Error analysis of the Legendre-Gauss collocation methods for the nonlinear distributed-order fractional differential equation

This paper is mainly concerned with the construction and convergence of Legendre-Gauss collocation methods for the nonlinear distributed-order fractional differential equation. Under some conditions, we first prove the existence and uniqueness of its exact solution. Then, we present the collocation methods for this problem. The main idea is to discretize the distributed-order fractional differential equation into a multi-term fractional differential equation by the Gauss-Legendre quadrature formula, and to approximate the multi-term one by the Legendre-Gauss collocation method. Moreover, we provide the convergence estimate for the proposed methods in the weighted L ² -norm. Finally, we give several numerical examples to demonstrate the high accuracy and the effectiveness of such methods.