**2.1. Fuzzy numbers**

well as the historical state of the system. For these reasons, it is intensively developed and advanced, and existence of its solution is studied by well‐known authors, Euler, Laplace, Liouville, Riemann, Fourier, Abel, Caputo, etc., to further widen its scope in describing various real‐world problems of science, for instance see [1–6]. Another wide‐spreading exploration of mathematics is theory of fuzzy calculus, which has a lot of interesting applications in physics, engineering, mechanics, and many others. It is the theory of a particular type of interval‐ valued functions, in which mapping is made in such a way that it takes all the possible values in 0, 1 and not only the crisp values as found in usual interval‐valued functions. After the inception of fuzzy set theory by Zadeh [7], its attributes have been extended and established to overcome impreciseness of parameters and structures in mathematical modeling, reasoning,

Advanced development of mathematical theories and techniques has gained very high standard. On the basis of classical theories, new theories are pioneered by undergoing its inadequacies and widening its scope in many disciplines. In a similar manner, the aforemen‐ tioned theories have been brought together in modeling different aspects of applied sciences, to analyze the change in the respective system at each fractional step with the uncertain parameters. Agarwal et al. [13] initiatively incorporated uncertainty into dynamical system, modeled fractional differential equations with uncertainty, and studied its possible solutions. Ahmad et al. [14] described the situation of impreciseness of initial values of fractional differential equations and discussed its solutions by utilizing Zadeh's extension principle. In [15, 16], authors considered the concept of Caputo and Riemann fractional derivative, respec‐ tively, together with the Hukuhara differentiability and demonstrated the fuzzy fractional

In light of noteworthy applications of above‐mentioned theories, in this chapter, we demon‐ strate fractional order dynamical models in fuzzy environment to depict unequivocal frac‐ tional differential equations of dynamical system. Moreover, we investigate its numerical solutions using the well‐known Grünwald‐Letnikov's fractional definition. This definition is widely applicable as a numerical scheme to solve linear and nonlinear differential equations of fractional order [24–26]. It is considered as an extended form of the classical Euler method. Here it will be utilized, for the first time, to solve fractional differential equations of imprecise functions. Sequentially, this chapter features description of fuzzy theory and fuzzy‐valued functions for the explanation of impreciseness, modeling of system of nonlinear fractional order differential equations with imprecise functions, deliberation of Grünwald‐Letnikov's fractional approach in conjunction with its truncation error for the proposed system, tabulated and pictorial investigations of some examples, and conclusive remarks of the undergone

Fuzzy calculus theory is the branch of mathematical analysis that deals with the interval analysis of imprecise functions. This section comprises some rudiments of fuzzy calculus

and computing [8–12].

differential equations and a lot of others [17–23].

168 Numerical Simulation - From Brain Imaging to Turbulent Flows

experiments and findings of the whole manuscript.

**2. Basic descriptions**

Let E be the set of subsets of the real axis R . If τ ∈E and *<sup>τ</sup>* : <sup>0</sup>, <sup>1</sup> <sup>→</sup> R such that, *τ* is normal, fuzzy convex, upper semi‐continuous membership function and compactly supported on the real axis R , then E is said to be the space of fuzzy numbers *τ*. Any τ∈E can be

represented in level sets explicitly, i.e. [ ] τ ττ ( ), ( ) <sup>=</sup> for ∈[ ] 0,1 , where τ ( ) and τ ( ) signify as the lower and upper branches of *τ*, respectively, that satisfy the following conditions:


$$\mathbf{c} \quad \underline{\mathbf{r}}(\lambda) \le \overline{\mathbf{r}}(\lambda)$$

The sum and scalar product of any fuzzy number is the consequence of Zadeh's extension principal. Let ⊕ , • and Θ be the symbols of addition, multiplication and subtraction, accordingly, for fuzzy numbers, which will be greatly used throughout the paper, then, for ∈[ ] 0,1 :

$$\mathbf{i.}$$

$$\begin{aligned} \textbf{i.} \qquad \left[\tau \oplus \nu\right]^\lambda = \left[\tau\right]^\lambda \oplus \left[\nu\right]^\lambda = \left[\underline{\tau}\left(\lambda\right) + \underline{\nu}\left(\lambda\right), \overline{\tau}\left(\lambda\right) + \overline{\nu}\left(\lambda\right)\right] \qquad \tau, \nu \in \mathbf{GL} \end{aligned}$$

**ii.**

$$a \in \mathfrak{G}, \left[a\overline{\tau}\right]^{\mathfrak{X}} = a\left[\overline{\tau}\right]^{\mathfrak{X}} = \begin{cases} \left[a\underline{\tau}\left(\lambda\right), a\overline{\tau}\left(\lambda\right)\right] & \text{if } a > 0\\ \{0\} & \text{if } a = 0\\ \left[a\,\overline{\tau}\left(\lambda\right), a\underline{\tau}\left(\lambda\right)\right] & \text{if } a < 0 \end{cases}$$

