Efficient Hermite Spectral Methods for Space Tempered Fractional Diffusion Equations

Abstract

Spectral and spectral collocation methods for tempered fractional diffusion equations on the real line $\mathbb{R}$ are developed. Applying the Fourier transform to the problem under consideration, we reduce it to systems of algebraic equations. Since Hermite functions are the eigenfunctions of the Fourier transform, they are used in the construction of spectral and spectral collocation methods for the algebraic equations obtained. The stability and convergence of the methods are studied. Numerical examples demonstrate the efficiency of the algorithms and confirm theoretical findings.