%0 Journal Article
%T Approximate Analytical Solution for Nonlinear System of Fractional Differential Equations by BPs Operational Matrices
%A Mohsen Alipour
%A Dumitru Baleanu
%J Advances in Mathematical Physics
%D 2013
%I Hindawi Publishing Corporation
%R 10.1155/2013/954015
%X We present two methods for solving a nonlinear system of fractional differential equations within Caputo derivative. Firstly, we derive operational matrices for Caputo fractional derivative and for Riemann-Liouville fractional integral by using the Bernstein polynomials (BPs). In the first method, we use the operational matrix of Caputo fractional derivative (OMCFD), and in the second one, we apply the operational matrix of Riemann-Liouville fractional integral (OMRLFI). The obtained results are in good agreement with each other as well as with the analytical solutions. We show that the solutions approach to classical solutions as the order of the fractional derivatives approaches 1. 1. Introduction Differential equations of fractional order have been subjected to many studies due to their frequent appearance in various applications in fluid mechanics, viscoelasticity, biology, physics, engineering, and so on. Recently, a large amount of literature was developed regarding the application of fractional differential equations in nonlinear dynamics (see, e.g., [1每11] and the references therein). Thus, a huge attention has been given to the solution of fractional ordinary differential equations, integral equations, and fractional partial differential equations of physical interest. As it is known, there exists no method that yields an exact solution for fractional differential equations. Various methods have been proposed in order to solve the fractional differential equations. These methods include the homotopy perturbation method [12每15], Adomian＊s decomposition method [16每20], variation iteration method [12每14, 21每23], homotopy analysis method [24], differential transform method [25], operational matrices [26每28], and nonstandard finite difference scheme [29]. In this paper, we investigate the nonlinear system of fractional differential equations as and the initial condition where and . Also, are multivariable polynomial functions. 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. 2. Basic Tools In this section, we recall some basic definitions and properties of
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