**iii.**

$$\begin{bmatrix} \tau \bullet \upsilon \end{bmatrix}^{\mathbb{X}} = \begin{bmatrix} \min\left\{ \underline{\tau}(\mathsf{x})\underline{\nu}(\mathsf{x}), \underline{\tau}(\mathsf{x})\overline{\nu}(\mathsf{x}), \overline{\tau}(\mathsf{x})\underline{\nu}(\mathsf{x}), \overline{\tau}(\mathsf{x})\overline{\nu}(\mathsf{x}) \right\}, \\\max\left\{ \underline{\tau}(\mathsf{x})\underline{\nu}(\mathsf{x}), \underline{\tau}(\mathsf{x})\overline{\nu}(\mathsf{x}), \overline{\tau}(\mathsf{x})\underline{\nu}(\mathsf{x}), \overline{\tau}(\mathsf{x})\overline{\nu}(\mathsf{x}) \right\} \end{bmatrix}$$

**iv.**

τυ τ υ τ υ τ υ τ υmin{ ( ) ( ), ( ) ( )} ,max{ ( ) ( ), ( ) ( )} Θ= − − ℘ − −

The distance between any two fuzzy numbers *τ* and *υ* is given by the Hausdorff metric as:

$$\mathbf{D}\left(\tau,\nu\right) = \sup\_{\lambda \in \left[0,1\right]} \mathbf{D}\left(\left[\tau\right]^{\lambda}, \left[\nu\right]^{\lambda}\right) = \sup\_{\lambda \in \left[0,1\right]} \max\left\{ \left| \underline{\tau}\left(\lambda\right) - \underline{\nu}\left(\lambda\right) \right|, \left| \overline{\tau}\left(\lambda\right) - \overline{\nu}\left(\lambda\right) \right| \right\} \tag{1}$$

Thus, (E , ) defines a complete metric space with the properties of Hausdorff metric for fuzzy numbers.

#### **2.2. Fuzzy-valued Function and its fractional derivative**

Any interval‐valued function is said to be a fuzzy‐valued function if is defined as . Its ‐level set can be represented by real‐valued functions and as its lower and upper branches, accordingly, i.e. , . Moreover, if and exist as finite fuzzy numbers, then exists. Consequently, let be the space of continuous fuzzy‐valued functions, then if and are continuous. The arithmetic for any two fuzzy‐valued functions and can be defined as previously mentioned in Section 2.1 for fuzzy numbers. Subsequent to existence of limit and continuity of , the fuzzy‐valued function (*t*) is said to be differentiable at each *t*0∈ *a*, *b* , if exists, such that

$$\mathbb{E}\_{\mathbb{C}^{\tilde{\mathbf{y}}^{\prime}}}(t\_{\mathbf{o}}) = \mathop{\mathrm{Lim}}\_{h \to \mathbf{0}} \frac{\mathbb{E}^{\tilde{\mathbf{y}}^{\tilde{\mathbf{y}}^{\prime}}}(t\_{\mathbf{o}} + h) \Theta\_{\mathbb{C}^{\tilde{\mathbf{y}}^{\prime}}}(t\_{\mathbf{o}})}{h} \tag{2}$$

where *h* is taken in a way that (*t*<sup>0</sup> + *h* )∈(*a*, *b*). For , is said to be differentiable at *t* ∈ *a*, *b* if its lower function and upper function are differentiable at *<sup>t</sup>* <sup>∈</sup> *<sup>a</sup>*, *<sup>b</sup>* , i.e. for all <sup>∈</sup> 0, 1 ,

$$\mathbf{L}\_{\mathbb{C}}\tilde{\varphi}^{\zeta}(t) = \left[ \min \left\{ \frac{d}{dt} \underline{\underline{\zeta}^{\zeta}}(t;\lambda), \frac{d}{dt} \overline{\underline{\zeta}^{\zeta}}(t;\lambda) \right\}, \max \left\{ \frac{d}{dt} \underline{\underline{\zeta}^{\zeta}}(t;\lambda), \frac{d}{dt} \overline{\underline{\zeta}^{\zeta}}(t;\lambda) \right\} \right] \tag{3}$$

In a similar manner, fractional order differential of can be defined as, for all <sup>∈</sup> 0, 1 , if and are differentiable of order *ω* >0, then is differentiable of order *ω* >0, i.e.

$$\underline{\sf d}^{o}\bigcirc\widetilde{\varphi}^{\epsilon}(t) = \left[\min\left\{D^{o}\_{\iota}\,\underline{\sf C^{\epsilon}}(t;\lambda),D^{o}\_{\iota}\,\overline{\preccurlyepsilon^{\epsilon}}(t;\lambda)\right\},\max\left\{D^{o}\_{\iota}\,\underline{\sf C^{\epsilon}}(t;\lambda),D^{o}\_{\iota}\,\overline{\preccurlyepsilon^{\epsilon}}(t;\lambda)\right\}\right] \tag{4}$$

where can be either fuzzy Riemann‐Liouville fractional differential operator or fuzzy Caputo-type fractional differential operator [15, 16, 19, 22, 23]. Here it is considered as fuzzy Caputo‐type fractional derivative that is approximated by Grünwald‐Letnikov's approach, illustrated in the next sequel.

