Regularity and spectral methods for two-sided fractional diffusion equations with a low-order term

TL;DR

This paper investigates the regularity of solutions to two-sided fractional diffusion equations with a lower-order term, and develops a spectral Petrov-Galerkin method with proven optimal convergence, verified by numerical experiments.

Contribution

It introduces a new spectral Petrov-Galerkin method for these equations and establishes optimal error estimates based on improved regularity analysis.

Findings

Regularity in weighted Sobolev spaces is significantly better than in standard spaces.

The spectral Petrov-Galerkin method achieves higher-order convergence.

Numerical results confirm the theoretical convergence rates.

Abstract

We study regularity and numerical methods for two-sided fractional diffusion equations with a lower-order term. We show that the regularity of the solution in weighted Sobolev spaces can be greatly improved compared to that in standard Sobolev spaces. With this regularity, we improve higher-order convergence of a spectral Galerkin method. We present a spectral Petrov-Galerkin method and provide an optimal error estimate for the Petrov-Galerkin method. Numerical results are presented to verify our theoretical convergence orders.