT <sup>M</sup>*ð Þ*t LMBT <sup>M</sup>*ðÞþ*<sup>t</sup> <sup>A</sup>*2*KM <sup>P</sup><sup>α</sup> <sup>M</sup>*�*<sup>M</sup>* � *<sup>Q</sup><sup>α</sup>*,*<sup>ϕ</sup> M*�*M* � �*G<sup>μ</sup>*<sup>2</sup> *<sup>M</sup>*�*<sup>M</sup>* <sup>þ</sup> *<sup>A</sup>*2*F*ð Þ<sup>1</sup> *<sup>M</sup> <sup>G</sup><sup>μ</sup>*<sup>2</sup> *<sup>M</sup>*�*<sup>M</sup>BT <sup>M</sup>*ð Þ*t* <sup>þ</sup> *<sup>B</sup>*2*LM <sup>P</sup><sup>β</sup> <sup>M</sup>*�*<sup>M</sup>* � *<sup>Q</sup><sup>β</sup>*,*<sup>ϕ</sup> M*�*M* � �*G<sup>ν</sup>*<sup>2</sup> *<sup>M</sup>*�*<sup>M</sup>* <sup>þ</sup> *<sup>B</sup>*2*F*ð Þ<sup>2</sup> *<sup>M</sup> <sup>G</sup><sup>ν</sup>*<sup>2</sup> *<sup>M</sup>*�*<sup>M</sup>BT <sup>M</sup>*ðÞþ*<sup>t</sup> <sup>C</sup>*2*KMP<sup>α</sup> <sup>M</sup>*�*MB<sup>T</sup> <sup>M</sup>*ð Þ*t* <sup>þ</sup> *<sup>C</sup>*2*F*ð Þ<sup>1</sup> *<sup>M</sup> BT <sup>M</sup>*ðÞ�*<sup>t</sup> <sup>C</sup>*2*KMQ<sup>α</sup>*,*<sup>ϕ</sup> <sup>M</sup>*�*<sup>M</sup>BT <sup>M</sup>*ð Þþ*<sup>t</sup> <sup>D</sup>*2*LMP<sup>β</sup> <sup>M</sup>*�*<sup>M</sup>BT <sup>M</sup>*ðÞþ*<sup>t</sup> <sup>D</sup>*2*F*ð Þ<sup>2</sup> *<sup>M</sup> BT <sup>M</sup>*ð Þ*t* � *<sup>D</sup>*2*LMQ<sup>β</sup>*,*<sup>ϕ</sup> <sup>M</sup>*�*<sup>M</sup>BT <sup>M</sup>*ðÞ¼ *<sup>t</sup> <sup>F</sup>*ð Þ <sup>4</sup> *BT <sup>M</sup>*ð Þ*t :* (46)

*x t*ðÞ¼ *KMP<sup>α</sup>*

*DOI: http://dx.doi.org/10.5772/intechopen.90302*

*y t*ðÞ¼ *LMP<sup>β</sup>*

**Example 4.1.** *Consider*

*f* <sup>1</sup>ðÞ¼ *t t*

þ

þ

*Figures 2 and* **3** *respectively.* **Example 4.2.** *Consider*

**179**

þ 100*t*

*f* <sup>2</sup>ðÞ¼ *t*

*f* <sup>3</sup>ðÞ¼ *t*

we get the required approximate solution.

*<sup>M</sup>*�*MBT*

*<sup>M</sup>*�*<sup>M</sup>BT*

**4. Applications of our method to some examples**

*<sup>D</sup><sup>α</sup>y t*ðÞþ *<sup>c</sup>*1*D<sup>ν</sup>*

subject to the boundary conditions

<sup>6</sup> ð Þ *<sup>t</sup>* � <sup>1</sup> <sup>3</sup> <sup>þ</sup> *<sup>t</sup>*

11147682583723703125*t*

<sup>6</sup> ð Þ *<sup>t</sup>* � <sup>1</sup> <sup>3</sup>

5081767996463981*t*

5081767996463981*t*

*The exact solution of the above problem is*

*<sup>D</sup><sup>α</sup>y t*ðÞ� <sup>2</sup>*D*<sup>0</sup>*:*<sup>9</sup>

278692064593092578125*t*

*<sup>M</sup>*ðÞþ*<sup>t</sup> <sup>F</sup>*ð Þ<sup>1</sup>

*Approximate Solutions of Some Boundary Value Problems by Using Operational Matrices…*

*<sup>M</sup>*ð Þþ*<sup>t</sup> <sup>F</sup>*ð Þ<sup>2</sup>

*<sup>M</sup> BT*

*<sup>M</sup> BT*

*y*ð Þ¼ 0 0, *y*ð Þ¼ 1 0*:*

*Solution: We solve this problem under the following parameters sets defined as S*<sup>1</sup> ¼ f g *α* ¼ 2, *ν* ¼ 1,*c*<sup>1</sup> ¼ 1,*c*<sup>2</sup> ¼ 1 , *S*<sup>2</sup> ¼ f g *α* ¼ 1*:*8, *ν* ¼ 0*:*8,*c*<sup>1</sup> ¼ 10,*c*<sup>2</sup> ¼ 100 , *S*<sup>3</sup> ¼ f g *α* ¼ 1*:*5, *ν* ¼ 0*:*5,*c*<sup>1</sup> ¼ 1*=*10,*c*<sup>2</sup> ¼ 1*=*100 , *and select source term for S*<sup>1</sup> *as*

<sup>6</sup>ðð Þþ <sup>72</sup>*<sup>t</sup>* � <sup>168</sup> <sup>126</sup>Þ � <sup>30</sup>*<sup>t</sup>*

406548945561989414912

25002760152062349017088

<sup>3</sup> � <sup>3360</sup>*<sup>t</sup>* ð Þ <sup>2</sup> <sup>þ</sup> <sup>2730</sup>*<sup>t</sup>* � <sup>715</sup>

<sup>6</sup> ð Þ *<sup>t</sup>* � <sup>1</sup> <sup>3</sup> *:*

*We solve this problem with the proposed method under different sets of parameters as*

<sup>3</sup> � <sup>3808</sup>*<sup>t</sup>* ð Þ <sup>2</sup> <sup>þ</sup> <sup>3570</sup>*<sup>t</sup>* � <sup>1105</sup> 22452467243837030400 þ

26 <sup>5</sup> 5250*t*

264146673456906240

*y t*ðÞ¼ *t*

*defined in S*1, *S*2, *S*3. *The observation and simulation demonstrate that the solution obtained with the proposed method is highly accurate. The comparison of exact solution with approximate solution obtained using the parameters set S*<sup>1</sup> *is displayed in Figure 1 subplot (a), while in Figure 1 subplot (b) we plot the absolute difference between the exact and approximate solutions using different scale levels. One can easily observe that the absolute error is much less than* 10�<sup>12</sup>*: The order of derivatives in this set is an integer. By solving the problem under parameters set S*<sup>2</sup> *and S*3, *we observe the same phenomena. The approximate solution matches very well with the exact solution. See*