#### **2.3. System of fractional order fuzzy differential equations**

( )

 τ υ

170 Numerical Simulation - From Brain Imaging to Turbulent Flows

**2.2. Fuzzy-valued Function and its fractional derivative**

to be differentiable at each *t*0∈ *a*, *b* , if exists, such that

differentiable at *<sup>t</sup>* <sup>∈</sup> *<sup>a</sup>*, *<sup>b</sup>* , i.e. for all <sup>∈</sup> 0, 1 ,

<sup>0</sup> <sup>0</sup> Lim*<sup>h</sup>*

∈ ∈

τυ

numbers.

i.e.

[ ] ( ) [] [] [ ] { ( ) ( ) ( ) ( ) } 0,1 0,1

Thus, (E , ) defines a complete metric space with the properties of Hausdorff metric for fuzzy

Any interval‐valued function is said to be a fuzzy‐valued function if is defined as

as its lower and upper branches, accordingly, i.e. ,

functions and can be defined as previously mentioned in Section 2.1 for fuzzy numbers. Subsequent to existence of limit and continuity of , the fuzzy‐valued function (*t*) is said

( ) ( ) () 0 0

where *h* is taken in a way that (*t*<sup>0</sup> + *h* )∈(*a*, *b*). For , is said to

be differentiable at *t* ∈ *a*, *b* if its lower function and upper function are

In a similar manner, fractional order differential of can be defined as, for all <sup>∈</sup> 0, 1 , if

and are differentiable of order *ω* >0, then is differentiable of order *ω* >0,

( ) min ; , ; ,max ; , ; ( ) ( ) ( ) ( ) *dd dd t tt tt dt dt dt dt* ′ <sup>=</sup>

*<sup>t</sup>* <sup>→</sup> *<sup>h</sup>*

. Its ‐level set can be represented by real‐valued functions and

exists. Consequently, let be the space of continuous fuzzy‐valued functions, then

if and are continuous. The arithmetic for any two fuzzy‐valued

*th t*

F FF FF (3)

+ Θ ′ <sup>=</sup> F F <sup>F</sup> (2)

, sup , sup max ,

**D D** = = −−

 τ  υ

. Moreover, if and exist as finite fuzzy numbers, then

 τ  υ

(1)

In particular, modeling of differential equations of fractional order in imprecise characteristics is obtained by encompassing fuzzy‐valued functions. Let , then fuzzy differ‐ ential equation of fractional order *ω* ∈(0, 1 , subjected to initial conditions, is structured as:

$$
\Delta \mathfrak{P}^\circ \circ \tilde{\mathbb{P}}(t) = \Psi \left( t, \Box \tilde{\mathbb{P}}(t) \right) \tag{5}
$$

$$
\Box \mathfrak{H}^{\tilde{\varphi}} \begin{pmatrix} t\_o \\ \end{pmatrix} = \Box \tilde{\ell} \tag{6}
$$

where the unknown fuzzy‐valued function can be written in form of ‐levels as, for all ∈ 0, 1 , , where as can be linear or nonlinear term in the form of fuzzy‐valued function and is the fuzzy number, which can also be expressed as , for all . Concisely, Eq. (5) is considered to have a unique and stable solution, for the reason that is continuous and satisfies the Lipschitz condition, i.e. there exists *L* >0 such that for

$$\mathbf{D}\left(\boldsymbol{\Psi}\left(\mathbf{t},\boldsymbol{\varepsilon}\bar{\boldsymbol{\varphi}}\right),\boldsymbol{\Psi}\left(\mathbf{t},\boldsymbol{\varepsilon}\bar{\boldsymbol{\varphi}}\right)\right) \leq L.\mathbf{D}\left(\boldsymbol{\varepsilon}\bar{\boldsymbol{\varphi}},\boldsymbol{\varepsilon}\bar{\boldsymbol{\varphi}}\right) \quad \forall\left(\mathbf{t},\boldsymbol{\varepsilon}\bar{\boldsymbol{\varphi}}\right),\left(\mathbf{t},\boldsymbol{\varepsilon}\bar{\boldsymbol{\varphi}}\right) \in \mathbf{Q},\ \boldsymbol{\varepsilon}\bar{\boldsymbol{\xi}}\cap\bar{\boldsymbol{\mathcal{W}}} \in \mathbf{Q}\tag{7}$$

Many papers [14, 15, 22] comprise the theorems of stability and uniqueness of the solution of Eq. (5).

