Analytical Solution of Nonlinear System of Fractional Differential Equations

In this paper, we apply the Adomian decomposition method (ADM) for solving nonlinear system of fractional differential equations (FDEs). The existence and uniqueness of the solution are proved. The convergence of the series solution and the error analysis are discussed. Some applications are solved such as fractional-order rabies model.


Introduction
This paper is concerned with the analytical solution of a nonlinear system of fractional differential equations. Systems of fractional differential equations (FDEs) have many applications in engineering and science, including electrical networks, fluid flow, control theory, fractals theory, electromagnetic theory, viscoelasticity, potential theory, chemistry, biology, optical and neural network systems ([1]- [16]). We use Adomain decomposition method ( [17]- [24]) for solving this type of equations. The existence and uniqueness of the solution are proved, the convergence of ADM series solution is discussed and the error analysis is given. This method has many advantages; it is efficiently working with different types of linear and nonlinear equations in deterministic or stochastic fields and gives an analytic solution for all these types of equations without linearization or discretization.
and the fractional derivative in this system is of sequential Caputo sense which defined as In the applications, the Caputo sense is preferred to be used because the initial conditions of ( ) i y t and its derivatives will be of integer orders and have a physical meaning. Now performing subsequently the fractional integration of order  (4) and has Adomian polynomials representation, where, Substitute from Equation (5) Finally, the solution is,

The Uniqueness of Solution
In the previous section, we find the series solution (10) of the system (1)-(2) and here we want to prove the existence and uniqueness of this series solution.
, then the series (10) is the solution of the system (1)- (2) and this solution is unique, where Proof. For existence, For uniqueness of the solution: Assume that y and z are two different solutions to the system (1)-(2) and hence, that, y z = and this completes the proof.

Error Analysis
For ADM, we can estimate the maximum absolute truncated error of the Adomian's series solution in the following theorem.

Numerical Examples
Example 1. Consider the following nonlinear system of FDEs, This system was discussed before in [25], it is solved by using the iterative method. Now, we will solve it by using ADM. Applying ADM to system (11) leads to the following recursive relations, ( ) where 1, j A and 2, j A represent the Adomian polynomials of the nonlinear terms 2 2 y and 2 3 y y respectively.
Using the relations (12)- (14), the first three terms of the series solution when 1 α = are,    subject to the initial conditions, Using ADM to system (18) A comparison between ADM solution and exact solution of 1 2 , y y and 3 y is given in Figures 2(a)-(c) ( 10 n = ).

Application: On Fractional-Order Rabies Model
The fractional-order rabies model, 0 1, 0 2, y y = = was discussed before in [27], it was solved by using Adams-type predictor-corrector method. Now, we will solve it by using ADM.
Applying ADM to the system (31) leads to the following scheme,

Conclusion
In this paper, we use a simple method to solve nonlinear system of FDEs, this method gives a good approximate analytical solution of this type of equation as we compare ADM solution with the exact solution and also by evaluating the maximum absolute error which results from using partial sum of the series ADM solution.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.