21 <sup>5</sup> 1750*t*

9 <sup>2</sup> 1344*t*

> 11 <sup>2</sup> 1344*t*

*<sup>M</sup>*ðÞ�*<sup>t</sup> KMQ<sup>α</sup>*,*<sup>ϕ</sup>*

*<sup>M</sup>*ðÞ�*<sup>t</sup> LMQ<sup>β</sup>*,*<sup>ϕ</sup>*

*<sup>M</sup>*�*<sup>M</sup>BT <sup>M</sup>*ð Þ*t*

*<sup>M</sup>*ð Þ*t* ,

<sup>5</sup> ð Þ <sup>3</sup>*<sup>t</sup>* � <sup>2</sup> ð Þ *<sup>t</sup>* � <sup>1</sup> <sup>2</sup> (49)

*t* <sup>6</sup> ð Þ *<sup>t</sup>* � <sup>1</sup> <sup>3</sup> <sup>100</sup> *:* (51)

*<sup>M</sup>*�*<sup>M</sup>BT*

*y t*ð Þþ *c*2*y t*ðÞ¼ *f t*ð Þ, 1<*α* <2 (48)

<sup>4</sup> <sup>þ</sup> <sup>3</sup>*<sup>t</sup>*

, (50)

<sup>3</sup> � <sup>14350</sup>*<sup>t</sup>* ð Þ <sup>2</sup> <sup>þ</sup> <sup>12915</sup>*<sup>t</sup>* � <sup>3813</sup>

*y t*ðÞ� 3*y t*ðÞ¼�4 cos 2ð Þ�*t* 7 sin 2ð Þ*t* (52)

<sup>3</sup> � <sup>4200</sup>*<sup>t</sup>* ð Þ <sup>2</sup> <sup>þ</sup> <sup>3255</sup>*<sup>t</sup>* � <sup>806</sup>

Rearranging the terms in the above system and using the following notation for simplicity in Eq. (46)

$$
\begin{split}
\hat{\mathbf{Q}}\_{M\times M}^{\alpha} &= A\_1 \Big(P\_{M\times M}^{\alpha} - \mathbf{Q}\_{M\times M}^{\alpha,\phi}\Big) G\_{M\times M}^{\mu\_1} + \mathbf{C}\_1 \Big(P\_{M\times M}^{\alpha} - \mathbf{Q}\_{M\times M}^{\alpha,\phi}\Big) \\
\hat{\mathbf{Q}}\_{M\times M}^{\beta} &= B\_1 \Big(P\_{M\times M}^{\beta} - \mathbf{Q}\_{M\times M}^{\beta,\phi}\Big) G\_{M\times M}^{\nu\_1} + D\_1 \Big(P\_{M\times M}^{\beta} - \mathbf{Q}\_{M\times M}^{\beta,\phi}\Big) \\
\hat{\mathbf{R}}\_{M\times M}^{\alpha} &= A\_2 \Big(P\_{M\times M}^{\alpha} - \mathbf{Q}\_{M\times M}^{\alpha,\phi}\Big) G\_{M\times M}^{\mu\_1} + C\_2 \Big(P\_{M\times M}^{\alpha} - \mathbf{Q}\_{M\times M}^{\alpha,\phi}\Big) \\
\hat{\mathbf{R}}\_{M\times M}^{\beta} &= B\_2 \Big(P\_{M\times M}^{\beta} - \mathbf{Q}\_{M\times M}^{\beta,\phi}\Big) G\_{M\times M}^{\nu\_2} + D\_2 \Big(P\_{M\times M}^{\beta} - \mathbf{Q}\_{M\times M}^{\beta,\phi}\Big) \\
F\_M &= A\_1 F\_M^{(1)} G\_{M\times M}^{\mu\_1} + B\_1 F\_M^{(2)} G\_{M\times M}^{\nu\_1} + C\_1 F\_M^{(1)} + F\_M^{(2)} - D\_1 F\_M^{(3)} \\
G\_M &= A\_2 F\_M^{(1)} G\_{M\times M}^{\mu\_2} + B\_2 F\_M^{(2)} G\_{M\times M}^{\nu\_2} + C\_2 F\_M^{(1)} + D\_2 F\_M^{(2)} - F\_M^{(4)},\$$

the above system (46) becomes

*KMBT <sup>M</sup>*ðÞþ*<sup>t</sup> KMQ*^ *<sup>α</sup> <sup>M</sup>*�*MBT <sup>M</sup>*ðÞþ*<sup>t</sup> LMQ*^ *<sup>β</sup> <sup>M</sup>*�*MBT <sup>M</sup>*ð Þþ*<sup>t</sup> FMBT <sup>M</sup>*ðÞ¼ *t* 0 *LMBT <sup>M</sup>*ðÞþ*<sup>t</sup> KMR*^*<sup>α</sup> <sup>M</sup>*�*MBT <sup>M</sup>*ðÞþ*<sup>t</sup> LMR*^*<sup>β</sup> <sup>M</sup>*�*MBT <sup>M</sup>*ðÞþ*<sup>t</sup> GMB<sup>T</sup> <sup>M</sup>*ðÞ¼ *t* 0 ½ � *KM LM BT <sup>M</sup>*ð Þ*t* 0 0 *B<sup>T</sup> <sup>M</sup>*ð Þ*t* " # þ ½ � *KM LM Q*^ *α <sup>M</sup>*�*<sup>M</sup>* 0 <sup>0</sup> *<sup>R</sup>*^*<sup>β</sup> M*�*M* 2 4 3 5 *BT <sup>M</sup>*ð Þ*t* 0 0 *BT <sup>M</sup>*ð Þ*t* " # þ½ � *KM LM* <sup>0</sup> *<sup>R</sup>*^*<sup>α</sup> M*�*M Q*^ *β <sup>M</sup>*�*<sup>M</sup>* 0 2 4 3 5 *BT <sup>M</sup>*ð Þ*t* 0 0 *B<sup>T</sup> <sup>M</sup>*ð Þ*t* " # þ ½ � *FM GM BT <sup>M</sup>*ð Þ*t* 0 0 *BT <sup>M</sup>*ð Þ*t* " # ¼ 0 ½ �þ *KM LM* ½ � *KM LM Q*^ *α <sup>M</sup>*�*<sup>M</sup> <sup>R</sup>*^*<sup>α</sup> M*�*M Q*^ *β <sup>M</sup>*�*<sup>M</sup> <sup>R</sup>*^*<sup>β</sup> M*�*M* 2 4 3 5 þ ½ �¼ *FM GM* 0, (47)

which is an algebraic equation that can be easily solved by using Matlab functional solver or Mathematica for unknown matrix ½ � *KM LM* . Calculating the coefficient matrix *KM*, *LM* and putting it in equations

