A spectral approach to non-linear weakly singular fractional integro-differential equations

TL;DR

This paper introduces a spectral Petrov-Galerkin numerical method for solving non-linear weakly singular fractional integro-differential equations, demonstrating exponential accuracy and efficient recurrence-based computation.

Contribution

It develops a novel spectral approach that handles non-smooth solutions and simplifies the algebraic system solving process for these complex equations.

Findings

Proves existence, uniqueness, and smoothness of solutions.

Establishes exponential convergence in $L^{2}$-norm.

Demonstrates effectiveness through numerical examples.

Abstract

In this work, a class of non-linear weakly singular fractional integro-differential equations is considered, and we first prove existence, uniqueness, and smoothness properties of the solution under certain assumptions on the given data. We propose a numerical method based on spectral Petrov-Galerkin method that handling to the non-smooth behavior of the solution. The most outstanding feature of our approach is to evaluate the approximate solution by means of recurrence relations despite solving complex non-linear algebraic system. Furthermore, the well-known exponential accuracy is established in $L^{2}$-norm, and we provide some examples to illustrate the theoretical results and the performance of the proposed method.