Analysis and Petrov-Galerkin numerical approximation for variable coefficient two-sided fractional diffusion, advection, reaction equations

TL;DR

This paper develops a Petrov-Galerkin method for variable coefficient two-sided fractional PDEs, proving well-posedness and inf-sup stability, with numerical results validating the approach and comparing model behaviors.

Contribution

It introduces a novel Petrov-Galerkin scheme for complex fractional equations with variable coefficients, addressing coercivity issues and establishing theoretical foundations.

Findings

Proved inf-sup condition for the fractional operators.

Established well-posedness and regularity of solutions.

Validated the numerical scheme with experiments.

Abstract

In this paper we investigate the variable coefficient two-sided fractional diffusion, advection, reaction equations on a bounded interval. It is known that the fractional diffusion operator may lose coercivity due to the variable coefficient, which makes both the mathematical and numerical analysis challenging. To resolve this issue, we design appropriate test and trial functions to prove the inf-sup condition of the variable coefficient fractional diffusion, advection, reaction operators in suitable function spaces. Based on this property, we prove the well-posedness and regularity of the solutions, as well as analyze the Petrov-Galerkin approximation scheme for the proposed model. Numerical experiments are presented to substantiate the theoretical findings and to compare the behaviors of different models.