Here, we consider the system of fractional order fuzzy differential equations of the following form:

$$\begin{aligned} \mathbf{d}\tilde{\mathbb{P}}^{\alpha} \odot \tilde{\mathbb{P}}\_{\varepsilon}(t) &= \Psi \left( \odot \tilde{\mathbb{P}}\_{\varepsilon}(t), \odot \tilde{\mathbb{P}}\_{\varepsilon}(t), \cdots, \odot \tilde{\mathbb{P}}\_{\varepsilon}(t) \right) \\ \mathbf{d}\tilde{\mathbb{P}}^{\alpha} \odot \tilde{\mathbb{P}}\_{\varepsilon}(t) &= \Psi \left( \odot \tilde{\mathbb{P}}\_{\varepsilon}(t), \odot \tilde{\mathbb{P}}\_{\varepsilon}(t), \cdots, \odot \tilde{\mathbb{P}}\_{\varepsilon}(t) \right) \end{aligned} \tag{8}$$

$$\mathbf{d}\tilde{\mathbb{A}}^{\alpha} \odot \tilde{\mathbb{A}}\_{\ast}(t) = \Psi \left( \odot \tilde{\mathbb{A}}\_{\prime}(t), \odot \tilde{\mathbb{A}}\_{\varepsilon}(t), \cdots, \odot \tilde{\mathbb{A}}\_{\ast}(t) \right),$$

with the initial conditions,

$$
\varepsilon \circ \tilde{\mathsf{K}}\_{\prime}^{\varphi} \left( t\_{0} \right) = \tilde{\nu}\_{1}, \varepsilon \circ \tilde{\mathsf{K}}\_{\varepsilon}^{\varphi} \left( t\_{0} \right) = \tilde{\nu}\_{2}, \dots, \varepsilon \circ \tilde{\mathsf{K}}\_{\ast}^{\varphi} \left( t\_{0} \right) = \tilde{\nu}\_{\ast} \tag{9}
$$

where *ν*˜ <sup>1</sup>, *ν*˜ 2, …, *ν*˜ *<sup>n</sup>* are the fuzzy numbers that can be written as, for all , *n*≥1, *ω1*, *ω2*, …, *ωn* are the fractional orders such that *ω<sup>n</sup>* ∈(0, 1 and the right hand side of Eq. (8) represent a system of fuzzy nonlinear equations with crisp coefficients *kij*, *i* ≥1, *j* ≤*n*, i.e.

$$\Psi\left(\ominus\tilde{\mathbb{K}}/(t), \ominus\tilde{\mathbb{K}}\_{\mathbb{Z}}(t), \dots, \ominus\tilde{\mathbb{K}}\_{\mathbb{w}}(t)\right) = \sum\_{j=1}^{n} k\_{ij} \ominus\tilde{\mathbb{K}}\_{\mathbb{C}}''\left(t\right), \qquad m \ge 1 \tag{10}$$

Therefore, Eq. (8) can be remodeled as:

$$\begin{aligned} \mathfrak{d}\_{l}^{\alpha\_{1}} \odot \widetilde{\mathcal{C}}\_{l}^{\*} \left( t \right) &= \sum\_{j=1}^{n} k\_{1,j} \widetilde{\mathcal{S}}\_{l}^{\*,\*} \left( t \right) = k\_{1,1} \widetilde{\mathcal{S}}\_{l}^{\*,\*} \left( t \right) \oplus k\_{1,2} \widetilde{\mathcal{S}}\_{l}^{\*,\*} \left( t \right) \oplus \dots \oplus k\_{1s} \widetilde{\mathcal{S}}\_{l}^{\*,\*} \left( t \right), \\\\ \mathfrak{d}\_{l}^{\alpha\_{2}} \odot \widetilde{\mathcal{C}}\_{l}^{\*} \left( t \right) &= \sum\_{j=1}^{n} k\_{2,j} \widetilde{\mathcal{S}}\_{l}^{\*,\*} \left( t \right) = k\_{2,1} \widetilde{\mathcal{S}}\_{l}^{\*,\*} \left( t \right) \oplus k\_{2,2} \widetilde{\mathcal{S}}\_{l}^{\*,\*} \left( t \right) \oplus \dots \oplus k\_{2s} \widetilde{\mathcal{S}}\_{l}^{\*,\*} \left( t \right), \\\\ & \vdots \\\\ \mathfrak{d}\_{1}^{\alpha\_{1}} \odot \widetilde{\mathcal{C}}\_{l}^{\*} \left( t \right) = \widetilde{\mathfrak{d}}\_{l}^{\*} \left( t \right) - k\_{1} \widetilde{\mathcal{S}}\_{l}^{\*,\*} \left( t \right) \oplus k\_{2} \widetilde{\mathcal{S}}\_{l}^{\*} \left( t \right) \oplus \dots \oplus k\_{2s} \widetilde{\mathcal{S}}\_{l}^{\*} \left( t \right) \end{aligned} \tag{11}$$

And as mentioned earlier, are taken as the fuzzy Caputo‐type fractional differential operators and are numerically interpreted using Grünwald‐Letnikov's fractional derivative definition.

