Approximate Analytical Solution for Nonlinear System of Fractional Differential Equations by BPs Operational Matrices

1 Department of Mathematics, Imam Khomeini International University, P.O. Box 34149-16818, Qazvin, Iran 2Department of Mathematics and Computer Sciences, Cankaya University, 06530 Ankara, Turkey 3 Institute of Space Sciences, P.O. Box MG-23, 077125 Magurele-Bucharest, Romania 4Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah 21589, Saudi Arabia

The structure of the paper is given later.In Section 2, we present some preliminaries and properties in fractional calculus and Bernstein polynomials.In Section 3, we make operational matrices for product, power, Caputo fractional derivative, and Riemann-Liouville fractional integral by BPs.In Section 4, we apply two methods for solving nonlinear system of fractional differential equations by BPs.In Section 5, numerical examples are simulated to demonstrate the high performance of the proposed method.Conclusions are presented in Section 6.

Basic Tools
In this section, we recall some basic definitions and properties of the fractional calculus and Bernstein polynomials.

Lemma 6. Suppose that the function
where ), then the error bound vanishes.

Operational Matrices of Bernstein Polynomials
In Section 3, we recall the operational matrices for product, power, Caputo fractional derivative and Riemann-Liouville fractional integral by BPs.
Theorem 10.One can get BPs operational matrix   from order ( + 1) × ( + 1) for the Caputo fractional derivative as follows: Proof.See [26] for details.Theorem 11.One can obtain the operational matrix   from order ( + 1) × ( + 1) for the Riemann-Liouville fractional integral on the basis of BPs from order  as Proof.See [28] for details.

Solving System of Fractional Differential Equations
In this section, we use two methods for solving system of fractional differential equations.In the first method, we use the operational matrix for Caputo fractional derivative (OMCFD), and in the second method, we apply the operational matrix for Riemann-Liouville fractional integral (OMRLFI).
Advances in Mathematical Physics From (17) and (15) we can write Therefore, problem (1) and (2) reduces to the following problem: and the initial condition Now, using Lemma 5 we can approximate all of the known functions in the system (19).Then, by using Lemma 7 and Corollaries 8 and 9, since functions   are polynomial, we obtain the following approximations: where   :  (+1)× →  1×(+1) .

Solving the Problem by OMRLFI. This method consists of two steps.
Step 1.Initial conditions are used to reduce a given initialvalue problem to a problem with zero initial conditions.Therefore we have a modified system, incorporating the initial values.

Examples
To demonstrate the applicability and to validate the numerical scheme, we apply the present method for the following examples.
Example 12. Consider the following linear system of fractional differential equations [24,25]: with initial condition For this problem we have the exact solution in the case of  1 =  2 = 1 as We solved this problem by OMCFD and OMRLFI.Figures 1 and 2 show the approximate solutions of  1 () and  2 (), respectively, as a function of time for  = 10, for different values of  1 ,  2 .The results show that numerical solutions are in good agreement with each other, in both methods.Also, these figures show that as  1 ,  2 approach close to 1, the numerical solutions approach to the solutions for  1 =  2 = 1 as expected.In Figures 3 and 4  both methods, for  = 10,  1 =  2 = 1.In these figures, we can see that obtained results using the presented methods agree well with the analytical solutions for  1 =  2 = 1.
Example 13.Let us consider the following nonlinear fractional system [24] as follows: such that The exact solution of this system, when  and OMRLFI.We conclude that as  1 ,  2 approach close to 1, the numerical solutions approach solutions for  1 =  2 = 1 as expected.Furthermore, in both methods, the results agree well with each other.Figures 7 and 8 show that, the absolute error of obtained results for  = 10 and  1 =  2 = 1 using OMCFD and OMRLFI is in good agreement with the exact solution.
Example 14.Consider the nonlinear system of fractional differential equations [24]: with the initial conditions given by Advances in Mathematical Physics (42) We can see the approximate solutions of  1 (),  2 () and  3 (), by OMCFD and OMRLFI for  = 10 and different values of  1 ,  2 and  3 , in Figures 9, 10, and 11.These figures show that, when  1 ,  2 , and  3 approach close to 1, the numerical solutions approach the solutions for  1 =  2 =  3 = 1 as expected.In Figures 9-11, we observe that results of OMCFD and OMRLFI overlap.In Figures 12, 13, and 14, we see the absolute error of the obtained results for  = 10 and  1 =  2 =  3 = 1 in both methods.

Conclusion
In this paper, we get operational matrices of the product, Caputo fractional derivative, and Riemann-Liouville fractional integral by Bernstein polynomials.Then by using these matrices, we proposed two methods that reduced the nonlinear systems of fractional differential equations to the two system of algebraic equations that can be solved easily.Finally, numerical examples are simulated to demonstrate the high performance of the proposed method.We saw that the results of both methods were in good agreement with each other, and the classical solutions were recovered when the order of the fractional derivative goes to 1.