*Approximate Solutions of Some Boundary Value Problems by Using Operational Matrices… DOI: http://dx.doi.org/10.5772/intechopen.90302*

$$\begin{cases} \boldsymbol{x}(t) = \boldsymbol{K}\_{M}\boldsymbol{P}\_{M\times M}^{a}\boldsymbol{B}\_{M}^{T}(t) + \boldsymbol{F}\_{M}^{(1)}\boldsymbol{B}\_{M}^{T}(t) - \boldsymbol{K}\_{M}\boldsymbol{Q}\_{M\times M}^{a,\phi}\boldsymbol{B}\_{M}^{T}(t),\\ \boldsymbol{y}(t) = \boldsymbol{L}\_{M}\boldsymbol{P}\_{M\times M}^{\beta}\boldsymbol{B}\_{M}^{T}(t) + \boldsymbol{F}\_{M}^{(2)}\boldsymbol{B}\_{M}^{T}(t) - \boldsymbol{L}\_{M}\boldsymbol{Q}\_{M\times M}^{\beta,\phi}\boldsymbol{B}\_{M}^{T}(t),\end{cases}$$

we get the required approximate solution.

#### **4. Applications of our method to some examples**

**Example 4.1.** *Consider*

Thus system (44) implies that

*<sup>M</sup>*�*<sup>M</sup>* � *<sup>Q</sup><sup>α</sup>*,*<sup>ϕ</sup>*

*<sup>M</sup>*�*<sup>M</sup>* � *<sup>Q</sup><sup>β</sup>*,*<sup>ϕ</sup>*

*<sup>M</sup>*�*<sup>M</sup>* � *<sup>Q</sup><sup>α</sup>*,*<sup>ϕ</sup>*

*<sup>M</sup>*�*<sup>M</sup>* � *<sup>Q</sup><sup>β</sup>*,*<sup>ϕ</sup>*

� �

*<sup>M</sup>*ðÞ�*<sup>t</sup> <sup>C</sup>*2*KMQ<sup>α</sup>*,*<sup>ϕ</sup>*

� �

*<sup>M</sup>*�*<sup>M</sup>BT*

*<sup>M</sup>*�*<sup>M</sup>BT*

*<sup>M</sup>*�*<sup>M</sup>* <sup>¼</sup> *<sup>A</sup>*<sup>1</sup> *<sup>P</sup><sup>α</sup>*

*<sup>M</sup>*�*<sup>M</sup>* <sup>¼</sup> *<sup>B</sup>*<sup>1</sup> *<sup>P</sup><sup>β</sup>*

*<sup>M</sup>*�*<sup>M</sup>* <sup>¼</sup> *<sup>A</sup>*<sup>2</sup> *<sup>P</sup><sup>α</sup>*

*<sup>M</sup>*�*<sup>M</sup>* <sup>¼</sup> *<sup>B</sup>*<sup>2</sup> *<sup>P</sup><sup>β</sup>*

*<sup>M</sup> <sup>G</sup>μ*<sup>1</sup>

*<sup>M</sup> <sup>G</sup>μ*<sup>2</sup>

*FM* <sup>¼</sup> *<sup>A</sup>*1*F*ð Þ<sup>1</sup>

*GM* <sup>¼</sup> *<sup>A</sup>*2*F*ð Þ<sup>1</sup>

the above system (46) becomes

*<sup>M</sup>*�*MBT*

*<sup>M</sup>*�*MBT*

*<sup>M</sup>*ð Þ*t* 0 0 *B<sup>T</sup>*

<sup>0</sup> *<sup>R</sup>*^*<sup>α</sup>*

*<sup>M</sup>*�*<sup>M</sup>* 0

" #

*BT*

*Q*^ *β*

2 4 � �

*<sup>M</sup>*ðÞ�*<sup>t</sup> <sup>C</sup>*1*KMQ<sup>α</sup>*,*<sup>ϕ</sup>*

� �

*M*�*M*

*M*�*M*

*<sup>M</sup>*ðÞ¼ *<sup>t</sup> <sup>F</sup>*ð Þ<sup>3</sup> *BT*

*M*�*M*

*M*�*M*

*<sup>M</sup>*ðÞ¼ *<sup>t</sup> <sup>F</sup>*ð Þ <sup>4</sup> *BT*

*<sup>M</sup>*�*<sup>M</sup>* � *<sup>Q</sup>α*,*<sup>ϕ</sup>*

*<sup>M</sup>*�*<sup>M</sup>* � *<sup>Q</sup>β*,*<sup>ϕ</sup>*

*<sup>M</sup>*�*<sup>M</sup>* � *<sup>Q</sup>α*,*<sup>ϕ</sup>*

*<sup>M</sup>*�*<sup>M</sup>* � *<sup>Q</sup>β*,*<sup>ϕ</sup>*

� �

*<sup>M</sup>*�*<sup>M</sup>* <sup>þ</sup> *<sup>B</sup>*1*F*ð Þ<sup>2</sup>

*<sup>M</sup>*�*<sup>M</sup>* <sup>þ</sup> *<sup>B</sup>*2*F*ð Þ<sup>2</sup>

*<sup>M</sup>*ðÞþ*<sup>t</sup> LMQ*^ *<sup>β</sup>*

*<sup>M</sup>*ðÞþ*<sup>t</sup> LMR*^*<sup>β</sup>*

*<sup>M</sup>*ð Þ*t*

*M*�*M*

coefficient matrix *KM*, *LM* and putting it in equations

3 5

*BT*

½ �þ *KM LM* ½ � *KM LM*

� �

� �

� �

*G<sup>μ</sup>*<sup>1</sup>

*G<sup>ν</sup>*<sup>1</sup>

*<sup>M</sup>*�*<sup>M</sup>BT*

*G<sup>μ</sup>*<sup>2</sup>

*G<sup>ν</sup>*<sup>2</sup>

*<sup>M</sup>*�*<sup>M</sup>BT*

*M*�*M*

*M*�*M*

*M*�*M*

*M*�*M*

*<sup>M</sup> <sup>G</sup>ν*<sup>1</sup>

*<sup>M</sup> <sup>G</sup>ν*<sup>2</sup>

*<sup>M</sup>*�*MBT*

*<sup>M</sup>*�*MBT*

þ ½ � *KM LM*

*<sup>M</sup>*ð Þ*t* 0 0 *B<sup>T</sup>*

which is an algebraic equation that can be easily solved by using Matlab functional solver or Mathematica for unknown matrix ½ � *KM LM* . Calculating the

