**5. Illustrative Examples**

In this section, to show the applicability and powerfulness of the introduced method, we present the solutions to five linear and nonlinear systems of fractional order differential equations.

**Example 1.** *We first considered the following linear system of fractional differential equations [7,8]:*

$$\begin{aligned} D^a u(t) &= u(t) + v(t) \\ D^a v(t) &= -u(t) + v(t) \end{aligned}$$

*subject to*

$$u(0) = 0, \ v(0) = 1.$$

*The exact solution of this system when α* = 1 *is known to be*

$$u(t) = e^t \sin t, \ v(t) = e^t \cos t.$$

This example was examined for *M* = 2, *k* = 0, and *α* = 0.9, 0.7, 0.5. When the obtained results were matched against the exact solution when *α* = 1, as demonstrated in Figure 1, we can clearly observe that when *α* approached 1, our results approached the exact solution. We also solved this problem by using Legendre polynomial operational matrix method (LPOMM), and we compared the results with the LWOMM. The numerical computations for *u*(*t*) and *v*(*t*) when *α* = 0.9 are revealed in Tables 1 and 2.

**Table 1.** Numerical solutions of *u*(*t*) when *α* = 0.9 attained by the introduced method and the LPOMM for Example 1.



**Table 2.** Numerical solutions of *v*(*t*) when *α* = 0.9 attained by the introduced method and the LPOMM for Example 1.

**Figure 1.** *Cont.*

**(b)** 

**Figure 1.** Comparison of our solutions and the exact solution when *α* = 0.9, 0.7, 0.5 in Example 1: (**a**) Our solution *u*(*t*); and (**b**) Our solution *v*(*t*).

**Example 2.** *We considered the following nonlinear system of fractional differential equations [13]:*

$$\begin{aligned} D^{\frac{3}{7}}u(t) &= -8u(t) + v^2(t) - 4t^6 + 4t^3 + \frac{8t^{\frac{3}{2}}}{\sqrt{\pi}} - 1 \\ D^{\frac{1}{7}}v(t) &= t^2Du(t) + v(t) - 3t^4 - 2t^3 + \frac{32t^{\frac{3}{2}}}{5\sqrt{\pi}} - 1 \end{aligned}$$

$$u(0) = 0, \ v(0) = 1, \ u(1) = 1, \ v(1) = 3, \ u'(0) = 0, \ u'(1) = 3.$$

The exact solution of this system is known to be

$$u(t) = t^3, \ v(t) = 2t^3 + 1$$

Using the parameters *M* = 3 and *k* = 0, we applied both the proposed method and the LPOMM to solve this problem and show that our approach is more efficient and useful. Our numerical results supported the idea that our solution approaches the exact solution more than the approximate solution LPOMM. Comparisons of the approximate and exact solutions are presented in Tables 3 and 4.


**Table 3.** The numerical results attained by using the introduced method in comparison to the approximate solution LPOMM and the exact solution *u*(*t*) in Example 2.

**Table 4.** The numerical results attained by using the introduced method in comparison with the approximate solution LPOMM and exact solution *v*(*t*) in Example 2.


**Example 3.** *We considered the following nonlinear system of fractional differential equations with the initial conditions [8]*

$$\begin{array}{l}D^{a}u(t) = \frac{u(t)}{2} \\ D^{a}v(t) = u^{2}(t) + v(t) \end{array} \text{ \textquotedblleft} 0 \rightleftharpoons \begin{array}{l} \frac{1}{2} \\ v(0) = 1, \ v(0) = 0. \end{array}$$

The exact solution of this system when *α* = 1 is known to be

$$u(t) = e^{\left(\frac{\hbar}{2}\right)}, \ v(t) = te^t.$$

The parameters *M* = 2, *k* = 0, and *α* = 0.5, 0.7, 0.9 were utilized. A comparison of our results and the exact solution when *α* = 1 is displayed in Figure 2. The figures support that when *α* approximated 1, our results approximated the exact solution. We also solved this problem by using the LPOMM, and compared the results to the LWOMM. Finally, we present the numerical computations for *u*(*t*) and *v*(*t*) when *α* = 0.9 in Tables 5 and 6.

