Numerical Solution of Nonlinear Multi-order Fractional Differential Equations by Implementation of the Operational Matrix of Fractional Derivative

The main aim of this article is to generalize the Legendre operational matrix to the fractional derivatives and implemented it to solve the nonlinear multi-order fractional differential equations. In this approach, a truncated Legendre series together with the Legendre operational matrix of fractional derivatives are used. The main characteristic behind the approach using this technique is that it reduces such problems to those of solving a system of algebraic equations thus greatly simplifying the problem. The proposed approach was based on the shifted Legendre tau and shifted Legendre collocation methods. In the limit, as approaches an integer value, the scheme provides solution for the integer-order differential equations. The fractional derivatives are described in the Caputo sense. A comparison between the proposed method and the Adomian Decomposition Method (ADM) is given. The study is conducted through illustrative example to demonstrate the validity and applicability of the presented method. The results reveal that the proposed method is very effective and simple. Moreover, only a small number of shifted Legendre polynomials is needed to obtain a satisfactory result.