**4. Numerical illustrations**

Subsequent to the algorithm demonstrated in Section 3, here numerical experiments of some system of fuzzy fractional differential equations are presented. Results for fuzzy‐valued functions are depicted in tabular form in the finite interval 0, 1 at different values of *ω* ∈(0, 1 . In addition, error bar pictorials are given for each respective example. All the exact values and calculations are carried out through *Mathematica 10*.

#### **4.1. Example 1**

Following nonlinear fractional system is solved in [27] using homotopy analysis method, here the system is restructured with imprecise functions and as:

$$
\underline{\sf d}\_{\gamma}^{\circ\_{\mathsf{q}}} \odot \widetilde{\mathsf{C}}\_{\gamma}^{\circ}(t) = 0.5 \odot \widetilde{\mathsf{C}}\_{\gamma}^{\circ}(t)
$$

$$
\underline{\sf d}\_{\gamma}^{\circ\_{\mathsf{q}}} \odot \widetilde{\mathsf{C}}\_{\gamma}(t) = \odot \widetilde{\mathsf{C}}\_{\gamma}(t) \oplus \odot \widetilde{\mathsf{C}}\_{\gamma}^{\circ}(t) \tag{27}
$$

with *ω*1, *ω*2∈(0, 1 and subjected to initial conditions

$$\text{Cov}\tilde{\mathbb{P}}(0) = \left[0.75 + 0.25\lambda, 1.125 - 0.125\lambda\right], \quad \text{s}\tilde{\mathbb{P}}\_z(0) = \left[\lambda - 1, 1 - \lambda\right] \tag{28}$$

On applying Grünwald‐Letnikov's fractional definition on left hand side of Eq. (27) and following the algorithm, the differential equations are reduced to nonlinear algebraic equa‐ tions as:

#### Numerical Solution of System of Fractional Differential Equations in Imprecise Environment http://dx.doi.org/10.5772/64150 177

$$\begin{aligned} \frac{1}{h^{a\_1}} \sum\_{i=0}^{\sigma} (-1)^i \binom{a\_1}{i} \mathfrak{z} \tilde{\xi}\_{\cdot}^{\cdot} ( (\sigma - i) h ) \Theta \frac{(\sigma h)^{-a\_1}}{\Gamma(1 - a\_1)} \mathfrak{z} \tilde{\xi}\_{\cdot}^{\cdot} (0) &= 0.5 \ominus \tilde{\xi}\_{\cdot}^{\cdot} ( \sigma h ), \\\\ \frac{1}{h^{a\_2}} \sum\_{i=0}^{\sigma} (-1)^i \binom{a\_2}{i} \mathfrak{z} \tilde{\xi}\_{\cdot}^{\cdot} ( (\sigma - i) h ) \Theta \frac{(\sigma h)^{-a\_2}}{\Gamma(1 - a\_2)} \mathfrak{z} \tilde{\xi}\_{\cdot}^{\cdot} (0) &= \mathfrak{z} \tilde{\xi}\_{\cdot} ( \sigma h) \oplus \mathfrak{z} \tilde{\xi}\_{\cdot}^{\cdot} ( \sigma h) \end{aligned} \tag{29}$$

which on expanding to ‐levels of and convert into system of four nonlinear equations, i.e. for all <sup>∈</sup> <sup>0</sup>, <sup>1</sup> ,

() () ( () () ()) ( () () ())

where defines Hausdroff distance. On using Lipschitz condition, i.e. Eq. (7), proof is

*n n t Xt h t t t Xt Xt Xt*

Θ =Ψ <sup>Ψ</sup>

( ) () () ( ) <sup>1</sup> 1 1 *n n Lh t X t O h n n n*

 σ

Subsequent to the algorithm demonstrated in Section 3, here numerical experiments of some system of fuzzy fractional differential equations are presented. Results for fuzzy‐valued functions are depicted in tabular form in the finite interval 0, 1 at different values of *ω* ∈(0, 1 . In addition, error bar pictorials are given for each respective example. All the exact

Following nonlinear fractional system is solved in [27] using homotopy analysis method, here

() () <sup>1</sup> 0.5 *<sup>t</sup> t t*

= ⊕ <sup>2</sup> <sup>D</sup> XXX <sup>221</sup> (27)

( ) 0 0.75 0.25 ,1.125 0.125 , 0 1,1 =+ − [ ] ( ) =− − [ ] X X 1 2 (28)

<sup>=</sup> <sup>D</sup> X X 1 1

() () () <sup>2</sup> *<sup>t</sup> tt t*

On applying Grünwald‐Letnikov's fractional definition on left hand side of Eq. (27) and following the algorithm, the differential equations are reduced to nonlinear algebraic equa‐

σ

 σ

**<sup>D</sup>** Xn XX X 12 n

σ

ω

176 Numerical Simulation - From Brain Imaging to Turbulent Flows

completed by obtaining the following equation:

ω

values and calculations are carried out through *Mathematica 10*.

