Galerkin Finite Element Method for Nonlinear Fractional Differential Equations

TL;DR

This paper develops a Galerkin finite element method to numerically solve nonlinear fractional differential equations involving Riemann-Liouville and Caputo derivatives, establishing theoretical properties and demonstrating accuracy through numerical experiments.

Contribution

It introduces a novel finite element approach for nonlinear fractional equations, including error estimates and stability analysis, with validation via numerical tests.

Findings

The method achieves accurate numerical solutions.

Theoretical error bounds are established.

Numerical experiments confirm stability and accuracy.

Abstract

In this paper, we study the existence, regularity, and approximation of the solution for a class of nonlinear fractional differential equations. {In order to do this}, suitable variational formulations are defined for a nonlinear boundary value problems with Riemann-Liouville and Caputo fractional derivatives together with the homogeneous Dirichlet condition. We {investigate} the well-posedness and also the regularity of the corresponding weak solutions. Then, we develop a Galerkin finite element approach {\color{blue}for} the numerical approximation of the weak formulations and {drive a priori error estimates and prove the stability of the schemes}. Finally, some numerical experiments are provided to {demonstrate} the accuracy of the proposed method.