### **3. Grünwald‐Letnikov's fractional derivative**

This section comprises the description of Grünwald‐Letnikov's fractional derivative in conjunction with the algorithm to solve the system of Eq. (11) and undergoes some requisite theorem and lemma of the governing approach.

Consider a function in finite interval [0, *T*], let the interval be divided into equidistant grids of step size *h* as:

$$0 = \eta\_0 < \eta\_1 < \dots < \eta\_{\sigma} = t = \sigma h \quad \text{with } \eta\_{\sigma} - \eta\_{\sigma - 1} = h \tag{12}$$

$$\sideset{}{^{GL}}{}{\mathop{D}}\_{i}^{\alpha} \odot \mathcal{F}\left(t\right) = \lim\_{h \to 0} \frac{1}{h^{\alpha}} \sum\_{i=0}^{\left\lfloor \frac{t}{h} \right\rfloor} (-1)^{i} \binom{\alpha}{i} \S^{\mathcal{F}}\left(t - ih\right) \tag{13}$$

where ( *ω i* ) are the binomial coefficients that are obtained by the formula:

$$
\binom{\alpha}{i} = \frac{\Gamma\left(\alpha + 1\right)}{i!\Gamma\left(\alpha - i + 1\right)}\tag{14}
$$

and *<sup>t</sup> <sup>h</sup>* represents the integral part.

#### **3.1. Lemma**

(11)

() () () () ( , ,, ) *<sup>n</sup> <sup>t</sup> t tt t*

() () 01 0 2 0 , ,, ( ) *<sup>n</sup> tt t* = =

where *ν*˜ <sup>1</sup>, *ν*˜ <sup>2</sup>, …, *ν*˜ *<sup>n</sup>* are the fuzzy numbers that can be written as, for all

the right hand side of Eq. (8) represent a system of fuzzy nonlinear equations with crisp

1 , ,, , 1 *n ij j t t t ktm* =

And as mentioned earlier, are taken as the fuzzy Caputo‐type fractional differential operators and are numerically interpreted using Grünwald‐Letnikov's fractional derivative

This section comprises the description of Grünwald‐Letnikov's fractional derivative in conjunction with the algorithm to solve the system of Eq. (11) and undergoes some requisite

Ψ =≥ ∑ <sup>m</sup> XX X X 12 n <sup>j</sup> (10)

 ν<sup>=</sup> XX X 12 n (9)

, *n*≥1, *ω1*, *ω2*, …, *ωn* are the fractional orders such that *ω<sup>n</sup>* ∈(0, 1 and

 D X XX X n 12 n = Ψ

νν

( () () ()) ( )

ω

172 Numerical Simulation - From Brain Imaging to Turbulent Flows

with the initial conditions,

coefficients *kij*, *i* ≥1, *j* ≤*n*, i.e.

definition.

Therefore, Eq. (8) can be remodeled as:

**3. Grünwald‐Letnikov's fractional derivative**

theorem and lemma of the governing approach.

Let be a smooth function in 0, *T* , such that it can be expressed as a power series for *t* <*T* , where *t* is the integral part of *t*, then the Grünwald‐Letnikov's approximation for each 0<*t* <*T* , a series of step size *h* and *t* =*σh* can be stated as:

$${}^{i\mathcal{C}}D\_{i}^{\alpha} \lozenge{}^{\varphi} \big( t \big) = \frac{1}{h^{\alpha}} \sum\_{i=0}^{\sigma} (-1)^{i} \binom{\alpha}{i} \big\{ \forall^{\mathcal{C}} \big( t\_{\sigma-i} \big) + O \big( h \big) \big} \qquad \qquad \left( h \rightarrow 0 \right) \tag{15}$$

This definition is considered to be equivalent to the definition of Riemann‐Liouville fractional derivative and for equivalence to Caputo's fractional definition the following term of initial value is added to the right hand side of Eq. (15), i.e.

$${}^{GL}D\_{i}^{\alpha} \heartsuit^{\varphi} \left(t\right) = \frac{1}{h^{\alpha}} \sum\_{i=0}^{\sigma} (-1)^{i} \binom{\alpha}{i} \heartsuit^{\varphi} \left(t\_{\sigma-i}\right) - \frac{t\_{\sigma}^{-\alpha}}{\Gamma\left(1-\alpha\right)} \heartsuit^{\varphi} \left(0\right) \tag{16}$$

That becomes zero if initial values of Caputo‐type differential equations are homogeneous and again reduces to that of Riemann‐Liouville definition. Since here the fuzzy Caputo‐type fractional differential equations are considered with inhomogeneous initial values, the definition in Eq. (16) will be used for the approximation of Eq. (11).