" #

*<sup>M</sup>*ð Þ*t :*

Rearranging the terms in the above system and using the following notation for

*Gμ*<sup>1</sup>

*Gν*<sup>1</sup>

*Gμ*<sup>2</sup>

*Gν*<sup>2</sup>

*<sup>M</sup>*ð Þ*t*

*<sup>M</sup>*�*<sup>M</sup>* <sup>þ</sup> *<sup>A</sup>*1*F*ð Þ<sup>1</sup>

*<sup>M</sup>*�*<sup>M</sup>* <sup>þ</sup> *<sup>B</sup>*1*F*ð Þ<sup>2</sup>

*<sup>M</sup>*�*<sup>M</sup>* <sup>þ</sup> *<sup>A</sup>*2*F*ð Þ<sup>1</sup>

*<sup>M</sup>*�*<sup>M</sup>* <sup>þ</sup> *<sup>B</sup>*2*F*ð Þ<sup>2</sup>

*<sup>M</sup> <sup>G</sup><sup>μ</sup>*<sup>1</sup>

*<sup>M</sup> <sup>G</sup><sup>ν</sup>*<sup>1</sup>

*<sup>M</sup> <sup>G</sup><sup>μ</sup>*<sup>2</sup> *<sup>M</sup>*�*<sup>M</sup>BT <sup>M</sup>*ð Þ*t*

*<sup>M</sup> <sup>G</sup><sup>ν</sup>*<sup>2</sup>

*<sup>M</sup>*�*<sup>M</sup>* <sup>þ</sup> *<sup>C</sup>*<sup>1</sup> *<sup>P</sup><sup>α</sup>*

*<sup>M</sup>*�*<sup>M</sup>* <sup>þ</sup> *<sup>D</sup>*<sup>1</sup> *<sup>P</sup><sup>β</sup>*

*<sup>M</sup>*�*<sup>M</sup>* <sup>þ</sup> *<sup>C</sup>*<sup>2</sup> *<sup>P</sup><sup>α</sup>*

*<sup>M</sup>*�*<sup>M</sup>* <sup>þ</sup> *<sup>D</sup>*<sup>2</sup> *<sup>P</sup><sup>β</sup>*

*<sup>M</sup>*�*<sup>M</sup>* <sup>þ</sup> *<sup>C</sup>*1*F*ð Þ<sup>1</sup>

*<sup>M</sup>*�*<sup>M</sup>* <sup>þ</sup> *<sup>C</sup>*2*F*ð Þ<sup>1</sup>

*<sup>M</sup>*ð Þþ*<sup>t</sup> FMBT*

*<sup>M</sup>*ðÞþ*<sup>t</sup> GMB<sup>T</sup>*

2 4

*<sup>M</sup>*ð Þ*t*

*Q*^ *α*

2 4

*Q*^ *β*

*Q*^ *α*

*<sup>M</sup>*ð Þþ*<sup>t</sup> <sup>D</sup>*2*LMP<sup>β</sup>*

*<sup>M</sup>*ð Þþ*<sup>t</sup> <sup>D</sup>*1*LMP<sup>β</sup>*

*<sup>M</sup>*�*<sup>M</sup>BT <sup>M</sup>*ð Þ*t*

*<sup>M</sup>*�*<sup>M</sup>BT*

*<sup>M</sup>*�*<sup>M</sup>BT*

*<sup>M</sup>*�*<sup>M</sup>BT*

*<sup>M</sup>*�*<sup>M</sup>* � *<sup>Q</sup>α*,*<sup>ϕ</sup>*

*<sup>M</sup>*�*<sup>M</sup>* � *<sup>Q</sup>β*,*<sup>ϕ</sup>*

*<sup>M</sup>*�*<sup>M</sup>* � *<sup>Q</sup>α*,*<sup>ϕ</sup>*

*<sup>M</sup>*�*<sup>M</sup>* � *<sup>Q</sup>β*,*<sup>ϕ</sup>*

*<sup>M</sup>* <sup>þ</sup> *<sup>F</sup>*ð Þ<sup>2</sup>

*<sup>M</sup>* <sup>þ</sup> *<sup>D</sup>*2*F*ð Þ<sup>2</sup>

*<sup>M</sup>*ðÞ¼ *t* 0

*<sup>M</sup>*ðÞ¼ *t* 0

*M*�*M*

*M*�*M*

*M*�*M*

þ ½ � *FM GM*

*<sup>M</sup>*�*<sup>M</sup> <sup>R</sup>*^*<sup>α</sup>*

*<sup>M</sup>*�*<sup>M</sup> <sup>R</sup>*^*<sup>β</sup>*

3 5

*BT*

*BT*

3

*<sup>M</sup>*ð Þ*t* 0 0 *BT*

*<sup>M</sup>*ð Þ*t* 0 0 *BT*

" #

" #

*<sup>M</sup>*ð Þ*t*

*<sup>M</sup>*ð Þ*t*

5 þ ½ �¼ *FM GM* 0,

¼ 0

(47)

*<sup>M</sup>*�*<sup>M</sup>* 0 <sup>0</sup> *<sup>R</sup>*^*<sup>β</sup>*

� �

*<sup>M</sup>* � *<sup>D</sup>*1*F*ð Þ<sup>3</sup> *M*

> *<sup>M</sup>* � *<sup>F</sup>*ð Þ <sup>4</sup> *<sup>M</sup>* ,

� �

� �

� �

*<sup>M</sup>*�*<sup>M</sup>BT*

*<sup>M</sup>*ðÞþ*<sup>t</sup> <sup>C</sup>*1*KMP<sup>α</sup>*

*<sup>M</sup>*ðÞþ*<sup>t</sup> <sup>C</sup>*2*KMP<sup>α</sup>*

*<sup>M</sup>*ðÞþ*<sup>t</sup> <sup>D</sup>*2*F*ð Þ<sup>2</sup>

*M*�*M*

*M*�*M*

*M*�*M*

*M*�*M*

*<sup>M</sup>*ð Þþ*<sup>t</sup> <sup>D</sup>*1*F*ð Þ<sup>2</sup>

*<sup>M</sup>*�*MBT <sup>M</sup>*ð Þ*t*

*<sup>M</sup>*�*MB<sup>T</sup> <sup>M</sup>*ð Þ*t*

(46)

