Regularity and spectral methods for two-sided fractional diffusion equations with a low-order term
Provides theoretical and numerical advances for solving fractional diffusion equations, which are important in modeling anomalous diffusion in physics and finance.
The paper improves convergence rates for spectral methods solving two-sided fractional diffusion equations with a low-order term by proving higher regularity in weighted Sobolev spaces, achieving optimal error estimates for a Petrov-Galerkin method.
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.