ω

with *ω*1, *ω*2∈(0, 1 and subjected to initial conditions

the system is restructured with imprecise functions and as:

ω

 σ

**4. Numerical illustrations**

σ

**4.1. Example 1**

tions as:

, ,, , , ,, *<sup>n</sup>*

 σ

( )

+ω

*n*

 σσ

1

*O h*

+

 ω

<sup>+</sup> <sup>−</sup> Θ ≤ ∀≥ Xn (26)

1 2

 σ

(25)

(30)


**Table 1.** Numerical results and absolute errors of for Example 1 at *ω*<sup>1</sup> =1, *ω*<sup>2</sup> =1, *h* =0.001 and *t* =1.

Solving this system, numerical approximations of Eq. (27) are obtained. **Tables 1** and **2** rep‐ resent absolute error of and , respectively, for *ω*<sup>1</sup> =*ω*<sup>2</sup> =1, *h* =0.001, *t* =1 and at dif‐ ferent values of , whereas **Table 3** shows the approximations of and for *<sup>ω</sup>*<sup>1</sup> =0.95, *ω*<sup>2</sup> =0.87, *<sup>h</sup>* =0.1 and *<sup>t</sup>* =1, at different values of . In **Figures 1** and **2**, the pointwise error variations of and *X*˜ 2 (*t*), accordingly, at each time within the given interval for *ω*<sup>1</sup> =*ω*<sup>2</sup> =1, *h* =0.1 and = 0.6, are plotted. In these graphs, each approximated point is plotted against the value of *σ* in a discrete manner and each bar line on respective approximated point illustrates the measure of the absolute error at that point. Absolute error is obtained by taking the point‐to‐point difference between exact and the solutions calculated by Grün‐ wald‐Letnikov's fractional approach. Since these variations show small differences, this im‐ plies our results are in good agreement with the exact solutions.


**Table 2.** Numerical results and absolute errors of for Example 1 at *ω*<sup>1</sup> =1, *ω*<sup>2</sup> =1, *h* =0.001 and *t* =1.


**Table 3.** Approximations of and of Example 1 for *ω*<sup>1</sup> =0.95, *ω*<sup>2</sup> =0.87, *h* =0.1 and *t* =1.

Numerical Solution of System of Fractional Differential Equations in Imprecise Environment http://dx.doi.org/10.5772/64150 179

**Figure 1.** Bar plot of *σ* of of Example 1 for *h* =0.1, *ω*<sup>1</sup> =*ω*<sup>2</sup> =1 and = 0.6.

**Figure 2.** Bar plot of approximate solutions and absolute error versus *σ* of of Example 1 for *h* =0.1, *ω*<sup>1</sup> =*ω*<sup>2</sup> =1 and = 0.6.

#### **4.2. Example 2**

ferent values of , whereas **Table 3** shows the approximations of and for *<sup>ω</sup>*<sup>1</sup> =0.95, *ω*<sup>2</sup> =0.87, *<sup>h</sup>* =0.1 and *<sup>t</sup>* =1, at different values of . In **Figures 1** and **2**, the pointwise

*ω*<sup>1</sup> =*ω*<sup>2</sup> =1, *h* =0.1 and = 0.6, are plotted. In these graphs, each approximated point is plotted against the value of *σ* in a discrete manner and each bar line on respective approximated point illustrates the measure of the absolute error at that point. Absolute error is obtained by taking the point‐to‐point difference between exact and the solutions calculated by Grün‐ wald‐Letnikov's fractional approach. Since these variations show small differences, this im‐

(*t*), accordingly, at each time within the given interval for

2

plies our results are in good agreement with the exact solutions.