*<sup>M</sup> BT <sup>M</sup>*ð Þ*t*

*<sup>M</sup> BT <sup>M</sup>*ð Þ*t*

*<sup>M</sup>*ð Þþ*<sup>t</sup> <sup>A</sup>*1*KM <sup>P</sup><sup>α</sup>*

<sup>þ</sup> *<sup>B</sup>*1*LM <sup>P</sup><sup>β</sup>*

� *<sup>D</sup>*1*LMQ<sup>β</sup>*,*<sup>ϕ</sup>*

<sup>þ</sup> *<sup>B</sup>*2*LM <sup>P</sup><sup>β</sup>*

� *<sup>D</sup>*2*LMQ<sup>β</sup>*,*<sup>ϕ</sup>*

<sup>þ</sup> *<sup>C</sup>*2*F*ð Þ<sup>1</sup> *<sup>M</sup> BT*

simplicity in Eq. (46)

*Q*^ *α*

*Q*^ *β*

*R*^*α*

*R*^*β*

*<sup>M</sup>*ðÞþ*<sup>t</sup> KMQ*^ *<sup>α</sup>*

*<sup>M</sup>*ðÞþ*<sup>t</sup> KMR*^*<sup>α</sup>*

½ � *KM LM*

þ½ � *KM LM*

**178**

*KMBT*

*LMBT*

<sup>þ</sup> *<sup>C</sup>*1*F*ð Þ<sup>1</sup> *<sup>M</sup> BT*

*<sup>M</sup>*ðÞþ*<sup>t</sup> <sup>A</sup>*2*KM <sup>P</sup><sup>α</sup>*

*KMBT*

*Functional Calculus*

*LMBT*

$$D^a y(t) + c\_1 D^\nu y(t) + c\_2 y(t) = f(t), \quad 1 < a < 2 \tag{48}$$

subject to the boundary conditions

$$\mathcal{y}(\mathbf{0}) = \mathbf{0}, \qquad \mathcal{y}(\mathbf{1}) = \mathbf{0}.$$

*Solution: We solve this problem under the following parameters sets defined as S*<sup>1</sup> ¼ f g *α* ¼ 2, *ν* ¼ 1,*c*<sup>1</sup> ¼ 1,*c*<sup>2</sup> ¼ 1 , *S*<sup>2</sup> ¼ f g *α* ¼ 1*:*8, *ν* ¼ 0*:*8,*c*<sup>1</sup> ¼ 10,*c*<sup>2</sup> ¼ 100 , *S*<sup>3</sup> ¼ f g *α* ¼ 1*:*5, *ν* ¼ 0*:*5,*c*<sup>1</sup> ¼ 1*=*10,*c*<sup>2</sup> ¼ 1*=*100 , *and select source term for S*<sup>1</sup> *as*

$$f\_1(t) = t^6(t-1)^3 + t^6((72t-168) + 126) - 30t^4 + 3t^5(3t-2)(t-1)^2 \tag{49}$$

$$f\_2(t) = \frac{1147682583723703125t^2(1780t^3 - 4200t^2 + 3255t - 806)}{406548945561999414912}$$

$$+ \frac{27869206459302578125t^2(5250t^3 - 14350t^2 + 12915t - 3813)}{25002760152062349017088} \tag{50}$$

$$f\_3(t) = \frac{5081767996463981t^2(1344t^3 - 3360t^2 + 2730t - 715)}{264146673456906240}$$

$$+ \frac{5081767996463981t^2(1344t^3 - 3808t^2 + 3570t - 1105)}{22452467243837030400} + \frac{t^6(t-1)^3}{100}.$$

*The exact solution of the above problem is*

$$y(t) = t^6(t-1)^3.$$

*We solve this problem with the proposed method under different sets of parameters as defined in S*1, *S*2, *S*3. *The observation and simulation demonstrate that the solution obtained with the proposed method is highly accurate. The comparison of exact solution with approximate solution obtained using the parameters set S*<sup>1</sup> *is displayed in Figure 1 subplot (a), while in Figure 1 subplot (b) we plot the absolute difference between the exact and approximate solutions using different scale levels. One can easily observe that the absolute error is much less than* 10�<sup>12</sup>*: The order of derivatives in this set is an integer.*

*By solving the problem under parameters set S*<sup>2</sup> *and S*3, *we observe the same phenomena. The approximate solution matches very well with the exact solution. See Figures 2 and* **3** *respectively.*

**Example 4.2.** *Consider*

$$D^a y(t) - \mathfrak{D}D^{0.9}y(t) - \mathfrak{H}y(t) = -4\cos\left(2t\right) - 7\sin\left(2t\right) \tag{52}$$

**Figure 1.**

*(a) Comparison of exact and approximate solution of Example 4.1, under parameters set* S*1. (b) Absolute error in the approximate solution of Example 4.1, under parameters set* S*1.*

*proposed method, the graph of exact and approximate solutions for different values of M and at α* ¼ 1*:*7 *is shown in* **Figure 4***. From the plot, we observe that the approximate solution becomes equal to the exact solution at α* ¼ 2*. We approximate the error of the method at different scale levels and record that when scale level increases the absolute error decreases as shown in* **Figure 4** *subplot (b) and accuracy approaches 10*�<sup>9</sup>

*(a) Comparison of exact and approximate solution of Example 4.1, under parameters set* S*3. (b) Absolute*

*Approximate Solutions of Some Boundary Value Problems by Using Operational Matrices…*

*DOI: http://dx.doi.org/10.5772/intechopen.90302*

*error in the approximate solution of Example 4.1, under parameters set* S*3.*

*a highly acceptable figure. For convergence of our proposed method, we examined the*

*decreases with a high speed with the increase of scale level M as shown in* **Figure 4b***.*

*(a) Comparison of exact and approximate solution of Example 4.2. (b) Absolute error for different scale level*

<sup>0</sup>∣*yexact* � *yapprox*∣*dt for different values of M and observed that the norm of error*

*quantity* Ð <sup>1</sup>

**Figure 4.**

**181**

*M of Example 4.2.*

**Figure 3.**

, *which is*

**Figure 2.**

*(a) Comparison of exact and approximate solution of Example 4.1, under parameters set* S*2. (b) Absolute error in the approximate solution of Example 4.1, under parameters set* S*2.*

subject to the boundary conditions

$$\mathcal{y}(\mathbf{0}) = \mathbf{0}, \qquad \mathcal{y}(\mathbf{1}) = \sin(2).$$

*Solution: The exact solution of the above problem is y t*ðÞ¼ sin 2ð Þ*t* , *when α* ¼ 2*. However the exact solution at fractional order is not known. We use the well-known property of FDEs that when α* ! 2, *the approximate solution approaches the exact solution for the evaluations of approximate solutions and check the accuracy by using different scale levels. By increasing the scale level M, the accuracy is also increased. By the* *Approximate Solutions of Some Boundary Value Problems by Using Operational Matrices… DOI: http://dx.doi.org/10.5772/intechopen.90302*

