Generalized fractional-order Legendre polynomials and its treatment for solving system of FDEs

This paper is devoted to introducing an efficient numerical method with introducing fractional-order Legendre polynomials (FLPs) and its treatment for solving systems of fractional differential equations (FDEs). The fractional derivative is described in the Caputo sense. The proposed method is based on the spectral collocation method using FLPs. This method reduces the system of FDEs to a system of algebraic equations which is solved by using a suitable numerical method. Special attention is given to studying the convergence analysis and deducing an error upper bound of the resulting approximate solution. We give a comparison with the fractional finite difference method (FDM), where we implement the Grünwald–Letnikov’s approach and state the stability of the obtained numerical scheme. Numerical illustrations are presented to demonstrate utility, validity, and the great potential of both proposed methods. Chaotic behavior is observed and the smallest fractional order for the chaotic behavior is obtained.