Discrete Galerkin Method for Fractional Integro-Differential Equations

TL;DR

This paper introduces a fully discrete Galerkin method using Generalized Jacobi Polynomials for solving fractional integro-differential equations, with convergence analysis and solvability investigation.

Contribution

It presents a novel Galerkin approach with GJPs for fractional equations, including convergence proof and algebraic system solvability analysis.

Findings

Method effectively approximates solutions to fractional integro-differential equations.

Convergence analysis under general regularity assumptions.

Numerical solvability demonstrated for specific cases.

Abstract

In this paper, we develop a fully discrete Galerkin method for solving initial value fractional integro-differential equations(FIDEs). We consider Generalized Jacobi polynomials(GJPs) with indexes corresponding to the number of homogeneous initial conditions as natural basis functions for the approximate solution. The fractional derivatives are used in the Caputo sense. The numerical solvability of algebraic system obtained from implementation of proposed method for a special case of FIDEs is investigated. We also provide a suitable convergence analysis to approximate solutions under a more general regularity assumption on the exact solution.