#### **Figure 3.**

*(a) Comparison of exact and approximate solution of Example 4.1, under parameters set* S*3. (b) Absolute error in the approximate solution of Example 4.1, under parameters set* S*3.*

*proposed method, the graph of exact and approximate solutions for different values of M and at α* ¼ 1*:*7 *is shown in* **Figure 4***. From the plot, we observe that the approximate solution becomes equal to the exact solution at α* ¼ 2*. We approximate the error of the method at different scale levels and record that when scale level increases the absolute error decreases as shown in* **Figure 4** *subplot (b) and accuracy approaches 10*�<sup>9</sup> , *which is a highly acceptable figure. For convergence of our proposed method, we examined the quantity* Ð <sup>1</sup> <sup>0</sup>∣*yexact* � *yapprox*∣*dt for different values of M and observed that the norm of error decreases with a high speed with the increase of scale level M as shown in* **Figure 4b***.*

#### **Figure 4.**

*(a) Comparison of exact and approximate solution of Example 4.2. (b) Absolute error for different scale level M of Example 4.2.*

subject to the boundary conditions

*error in the approximate solution of Example 4.1, under parameters set* S*2.*

**Figure 1.**

*Functional Calculus*

**Figure 2.**

**180**

*y*ð Þ¼ 0 0, *y*ð Þ¼ 1 sin 2ð Þ*:*

*(a) Comparison of exact and approximate solution of Example 4.1, under parameters set* S*2. (b) Absolute*

*(a) Comparison of exact and approximate solution of Example 4.1, under parameters set* S*1. (b) Absolute*

*error in the approximate solution of Example 4.1, under parameters set* S*1.*

*Solution: The exact solution of the above problem is y t*ðÞ¼ sin 2ð Þ*t* , *when α* ¼ 2*. However the exact solution at fractional order is not known. We use the well-known property of FDEs that when α* ! 2, *the approximate solution approaches the exact solution for the evaluations of approximate solutions and check the accuracy by using different scale levels. By increasing the scale level M, the accuracy is also increased. By the*

**Example 4.3.** *Consider the following coupled system of fractional differential equations*

$$\begin{aligned} D^{1.8}\mathbf{x}(t) + D\mathbf{x}(t) + \mathfrak{B}D^{0.8}\mathbf{y}(t) + 2\mathbf{x}(t) - \mathbf{y}(t) &= \mathbf{f}(t) \\ D^{1.8}\mathbf{y}(t) - \mathfrak{G}D^{0.8}\mathbf{x}(t) + D\mathbf{y}(t) - \mathbf{x}(t) &= \mathbf{g}(t) \end{aligned} \tag{53}$$

*In the given* **Figure 5***, we have shown the comparison of exact x t*ð Þ, *y t*ð Þ *and approx-*

*As expected, the method provides a very good approximation to the solution of the problem. At first, we approximate the solutions of the problem at α* ¼ 2 *because the exact solution at α* ¼ 2 *is known. We observe that at very small scale levels, the method provides a very good approximation to the solution. We approximate the absolute error by the*

*Approximate Solutions of Some Boundary Value Problems by Using Operational Matrices…*

*Xerror* ¼ ∣*xexact* � *xapprox*∣*:*

*Yerror* ¼ ∣*yexact* � *yapprox*∣*:*

*absolute error is much less than* <sup>10</sup>�<sup>10</sup> *at scale level M* <sup>¼</sup> <sup>7</sup>*, see* **Figure 6**. *We also approximate the solution at some fractional value of α and observe that as α* ! 2 *the approximate solution approaches the exact solution*, *which guarantees the accuracy of the solution at fractional value of α.* **Figure 6** *shows this phenomenon. In* **Figure 6***, the subplot (a) represents the absolute error of x t*ð Þ *and subplot (b) represents the absolute*

*<sup>D</sup>*<sup>1</sup>*:*<sup>8</sup>*x t*ð Þ� *x t*ðÞþ <sup>3</sup>*y t*ðÞ¼ *f t*ð Þ

*x*ð Þ¼� 0 1, *x*ð Þ¼� 1 1 *and y*ð Þ¼� 0 1, *y*ð Þ¼� 1 1*:*

*<sup>D</sup>*<sup>1</sup>*:*<sup>8</sup>*y t*ðÞþ <sup>4</sup>*x t*ðÞ� <sup>2</sup>*y t*ðÞ¼ *g t*ð Þ, (56)

**Example 4.4.** *Consider the following coupled system*

subject to the boundary conditions

*Solution: The exact solution for α* ¼ *β* ¼ 2 *is*

*Absolute error in approximate solutions at different scale level* M ¼ *3:7 for Example 4.3.*

*We approximate the absolute error at different scale level of M, and observe that the*

*imate x t*ð Þ, *y t*ð Þ *in subplot (a) and (b) respectively.*

*DOI: http://dx.doi.org/10.5772/intechopen.90302*

*formula*

and

*error of y t*ð Þ*.*

**Figure 6.**

**183**

subject to the boundary conditions

*x*ð Þ¼ 0 1, *x*ð Þ¼ 1 2 *and y*ð Þ¼ 0 2, *y*ð Þ¼ 1 2*:*

*Solution: The exact solution is*

$$x(t) = t^5(1-t), \qquad y(t) = t^4(1-t).$$

*We approximate the solution of this problem with this new method. The source terms are given by*

$$\begin{split} f(t) &= 2t^5(t-1) - t^4(t-1) + t^4(6t-5) - \frac{229536516744740625t^8(10t-7)}{1008806316530991104} \\ &+ \frac{1337721910046844375t^8(25t-21)}{1008806316530991104} \end{split} \tag{54}$$

$$\begin{split} g(t) &= t^{\circ}(5t - 4) - t^{\circ}(t - 1) - \frac{11147682583723703125t^{\sharp}(15t - 13)}{6557241057451442176} \\ &- \frac{89181460669789625t^{\sharp}(25t - 16)}{144115188075855872} . \end{split} \tag{55}$$

**Figure 5.** *Comparison of exact and approximate solution of Example 4.3 for different scale level* M*.*

*Approximate Solutions of Some Boundary Value Problems by Using Operational Matrices… DOI: http://dx.doi.org/10.5772/intechopen.90302*

*In the given* **Figure 5***, we have shown the comparison of exact x t*ð Þ, *y t*ð Þ *and approximate x t*ð Þ, *y t*ð Þ *in subplot (a) and (b) respectively.*