Now let be a fuzzy‐valued function such that , then Grünwald‐Letnikov's fractional derivative of (*t*) is expressed as:

$$\stackrel{\circ}{\Gamma}^{\infty} \blacksquare \mathsf{D}\_{\tau}^{\mu \nu} \stackrel{\circ}{\lhd}^{\tau} \left( t \right) = \frac{1}{h^{\nu}} \sum\_{i=0}^{\tau} \left( -1 \right)^{i} \binom{\alpha}{i} \stackrel{\circ}{\lhd}^{\tau} \left( t\_{\sigma \tau \nu} \right) \oplus \frac{\stackrel{\circ}{\nu \zeta}^{\tau \nu}}{\Gamma \left( 1 - \alpha \nu \right)} \stackrel{\circ}{\lhd}^{\tau} \left( 0 \right) \tag{17}$$

and in ‐level sets it is sorted out as, for all ∈ 0, 1 ,

$$\mathbf{f}^{GL}\_{\tau} \square \widetilde{\mathbf{y}^{\tau}}(t) = \begin{bmatrix} \frac{1}{h^{\alpha}} \sum\_{i=0}^{\sigma} (-1)^{i} \binom{\alpha}{i} \widetilde{\mathbf{y}^{\tau}}(t\_{\sigma-i}, \lambda) - \frac{t\_{\sigma}^{-\alpha}}{\Gamma \{1 - \alpha\}} \widetilde{\mathbf{y}^{\tau}}(0, \lambda),\\ \frac{1}{h^{\alpha}} \sum\_{i=0}^{\sigma} (-1)^{i} \binom{\alpha}{i} \widetilde{\mathbf{y}^{\tau}}(t\_{\sigma-i}, \lambda) - \frac{t\_{\sigma}^{-\alpha}}{\Gamma \{1 - \alpha\}} \widetilde{\mathbf{y}^{\tau}}(0, \lambda) \end{bmatrix} \tag{18}$$

Next consider the fractional system in Eq. (11), for the cases of inhomogeneous initial values. Assume the uniform grids *t<sup>σ</sup>* =*σ h* , where *σ* =1, …, *M* , such that *Mh* =*T* , *M* ∈ . Applying Grünwald‐Letnikov's fractional derivative on left hand sides of Eq. (11) we get,

$$\begin{split} \frac{1}{h^{\alpha\_1}} \sum\_{i=0}^{\sigma} (-1)^i \binom{\alpha\_1}{i} \circledast\_{\widetilde{\gamma}\_i}^{\widetilde{\gamma}\_i} ((\sigma - i)h) \Theta \frac{(\sigma h)^{-\alpha\_1}}{\Gamma(1 - \alpha\_1)} \circledast\_{\widetilde{\gamma}\_i}^{\widetilde{\gamma}\_i} (0) &= \sum\_{j=1}^n k\_{1,j} \circledast\_{\widetilde{\gamma}\_j}^{\widetilde{\gamma}\_n} (\sigma h) \,. \\\\ \frac{1}{h^{\alpha\_2}} \sum\_{i=0}^{\sigma} (-1)^i \binom{\alpha\_2}{i} \circledast\_{\widetilde{\gamma}\_i}^{\widetilde{\gamma}\_i} ((\sigma - i)h) \Theta \frac{(\sigma h)^{-\alpha\_2}}{\Gamma(1 - \alpha\_2)} \circledast\_{\widetilde{\gamma}\_i}^{\widetilde{\gamma}\_i} (0) &= \sum\_{j=1}^n k\_{1,j} \circledast\_{\widetilde{\gamma}\_j}^{\widetilde{\gamma}\_n} (\sigma h) \,. \\\\ \frac{1}{h^{\alpha\_n}} \sum\_{i=0}^{\sigma} (-1)^i \binom{\alpha\_n}{i} \circledast\_{\widetilde{\gamma}\_n}^{\widetilde{\gamma}\_n} ((\sigma - i)h) \Theta \frac{(\sigma h)^{-\alpha\_n}}{\Gamma(1 - \alpha\_n)} \circledast\_{\widetilde{\gamma}\_n}^{\widetilde{\gamma}\_n} (0) &= \sum\_{j=1}^n k\_{\alpha\_n} \circledast\_{\widetilde{\gamma}\_j}^{\widetilde{\gamma}\_n} (\sigma h) \end{split} \tag{19}$$

Solving above system fuzzy-valued functions of respective fuzzy functions are generated at different grid points.

#### **3.2. Theorem: truncation error**

Let fuzzy‐valued functions be the approximations to the true solutions *X*˜ 1(*tσ*), *<sup>X</sup>*˜ 2(*tσ*), <sup>⋯</sup>, *<sup>X</sup>*˜ *<sup>n</sup>*(*tσ*), respectively and consider *Ψ* satisfies Lipchitz condition, then the local truncation error of the proposed numerical approach is *O*(*h* 1+*ωn*), for *n* ≥1, i.e.

