A Space-Time Spectral Method for the Time Fractional Diffusion Equation

Abstract

In this paper, we consider the numerical solution of the time fractional diffusion equation. Essentially, the time fractional diffusion equation differs from the standard diffusion equation in the time derivative term. In the former case, the first-order time derivative is replaced by a fractional derivative, making the problem global in time. We propose a spectral method in both temporal and spatial discretizations for this equation. The convergence of the method is proven by providing a priori error estimate. Numerical tests are carried out to confirm the theoretical results. Thanks to the spectral accuracy in both space and time of the proposed method, the storage requirement due to the "global time dependence" can be considerably relaxed, and therefore calculation of the long-time solution becomes possible.

Efficient spectral–Galerkin methods for fractional partial differential equations with variable coefficients

“…Moreover, some fast solvers for FD/FE approximations to FPDEs have been developed in [27,28,29] and [11,20] by exploiting their Toepliz structures. On the other hand, spectral methods for some fractional PDEs have been proposed in [12,13] where the wellposedness of some FPDEs and their spectral approximations have been established. Recently, some efficient spectral/spectral-element DG methods for a class of one-dimensional FPDEs with constantcoefficients and one-sided fractional derivatives have been proposed in [31,33] by using eigenfunctions of fractional Sturm-Liouville problems as basis functions.…”

Numerical schemes for integro-differential equations with Erdélyi-Kober fractional operator

“…Lastly, we would like to mention several papers discussing numerical approaches to time-fractional diffusion. A very thorough treatment has been given in [30] where a combined space-time spectral method was used. A similar setting of finite differences was also applied in [31].…”

Fractional Laguerre spectral methods and their applications to fractional differential equations on unbounded domain

“…Many scholars are today concerned with spectral methods for solving FPDEs. Lia and Xu [11] and [12] published The great difficulties to obtain the numerical solutions of differential problems on unbounded domains [45]- [58]. Finite difference methods and finite element methods cannot be used directly.…”

“…From the numerical viewpoint, the FPDEs have been extensively concerned. Many powerful methods have been proposed, such as spectral method [11], [12], finite difference method [13]- [16], finite element method [17], random walk approach [18], the decomposition method [19], [20], the homotopy perturbation method [21], [22], the integral equation method [23] and so many others. Many scholars are today concerned with spectral methods for solving FPDEs.…”