*As expected, the method provides a very good approximation to the solution of the problem. At first, we approximate the solutions of the problem at α* ¼ 2 *because the exact solution at α* ¼ 2 *is known. We observe that at very small scale levels, the method provides a very good approximation to the solution. We approximate the absolute error by the formula*

$$X\_{errr} = |\mathbf{x}\_{exact} - \mathbf{x}\_{appprox}|.$$

and

**Example 4.3.** *Consider the following coupled system of fractional differential*

*x*ð Þ¼ 0 1, *x*ð Þ¼ 1 2 *and y*ð Þ¼ 0 2, *y*ð Þ¼ 1 2*:*

ð Þ 1 � *t* , *y t*ðÞ¼ *t*

*We approximate the solution of this problem with this new method. The source*

<sup>5</sup> ð Þ 25*t* � 21

<sup>5</sup> ð Þ� *<sup>t</sup>* � <sup>1</sup> <sup>11147682583723703125</sup>*<sup>t</sup>*

<sup>5</sup> ð Þ 25*t* � 16

16

11

*Comparison of exact and approximate solution of Example 4.3 for different scale level* M*.*

*y t*ðÞþ 2*x t*ð Þ� *y t*ðÞ¼ *f t*ð Þ

<sup>4</sup>ð Þ <sup>1</sup> � *<sup>t</sup> :*

16

1008806316530991104

<sup>5</sup> ð Þ 15*t* � 13

21

<sup>5</sup> ð Þ 10*t* � 7

(54)

<sup>4</sup> ð Þ� <sup>6</sup>*<sup>t</sup>* � <sup>5</sup> <sup>2229536516744740625</sup>*<sup>t</sup>*

6557241057451442176

<sup>144115188075855872</sup> *:* (55)

*<sup>D</sup>*1*:*8*y t*ð Þ� <sup>6</sup>*D*0*:*8*x t*ðÞþ *Dy t*ð Þ� *x t*ðÞ¼ *g t*ð Þ (53)

*<sup>D</sup>*1*:*8*x t*ðÞþ *Dx t*ðÞþ <sup>9</sup>*D*0*:*<sup>8</sup>

*x t*ðÞ¼ *t* 5

<sup>4</sup> ð Þþ *<sup>t</sup>* � <sup>1</sup> *<sup>t</sup>*

1008806316530991104

1337721910046844375*t*

� <sup>89181460669789625</sup>*<sup>t</sup>*

subject to the boundary conditions

*Solution: The exact solution is*

<sup>5</sup> ð Þ� *<sup>t</sup>* � <sup>1</sup> *<sup>t</sup>*

<sup>3</sup> ð Þ� <sup>5</sup>*<sup>t</sup>* � <sup>4</sup> *<sup>t</sup>*

*equations*

*Functional Calculus*

*terms are given by*

*f t*ðÞ¼ 2*t*

þ

*g t*ðÞ¼ *t*

**Figure 5.**

**182**

$$Y\_{error} = |\mathcal{y}\_{exact} - \mathcal{y}\_{approx}|.$$

*We approximate the absolute error at different scale level of M, and observe that the absolute error is much less than* <sup>10</sup>�<sup>10</sup> *at scale level M* <sup>¼</sup> <sup>7</sup>*, see* **Figure 6**. *We also approximate the solution at some fractional value of α and observe that as α* ! 2 *the approximate solution approaches the exact solution*, *which guarantees the accuracy of the solution at fractional value of α.* **Figure 6** *shows this phenomenon. In* **Figure 6***, the subplot (a) represents the absolute error of x t*ð Þ *and subplot (b) represents the absolute error of y t*ð Þ*.*

**Example 4.4.** *Consider the following coupled system*

$$\begin{aligned} D^{1.8}\mathfrak{x}(t) - \mathfrak{x}(t) + \mathfrak{z}y(t) &= f(t) \\ D^{1.8}\mathfrak{y}(t) + 4\mathfrak{x}(t) - 2\mathfrak{y}(t) &= \mathfrak{g}(t), \end{aligned} \tag{56}$$

subject to the boundary conditions

$$\varkappa(\mathbf{0}) = -\mathbf{1}, \quad \varkappa(\mathbf{1}) = -\mathbf{1} \quad \text{and} \quad \jmath(\mathbf{0}) = -\mathbf{1}, \quad \jmath(\mathbf{1}) = -\mathbf{1}.$$

*Solution: The exact solution for α* ¼ *β* ¼ 2 *is*

**Figure 6.** *Absolute error in approximate solutions at different scale level* M ¼ *3:7 for Example 4.3.*

$$\mathbf{x}(t) = t^5 - t^4 - 1, \quad \text{and} \quad \mathbf{y}(t) = t^4 - t^3 - 2. \tag{57}$$

*The source terms are given by*

$$f(t) = \frac{445907303348948125t^{32}(25t - 21)}{3026418949592973312} + t^3 - 4t^4 + 3t^5 - 2,$$

$$g(t) = \frac{89181460669789625t^{2.5}(5t - 4)}{14411518807585872} - 4t^3 + 6t^4 - 2t^5 - 2.$$

*Approximating the solution with the proposed method, we observe that our scheme gives high accuracy of approximate solution. In* **Figure 7***, we plot the exact solutions together with the approximate solutions in* **Figure 7(a)** and **(b)** *for x t*ð Þ *and y t*ð Þ*, respectively. We see from the subplots (a) and (b) that our approximations have close agreement to that of exact solutions*. *This accuracy may be made better by increasing scale level. Further, one can observe that absolute error is below* 10�<sup>10</sup> *in* **Figure 8***, which indicates better accuracy of our proposed method for such types of practical problems of applied sciences.*

*In* **Figure 8***, the subplot (a) represents absolute error for x t*ð Þ *while subplot (b) represents the same quantity for y t*ð Þ*. From the subplots, we see that maximum absolute error for our proposed method for the given problem (4.4) is below* 10�<sup>10</sup>*: This is very*

*Comparison of solution between Legendre wavelet method (LWM) [47], Jacobi polynomial method (JM) and*

*(a) Comparison of exact and approximate solution at scale level* M ¼ 10,*ω* ¼ 3*:*5, *α* ¼ 2, *ν* ¼ 1 *for*

*Approximate Solutions of Some Boundary Value Problems by Using Operational Matrices…*

*ω M α ν* **∥***x***app** � *x***ex∥ at BM ∥***x***app** � *x***ex∥ at WM ∥***x***app** � *x***ex∥ at JM** 0.5 10 2 1 7*:*000ð Þ �3 2*:*966ð Þ �1 1*:*500ð Þ �2 1.5 15 1.6 0.9 6*:*091ð Þ �3 4*:*918ð Þ �2 1*:*623ð Þ �1 2.0 20 1.8 0.8 1*:*237ð Þ �3 2*:*108ð Þ �2 2*:*723ð Þ �2 3.5 25 1.9 0.7 1*:*008ð Þ �3 5*:*795ð Þ �2 1*:*813ð Þ �3