#### Numerical Solution of System of Fractional Differential Equations in Imprecise Environment http://dx.doi.org/10.5772/64150 175

$$\begin{aligned} \varepsilon \check{\boldsymbol{\xi}}\_{\cdot}^{\*} \left( \boldsymbol{t}\_{\sigma} \right) \Theta \check{\boldsymbol{X}}\_{1} \left( \boldsymbol{t}\_{\sigma} \right) &= O \left( \boldsymbol{h}^{1 \circ \boldsymbol{\alpha}\_{1}} \right), \\ \varepsilon \check{\boldsymbol{\xi}}\_{\cdot}^{\*} \left( \boldsymbol{t}\_{\sigma} \right) \Theta \check{\boldsymbol{X}}\_{2} \left( \boldsymbol{t}\_{\sigma} \right) &= O \left( \boldsymbol{h}^{1 \circ \boldsymbol{\alpha}\_{2}} \right), \\ \vdots & \quad \vdots \\ \varepsilon \check{\boldsymbol{\xi}}\_{\cdot}^{\*} \left( \boldsymbol{t}\_{\sigma} \right) \Theta \check{\boldsymbol{X}}\_{\kappa} \left( \boldsymbol{t}\_{\sigma} \right) &= O \left( \boldsymbol{h}^{1 \circ \boldsymbol{\alpha}\_{\*}} \right). \end{aligned} \tag{20}$$

#### **Proof:**

(17)

(19)

Now let be a fuzzy‐valued function such that , then Grünwald‐Letnikov's

( ) ( ) ( ) ( )

F F

F F D (18)

*t*

σ

−

ω

ω

ω

ω

*t*

σ

−

<sup>1</sup> <sup>1</sup> , 0, , <sup>1</sup>

− −

*i*

−

*t*

σ

*t*

σ

Next consider the fractional system in Eq. (11), for the cases of inhomogeneous initial values. Assume the uniform grids *t<sup>σ</sup>* =*σ h* , where *σ* =1, …, *M* , such that *Mh* =*T* , *M* ∈ . Applying

Solving above system fuzzy-valued functions of respective fuzzy functions are generated at

Let fuzzy‐valued functions be the approximations to the true

then the local truncation error of the proposed numerical approach is *O*(*h* 1+*ωn*), for *n* ≥1, i.e.

*<sup>n</sup>*(*tσ*), respectively and consider *Ψ* satisfies Lipchitz condition,

<sup>1</sup> <sup>1</sup> , 0, <sup>1</sup>

−

*i*

Γ −

( ) ( ) ( ) ( )

fractional derivative of (*t*) is expressed as:

174 Numerical Simulation - From Brain Imaging to Turbulent Flows

and in ‐level sets it is sorted out as, for all ∈ 0, 1 ,

0

σ

∑

=

σ

∑

ω

ω

*h i*

*h i*

*i*

ω

ω

Grünwald‐Letnikov's fractional derivative on left hand sides of Eq. (11) we get,

Γ − <sup>=</sup> − −

*i*

0

*i*

=

( )

*t*

*i GL*

F

*t*

different grid points.

solutions *X*˜

**3.2. Theorem: truncation error**

1(*tσ*), *<sup>X</sup>*˜

2(*tσ*), <sup>⋯</sup>, *<sup>X</sup>*˜

ω

Assume the *n*th equation of the system (19) and on applying Grünwald‐Letnikov's fractional derivative we have,

$$\frac{1}{h^{\alpha\_s}} \sum\_{i=0}^{\sigma} (-1)^i \binom{\alpha\_s}{i} \mathfrak{s} \tilde{\xi}\_{\boldsymbol{\sigma}}^{\boldsymbol{\sigma}}(t\_{\sigma-i}) \Theta \frac{t\_{\sigma}^{-\alpha\_s}}{\Gamma(1-\alpha\_s)} \odot \tilde{\xi}\_{\boldsymbol{\sigma}}^{\boldsymbol{\sigma}}(0) = \Psi \left( \odot \breve{\xi}\_{\boldsymbol{\prime}}^{\boldsymbol{\sigma}}(t\_{\sigma}), \odot \breve{\xi}\_{\boldsymbol{\sigma}}^{\boldsymbol{\sigma}}(t\_{\sigma}), \cdot, \cdot, \odot \breve{\xi}\_{\boldsymbol{\sigma}}^{\boldsymbol{\sigma}}(t\_{\sigma}) \right) \tag{21}$$

for *n* ≥1 and from Lemma 3.1 we can attain,

$$\frac{1}{h^{\alpha\_s}} \sum\_{i=0}^{\sigma} (-1)^i \binom{\alpha\_s}{i} \tilde{X}\_{\pi}(t\_{\sigma-i}) \Theta \frac{t\_{\sigma}^{-\alpha\_s}}{\Gamma(1-\alpha\_s)} \tilde{X}\_{\pi}(0) + O(h) = \Psi \left( \tilde{X}\_1(t\_{\sigma}), \tilde{X}\_2(t\_{\sigma}), \dots, \tilde{X}\_s(t\_{\sigma}) \right) \tag{22}$$