0 [1.2745, 1.9117] [-1.0584, 8.2274] 0.2 [1.3594, 1.8692] [-0.1017, 7.3564] 0.4 [1.4444, 1.8267] [0.8747, 6.4903] 0.6 [1.5294, 1.7842] [1.8707, 5.6291] 0.8 [1.6143, 1.7418] [2.8863, 4.7728] 1 [1.6993, 1.6993] [3.9215, 3.9215]

**Exact solutions Approx. solutions Absolute error** 0 [-1.0426, 1.2424] [-1.0426, 1.2426] [9.1289×10-6, 1.9828×10-4] 0.2 [-0.8133, 1.0155] [-0.8133, 1.0157] [2.8859×10-5, 1.8104×10-4] 0.4 [-0.5835, 0.7888] [-0.5834, 0.7889] [4.9162×10-5, 1.6393×10-4] 0.6 [-0.3531, 0.5621] [-0.3530, 0.5623] [7.0025×10-5, 1.4696×10-4] 0.8 [-0.1222, 0.3356] [-0.1221, 0.3358] [9.1457×10-5, 1.3014×10-4] 1 [0.1093, 0.1093] [0.1094, 0.1094] [1.1346×10-4, 1.1346×10-4]

**Table 2.** Numerical results and absolute errors of for Example 1 at *ω*<sup>1</sup> =1, *ω*<sup>2</sup> =1, *h* =0.001 and *t* =1.

**Table 3.** Approximations of and of Example 1 for *ω*<sup>1</sup> =0.95, *ω*<sup>2</sup> =0.87, *h* =0.1 and *t* =1.

error variations of and *X*˜

178 Numerical Simulation - From Brain Imaging to Turbulent Flows

Consider the following nonlinear fractional system [27] with imprecise functions , and as:

$$\begin{aligned} \mathfrak{gl}^{\alpha} \circ \check{\mathbb{K}}\_{\prime}^{\gamma}(t) &= \infty \check{\mathbb{K}}\_{\prime}^{\gamma}(t), \\\\ \mathfrak{gl}^{\alpha} \circ \check{\mathbb{K}}\_{\varepsilon}^{\gamma}(t) &= 2 \circ \check{\mathbb{K}}\_{\prime}^{\gamma}(t) \end{aligned}$$

$$\mathfrak{gl}^{\alpha\_0} \circ \breve{\mathbb{K}}\_{\varepsilon}(t) = \mathfrak{Z} \circ \breve{\mathbb{K}}\_{\prime}(t) \bullet \circ \breve{\mathbb{K}}\_{\varepsilon}(t) \tag{31}$$

with *ω*1, *ω*2, *ω*3∈(0, 1 and subjected to initial conditions

$$\varepsilon \circ \tilde{\mathsf{F}}\_{\cdot}(0) = \varepsilon \circ \tilde{\mathsf{F}}\_{\cdot}(0) = \left[ 0.75 + 0.25 \aleph, 1.125 - 0.125 \aleph \right], \quad \varepsilon \circ \tilde{\mathsf{F}}\_{\cdot}(0) = \left[ \aleph - 1, 1 - \aleph \right] \tag{32}$$

On employing Grünwald‐Letnikov's approach, the differential equations are converted into nonlinear algebraic equations as:

$$\begin{aligned} \frac{1}{h^{\alpha\_1}} \sum\_{i=0}^{\sigma} (-1)^i \binom{\alpha\_1}{i} \mathsf{c} \tilde{\psi}\_{\gamma}^{\gamma} ( (\sigma - i) h ) \Theta \frac{(\sigma h)^{-\alpha\_1}}{\Gamma(1 - \alpha\_1)} \mathsf{c} \tilde{\psi}\_{\gamma}^{\gamma} ( 0 ) &= \mathsf{c} \tilde{\psi}\_{\gamma}^{\gamma} ( \sigma \, h ), \\\\ \frac{1}{h^{\alpha\_2}} \sum\_{i=0}^{\sigma} (-1)^i \binom{\alpha\_2}{i} \mathsf{c} \tilde{\psi}\_{\gamma}^{\gamma} ( (\sigma - i) h ) \Theta \frac{(\sigma h)^{-\alpha\_2}}{\Gamma(1 - \alpha\_2)} \mathsf{c} \tilde{\psi}\_{\gamma}^{\gamma} ( 0 ) &= 2 \mathsf{c} \tilde{\psi}\_{\gamma}^{\gamma} ( \sigma \, h ) \end{aligned}$$
 