**Table 5.** Our solutions *u*(*t*) when *α* = 0.9 attained by the presented method and the LPOMM for Example 3.


**Figure 2.** Comparison of our solutions to the exact solution when *α* = 0.9, 0.7, 0.5 for Example 3: (**a**) Our solution *u*(*t*); and (**b**) Our solution *v*(*t*).

**Table 6.** Our solutions *v*(*t*) when *α* = 0.9 attained by the presented method and the LPOMM for Example 3.


**Example 4.** *We considered the following nonlinear system of FDEs with initial conditions [13]*

$$\begin{aligned} D^{\alpha}u(t) &= -1002u(t) + 1000v^2(t) \\ D^{\alpha}v(t) &= u(t) - v(t) - v^2(t) \end{aligned}$$

$$u(0) = 1, \ v(0) = 1$$

The exact solution of this system when *α* = 1 is known to be

$$u(t) = e^{-2t}, \ v(t) = e^{-t}$$

This example was analyzed for *M* = 3, *k* = 0, and *α* = 0.9, 0.7, 0.5. When the obtained results were matched against the exact solution when *α* = 1, as demonstrated in Figure 3, we can clearly observe that when *α* approached 1, our results approached the exact solution. We also solved this problem by using the LPOMM, and compared the results with the LWOMM. The numerical computations for *u*(*t*) and *v*(*t*) when *α* = 0.99 are also revealed in Tables 7 and 8.

**Table 7.** Numerical solutions of *u*(*t*) when *α* = 0.99 for Example 4.


**Table 8.** Numerical solutions of *v*(*t*) when *α* = 0.99 for Example 4.


**Figure 3.** Comparison of our solutions to the exact solution when *α* = 0.9, 0.7, 0.5 for Example 4: (**a**) Our solution *u*(*t*); and (**b**) Our solution *v*(*t*).

**Example 5.** *We considered the following fractional order Brusselator system [16,17]:*

$$\begin{aligned} D^\mathfrak{u}u(t) &= -2\mathfrak{u}(t) + \mathfrak{u}^2(t)v(t) \\ D^\mathfrak{v}v(t) &= \mathfrak{u}(t) - \mathfrak{u}^2(t)v(t) \end{aligned}$$

$$\mathfrak{u}(0) = 1, \ v(0) = 1.$$

The approximate solutions of this system when *α* = 1 and *α* = 0.98 were presented by Chang and Isah using the LWPT [17] and by Bota and Caruntu using the PLSM [16]. These solutions when *α* = 98 are given by

```
uLWPT(t) = 1 − 1.0791t + 0.2711t
                                 2 − 0.0638t
                                             3, vLWPT(t) = 1 + 0.0151t + 0.4185t
                                                                                  2 − 0.2624t
                                                                                             3
uPLSM(t) = 1 − 1.08655t + 0.311138t
                                     2 + 0.0243682t
                                                    3, vPLSM(t) = 1 + 0.0349127t + 0.333424t
                                                                                              2 − 0.184414t
                                                                                                            3 .
```
The parameters *M* = 2, *k* = 0, and *α* = 0.98 were used. A comparison of our results to the approximate solutions introduced by Bota and Caruntu [16] and Chang and Isah [17] when *α* = 0.98 is displayed in Figure 4. Finally, we also present the numerical computations for *u*(*t*) and *v*(*t*) when *α* = 0.98 in Tables 9 and 10.


**Table 9.** Numerical solutions of *u*(*t*) when *α* = 0.98 obtained by the introduced method, the LWPT, and the PLSM for Example 5.


**Table 10.** Numerical solutions of *v*(*t*) when *α* = 0.98 obtained by the introduced method, the LWPT, and the PLSM for Example 5.

**Figure 4.** Comparison of our solutions to the approximate solution LWPT and the approximate solution PLSM when *α* = 0.98 for Example 5: (**a**) Our solution *u*(*t*); and (**b**) Our solution *v*(*t*).