Subtracting Eq. (22) from Eq. (21),

$$\begin{split} \frac{1}{h^{a}} \sum\_{i=0}^{\sigma} (-1)^{i} \binom{\boldsymbol{\alpha}\_{\star}}{i} \check{X}\_{\star} (\boldsymbol{t}\_{\sigma-i}) \Theta \frac{\boldsymbol{t}\_{\sigma}^{-\boldsymbol{\alpha}\_{\star}}}{\Gamma(1-\boldsymbol{\alpha}\_{\star})} \check{X}\_{\star} (0) \Theta \frac{1}{h^{a}} \sum\_{i=0}^{\sigma} (-1)^{i} \binom{\boldsymbol{\alpha}\_{\star}}{i} \check{\mathsf{P}}\_{\star}^{\boldsymbol{\varepsilon}} (\boldsymbol{t}\_{\sigma-i}) \\ \Theta \frac{\boldsymbol{t}\_{\sigma}^{-\boldsymbol{\alpha}\_{\star}}}{\Gamma(1-\boldsymbol{\alpha}\_{\star})} \check{\mathsf{P}}\_{\star}^{\boldsymbol{\varepsilon}} (0) + O(h) = \Psi \Big( \check{X}\_{1} (\boldsymbol{t}\_{\sigma}), \check{X}\_{2} (\boldsymbol{t}\_{\sigma}), \dots, \check{X}\_{\star} (\boldsymbol{t}\_{\sigma}) \Big) \\ \Theta \Psi \Big( \operatorname{c} \check{\mathsf{P}}\_{\uparrow} (\boldsymbol{t}\_{\sigma}), \operatorname{c} \check{\mathsf{P}}\_{\downarrow} (\boldsymbol{t}\_{\sigma}), \dots, \operatorname{c} \check{\mathsf{P}}\_{\star} (\boldsymbol{t}\_{\sigma}) \Big) \end{split} \tag{23}$$

Let, for *i* =0, 1, …*σ* −1, , then on further manipulation we get,

$$\begin{split} \frac{1}{h^{a\_s}} \Big[ \tilde{X}\_s(t\_\sigma) \Theta \circ \tilde{\mathbb{W}}\_s(t\_\sigma) \Big] + O(h) &= \mathbb{V} \Big( \tilde{X}\_1(t\_\sigma), \tilde{X}\_2(t\_\sigma), \dots, \tilde{X}\_s(t\_\sigma) \Big) \\ &\qquad \qquad \Theta \mathbb{V} \Big( \mathrm{c} \tilde{\mathbb{V}}\_\prime(t\_\sigma), \mathrm{c} \tilde{\mathbb{V}}\_\sim(t\_\sigma), \dots, \mathrm{c} \tilde{\mathbb{V}}\_s(t\_\sigma) \Big) \Big] \end{split} \tag{24}$$

or it can be rearranged as:

$$\begin{aligned} \left[\Box\tilde{\mathbb{K}}\_{\boldsymbol{s}}(\boldsymbol{t}\_{\sigma})\Theta\bar{\boldsymbol{X}}\_{\boldsymbol{s}}(\boldsymbol{t}\_{\sigma})\right] &= h^{\alpha}\mathcal{D}\Big[\Psi\Big(\operatorname{c}\tilde{\mathbb{K}}(\boldsymbol{t}\_{\sigma}), \operatorname{c}\tilde{\mathbb{K}}\_{\boldsymbol{z}}(\boldsymbol{t}\_{\sigma}), \dots, \operatorname{c}\tilde{\mathbb{K}}\_{\boldsymbol{s}}(\boldsymbol{t}\_{\sigma})\Big), \Psi\Big(\bar{\boldsymbol{X}}\_{1}(\boldsymbol{t}\_{\sigma}), \bar{\boldsymbol{X}}\_{2}(\boldsymbol{t}\_{\sigma}), \dots, \bar{\boldsymbol{X}}\_{\boldsymbol{s}}(\boldsymbol{t}\_{\sigma})\Big)\Big] \\ &+ O\Big(h^{\mathrm{l} + \alpha\_{\boldsymbol{s}}}\Big) \end{aligned} \tag{25}$$

where defines Hausdroff distance. On using Lipschitz condition, i.e. Eq. (7), proof is completed by obtaining the following equation:

$$\left(1 - L\_{\boldsymbol{\pi}} h^{a\_s}\right) \left[\bigcirc \tilde{\mathsf{K}}\_{\boldsymbol{\pi}}\left(t\_{\boldsymbol{\sigma}}\right) \Theta \tilde{X}\_{\boldsymbol{\pi}}\left(t\_{\boldsymbol{\sigma}}\right)\right] \leq O\left(h^{1+a\_s}\right) \quad \forall \; n \geq 1\tag{26}$$