$$\frac{1}{h^{\alpha\_1}} \sum\_{i=0}^{\sigma} (-1)^i \binom{\alpha\_3}{i} \mathsf{c} \tilde{\psi}\_{\gamma}^{\gamma} ( (\sigma - i) h ) \Theta \frac{(\sigma h)^{-\alpha\_1}}{\Gamma(1 - \alpha\_3)} \mathsf{c} \tilde{\psi}\_{\gamma}^{\gamma} ( 0 ) = 3 \mathsf{c} \tilde{\psi}\_{\gamma}^{\gamma} ( \sigma h) \bullet \bar{\psi}\_{\gamma}^{\gamma} ( \sigma h) \tag{33}$$

and in ‐levels of , , and *X*˜ 3 (*t*) the system above converts into six nonlinear equations, i.e. for all ,

$$\begin{split} &\frac{1}{h^{n}}\sum\_{i=0}^{\sigma}(-1)^{\binom{\alpha\_{i}}{i}}\dfrac{\underset{i\geq\lambda^{\prime}\_{i}}{\rightleftarrows^{\prime}\_{i}}}{i}((\sigma-i)!\kappa;\lambda)-\dfrac{(\sigma h)^{-\alpha\_{i}}}{\Gamma(1-\alpha\_{i})}\dfrac{\right(\sigma\lambda)}{\right(-\alpha\_{i})}\dfrac{\right(\sigma\lambda)}+\dfrac{\right(\sigma\lambda)}{\right(\sigma\lambda)}\left(\sigma\lambda;\lambda), \\\\ &\frac{1}{h^{n\_{1}}}\dfrac{\right(\sigma\lambda)}{i-1}(-1)^{\binom{\alpha\_{1}}{i}}\dfrac{\left((\sigma-i)!\kappa;\lambda\right)-\dfrac{(\sigma\sigma\lambda)^{-\alpha\_{i}}}{\Gamma(1-\alpha\_{i})}\dfrac{\right(\sigma\lambda)}{\Gamma(1-\alpha\_{i})}\dfrac{\right(\sigma\lambda)}=\overlef(\sigma\lambda;\lambda\right), \\\\ &\frac{1}{h^{\alpha\_{1}}}\dfrac{\left(-1\right)^{\binom{\alpha\_{1}}{i}}\Bigl{(}\kappa\lambda\overset{\alpha\_{i}\cdot}{\right)}\big((\sigma-i)\kappa;\lambda\right)-\dfrac{(\sigma\lambda)^{-\alpha\_{i}}}{\Gamma(1-\alpha\_{i})}\dfrac{\underset{i\geq\lambda^{\prime}\_{i}}{\longrightarrow}(0;\lambda)}{\left(1-\alpha\_{i}\right)}=2\underbrace{\underset{i\geq\lambda^{\prime}\_{i}}{\longrightarrow}(\sigma\lambda;\lambda)}, \\\\ &\frac{1}{h^{\alpha\_{1}}}\dfrac{\left(-1\right)^{\binom{\alpha\_{1}}{i}}\Bigl{(}\kappa\lambda\overset{\alpha\_{1}}{i}\Bigr{(}(\sigma-i)\kappa;\lambda)-\dfrac{(\sigma\lambda)^{-\alpha\_{i}}}{\Gamma(1-\alpha\_{i})}\dfrac{\right)}{\left(\Gamma(1-\alpha\_{i})}\Bigr{(}(0;\lambda)=2\overlef(-\overlef$$

Numerical Solution of System of Fractional Differential Equations in Imprecise Environment http://dx.doi.org/10.5772/64150 181

$$\frac{1}{h^{\alpha\_{\flat}}} \sum\_{i=0}^{\sigma} (-1)^{i} \binom{\alpha\_{\flat}}{i} \overline{\circledast}^{\upsilon}\_{\circ}((\sigma - i)k; \lambda) - \frac{(\sigma h)^{-\upsilon\_{\flat}}}{\Gamma(1 - \alpha\_{\flat})} \overline{\circledast}^{\upsilon}\_{\circ}(0; \lambda) = 3 \overline{\circledast}^{\upsilon}\_{\circ}(\sigma h; \lambda) \overline{\circledast}^{\upsilon}\_{\circ}(\sigma h; \lambda) \tag{34}$$

Thus, numerical results of Eq. (31) are obtained from the above system. **Tables 4**–**6** present absolute error of , and *X*˜ 3 (*t*), respectively, for *ω*<sup>1</sup> =*ω*<sup>2</sup> =*ω*<sup>3</sup> =1, *h* =0.001, *t* =1 and at different values of . In **Table 7**, the approximations of , and *X*˜ 3 (*t*) are rendered for *h* =0.1, *ω*<sup>1</sup> =0.95, *ω*<sup>2</sup> =0.87, *ω*<sup>3</sup> =0.79 and *t* =1, at different values of . Additionally, the pointwise error variations between approximated and exact solutions of , and *X*˜ 3 (*t*) at each time within the given interval for *ω*<sup>1</sup> =*ω*<sup>2</sup> =*ω*<sup>3</sup> =1 and = 0.6 are plotted in **Figures 3**–**5**, respectively. It is to be noted that the small length of bar lines on each point is illustrating small differences between the exact and the result obtained by the proposed approach that shows the acceptable convergence of the solution towards the exact values.