> *x t*ðÞþ *x t*ðÞ¼�*ωπ*ð Þ *sin* ð Þþ *ωπt ωπ <sup>x</sup>*ð Þ¼ <sup>0</sup> 0, *<sup>x</sup>*ð Þ¼� <sup>1</sup> <sup>2</sup>*:* (58)

*small and justifies the efficiency of our constructed method.* **Example 4.5.** *Consider the boundary value problem*

> 2 *Dν*

*Taking α* ¼ 2, *ν* ¼ 1 *and ω* ¼ 1, 3, 5, … , *the exact solution is given by*

*(JM) and Bernstein polynomials method (BM), as shown in* **Figure 9***.*

*x t*ðÞ¼ cosð Þ� *ωπt* 1*:*

*We plot the comparison between exact and approximate solutions to the given example at M* ¼ 10 *and corresponding to ω* ¼ 3*:*5, *α* ¼ 2, *β* ¼ 1*. Further, we approximate the solution through Legendre wavelet method (LWM)* [47], *Jacobi polynomial method*

From **Table 1**, we see that Bernstein polynomials also provide excellent solutions

The above analysis and discussion take us to the conclusion that the new method is very efficient for the solution of boundary value problems as well as initial value

*<sup>D</sup>αx t*ð Þþ ð Þ *ωπ*

*Bernstein polynomials method (BM) for Example 4.5.*

**Figure 9.**

**Table 1.**

**185**

*Example 4.5. (b) Absolute error at* M ¼ 10*.*

*DOI: http://dx.doi.org/10.5772/intechopen.90302*

to fractional differential equations [48].

**5. Conclusion and future work**

**Figure 8.** *Absolute error for different scale level* M ¼ *3:7 for Example 4.4.*

*Approximate Solutions of Some Boundary Value Problems by Using Operational Matrices… DOI: http://dx.doi.org/10.5772/intechopen.90302*

#### **Figure 9.**

*x t*ðÞ¼ *t*

*f t*ðÞ¼ <sup>445907303348948125</sup>*<sup>t</sup>*

*g t*ðÞ¼ <sup>89181460669789625</sup>*<sup>t</sup>*

*The source terms are given by*

*Functional Calculus*

**Figure 7.**

**Figure 8.**

**184**

<sup>5</sup> � *<sup>t</sup>*

<sup>4</sup> � 1, *and y t*ðÞ¼ *<sup>t</sup>*

<sup>3026418949592973312</sup> <sup>þ</sup> *<sup>t</sup>*

<sup>14411518807585872</sup> � <sup>4</sup>*<sup>t</sup>*

*our proposed method for such types of practical problems of applied sciences.*

*Comparison of exact and approximate solution at scale level* M ¼ *3, 7 for Example 4.4.*

*Absolute error for different scale level* M ¼ *3:7 for Example 4.4.*

<sup>3</sup>*:*2ð Þ <sup>25</sup>*<sup>t</sup>* � <sup>21</sup>

<sup>2</sup>*:*5ð Þ <sup>5</sup>*<sup>t</sup>* � <sup>4</sup>

*Approximating the solution with the proposed method, we observe that our scheme gives high accuracy of approximate solution. In* **Figure 7***, we plot the exact solutions together with the approximate solutions in* **Figure 7(a)** and **(b)** *for x t*ð Þ *and y t*ð Þ*, respectively. We see from the subplots (a) and (b) that our approximations have close agreement to that of exact solutions*. *This accuracy may be made better by increasing scale level. Further, one can observe that absolute error is below* 10�<sup>10</sup> *in* **Figure 8***, which indicates better accuracy of*

<sup>4</sup> � *<sup>t</sup>*

<sup>3</sup> � <sup>4</sup>*<sup>t</sup>*

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

<sup>4</sup> <sup>þ</sup> <sup>3</sup>*<sup>t</sup>*

<sup>4</sup> � <sup>2</sup>*<sup>t</sup>*

<sup>3</sup> � <sup>2</sup>*:* (57)

<sup>5</sup> � 2,

<sup>5</sup> � <sup>2</sup>*:*

*(a) Comparison of exact and approximate solution at scale level* M ¼ 10,*ω* ¼ 3*:*5, *α* ¼ 2, *ν* ¼ 1 *for Example 4.5. (b) Absolute error at* M ¼ 10*.*


#### **Table 1.**

*Comparison of solution between Legendre wavelet method (LWM) [47], Jacobi polynomial method (JM) and Bernstein polynomials method (BM) for Example 4.5.*

*In* **Figure 8***, the subplot (a) represents absolute error for x t*ð Þ *while subplot (b) represents the same quantity for y t*ð Þ*. From the subplots, we see that maximum absolute error for our proposed method for the given problem (4.4) is below* 10�<sup>10</sup>*: This is very small and justifies the efficiency of our constructed method.*

**Example 4.5.** *Consider the boundary value problem*

$$\begin{aligned} D^a \mathfrak{x}(t) + (a\mathfrak{\pi})^2 D^b \mathfrak{x}(t) + \mathfrak{x}(t) &= -a\mathfrak{\pi} \left( \sin \left( a\mathfrak{\pi} t \right) + a\mathfrak{\pi} \right) \\ \mathfrak{x}(\mathbf{0}) = \mathbf{0}, \quad \mathfrak{x}(\mathbf{1}) &= -\mathbf{2} \end{aligned} \tag{58}$$

*Taking α* ¼ 2, *ν* ¼ 1 *and ω* ¼ 1, 3, 5, … , *the exact solution is given by*

$$\mathbf{x}(t) = \cos\left(a\pi t\right) - \mathbf{1}.$$

*We plot the comparison between exact and approximate solutions to the given example at M* ¼ 10 *and corresponding to ω* ¼ 3*:*5, *α* ¼ 2, *β* ¼ 1*. Further, we approximate the solution through Legendre wavelet method (LWM)* [47], *Jacobi polynomial method (JM) and Bernstein polynomials method (BM), as shown in* **Figure 9***.*

From **Table 1**, we see that Bernstein polynomials also provide excellent solutions to fractional differential equations [48].

#### **5. Conclusion and future work**

The above analysis and discussion take us to the conclusion that the new method is very efficient for the solution of boundary value problems as well as initial value

## *Functional Calculus*

problems including coupled systems of fractional differential equations. One can easily extend the method for obtaining the solution of such types of problems with other kinds of boundary and initial conditions. Bernstein polynomials also give best approximate solutions to fractional order differential equations like Legendre wavelet method (LWM), approximation by Jacobi polynomial method (JPM), etc. The new operational matrices obtained in this method can easily be extended to two-dimensional and higher dimensional cases, which will help in the solution of fractional order partial differential equations. Also, we compare our result to that of approximate methods for different scale levels. We observed that the proposed method is also an accurate technique to handle numerical solutions.

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