A novel spectral Galerkin/Petrov–Galerkin algorithm for the multi-dimensional space–time fractional advection–diffusion–reaction equations with nonsmooth solutions

Abstract

The usual classical polynomials-based spectral Galerkin and Petrov–Galerkin methods enjoy high-order accuracy for problems with smooth solutions. However, their accuracy and fidelity can be deteriorated when the solutions exhibit weakly singular behaviors and this issue becomes much more severe for polynomial-based spectral methods. The eigenfunctions of the Sturm–Liouville problems of fractional order serve as basis functions for constructing efficient spectral approximations for fractional differential models with nonsmooth solutions. In this paper, the Petrov–Galerkin spectral method is adopted to deal with the initial singularity in the temporal direction in which the first kind Jacobi poly-fractonomials are utilized as temporal trial functions and the second kind Jacobi poly-fractonomials as temporal test functions. Along the spatial direction, the Galerkin spectral method is adopted for the first time to deal with the boundary singularity in the spatial direction in which weighted Jacobi functions are utilized as bases in multi-dimensions. Various numerical experiments are provided to demonstrate the performance of the proposed schemes.