A New Spectral-Collocation Method Using Legendre Multi-wavelets for Solving of Nonlinear Fractional Differential Equations

Abstract

In this paper, a novel spectral collocation method using Legendre multi-wavelets as the basis functions is presented to obtain the numerical solution of nonlinear fractional differential equations. The fractional derivative is described in the Caputo sense. The two-scale relations of Legendre multi-wavelets and the properties of block pulse functions have been used in the evaluation of the fractional integral operational matrix and expansion coefficients of the nonlinear terms for the Legendre multi-wavelets. Due to the aforementioned properties, the original differential equation is converted into a nonlinear system of algebraic equations which can be solved by existing tools. The numerical results are compared with exact solutions and existing numerical solutions found in the literature and demonstrate the validity and applicability of the proposed method.