The convergence of operational Tau method for solving a class of nonlinear Fredholm fractional integro-differential equations on Legendre basis

TL;DR

This paper develops a Legendre-based Tau method to approximate solutions for nonlinear fractional Fredholm integro-differential equations, proving convergence in L^2-norm.

Contribution

It introduces a novel operational Tau method using Legendre basis for solving a specific class of nonlinear fractional equations, with proven convergence.

Findings

Method converges in L^2-norm

Effective for nonlinear fractional Fredholm equations

Provides a new computational approach

Abstract

In this paper, we investigate approximate solutions for nonlinear Fredholm integro-differential equations of fractional order. We present an operational Tau method by obtaining the Tau matrix representation. We solve a special class of nonlinear Fredholm integro-differential equations based on Legendre-Tau method. By using the Sobolev inequality and some of Banach algebra properties, we prove that our proposed method converges to the exact solution in L^2-norm.