Two computational algorithms for the numerical solution for system of fractional differential equations

• Main Authors: M.M. Khader, N.H. Sweilam, A.M.S. Mahdy
• Format: Article
• Language: English
• Published: Emerald Publishing 2015-01-01
• Series: Arab Journal of Mathematical Sciences
• Subjects: System of fractional differential equations; Caputo derivative; Chebyshev approximation; Convergence analysis; Grünwald–Letnikov’s approach; Fractional FDM
• Online Access: http://www.sciencedirect.com/science/article/pii/S1319516613000522

In this paper, two efficient numerical methods for solving system of fractional differential equations (SFDEs) are considered. The fractional derivative is described in the Caputo sense. The first method is based upon Chebyshev approximations, where the properties of Chebyshev polynomials are utilized to reduce SFDEs to system of algebraic equations. Special attention is given to study the convergence and estimate the error of the presented method. The second method is the fractional finite difference method (FDM), where we implement the Grünwald–Letnikov’s approach. We study the stability of the obtained numerical scheme. The numerical results show that the approaches are easy to implement implement for solving SFDEs. The methods introduce a promising tool for solving many systems of linear and non-linear fractional differential equations. Numerical examples are presented to illustrate the validity and the great potential of both proposed techniques.