() () () <sup>3</sup> 3 *<sup>t</sup> t tt*

() () 0 0 0.75 0.25 ,1.125 0.125 , 0 1,1 = =+ − [ ] ( ) =− − [ ] X X 1 2 X3 (32)

*h*

σ

1

σ

*h*

ω

−

ω

ω

−

ω

*i h h h*

Γ − <sup>∑</sup> X3 XXX 31 2 (33)

On employing Grünwald‐Letnikov's approach, the differential equations are converted into

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

− −Θ <sup>=</sup> Γ − <sup>∑</sup> X1 X X 1 1

− −Θ <sup>=</sup> Γ − <sup>∑</sup> <sup>2</sup> X2 X X 2 1

ω

−

ω

() ( ) ( ) ( )

σ

0 1

() ( ) ( ) ( )

<sup>1</sup> <sup>1</sup> 0 2

1

σ

− −Θ = •

*h*

0 2

σ

1

ω

2

ω

() ( ) ( ) ( )

<sup>1</sup> <sup>1</sup> 0 3

3

0 3

σ

*i*

*i*

= • <sup>D</sup> X XX 3 12 (31)

( ) () ( ) <sup>1</sup>

( ) () ( ) <sup>2</sup>

σ

(*t*) the system above converts into six nonlinear equations,

 σ

> σ

> > σ

*i h h*

*i h h*

( ) () ( ) ( ) <sup>3</sup>

ω

with *ω*1, *ω*2, *ω*3∈(0, 1 and subjected to initial conditions

180 Numerical Simulation - From Brain Imaging to Turbulent Flows

nonlinear algebraic equations as:

1

ω

2

*i*

ω

3

ω

i.e. for all ,

*i*

=

σ

*h i*

and in ‐levels of , , and *X*˜

*i*

=

σ

*h i*

3

ω

*i*

=

σ

*h i*


**Table 4.** Numerical results and absolute errors of for Example 2 at *ω*<sup>1</sup> =*ω*<sup>2</sup> =*ω*<sup>3</sup> =1, *h* =0.001 and *t* =1.


**Table 5.** Numerical results and absolute errors of for Example 2 at *ω*<sup>1</sup> =*ω*<sup>2</sup> =*ω*<sup>3</sup> =1, *h* =0.001 and *t* =1.


**Table 6.** Numerical results and absolute errors of (*t*) X3 for Example 2 at *ω*<sup>1</sup> <sup>=</sup>*ω*<sup>2</sup> <sup>=</sup>*ω*<sup>3</sup> =1, *<sup>h</sup>* =0.001 and *<sup>t</sup>* =1.


**Table 7.** Approximations of , and (*t*) X3 of Example 2 for *ω*<sup>1</sup> =0.95, *ω*<sup>2</sup> =0.87 *<sup>ω</sup>*<sup>3</sup> =0.79, *<sup>h</sup>* =0.1 and *t* =1.

**Figure 3.** Bar plot of approximate solutions and absolute error versus *σ* of of Example 2 for *h* =0.1, *ω*<sup>1</sup> =*ω*<sup>2</sup> =*ω*<sup>3</sup> =1 and = 0.6.

Numerical Solution of System of Fractional Differential Equations in Imprecise Environment http://dx.doi.org/10.5772/64150 183

**Figure 4.** Bar plot of approximate solutions and absolute error versus *σ* of of Example 2 for *h* =0.1, *ω*<sup>1</sup> =*ω*<sup>2</sup> =*ω*<sup>3</sup> =1 and = 0.6.

**Figure 5.** Bar plot of approximate solutions and absolute error versus *σ* of *<sup>X</sup>*˜ 3 (*t*) of Example 2 for *h* =0.1, *ω*<sup>1</sup> =*ω*<sup>2</sup> =*ω*<sup>3</sup> =1 and = 0.6.
