Efficient Spectral Collocation Method for Tempered Fractional Differential Equations

Abstract

Transient anomalous diffusion may be modeled by a tempered fractional diffusion equation. In this paper, we present a spectral collocation method with tempered fractional Jacobi functions (TFJFs) as basis functions and obtain an efficient algorithm to solve tempered-type fractional differential equations. We set up the approximation error as O(Nμ−ν) for projection and interpolation by the TFJFs, which shows “spectral accuracy” for a certain class of functions. We derive a recurrence relation to evaluate the collocation differentiation matrix for implementing the spectral collocation algorithm. We demonstrate the effectiveness of the new method for the nonlinear initial and boundary problems, i.e., the fractional Helmholtz equation, and the fractional Burgers equation.