Müntz Spectral Method for Two-Dimensional Space-Fractional Convection-Diffusion Equation

Abstract

In this paper, we propose and analyze a Müntz spectral method for a class of two-dimensional space-fractional convection-diffusion equations. The proposed methods make new use of the fractional polynomials, also known as Müntz polynomials, which can be regarded as continuation of our previous work. The extension is twofold. Firstly, the existing Müntz spectral method for fractional differential equation with fractional derivative order 0<µ<1 is generalized to 0<µ≤2, which is nontrivial since the classical Müntz polynomials only have no more than H^1/2+λ regularity, where 0 < λ ≤ 1 is the characteristic parameter of the Müntz polynomial. Secondly, 1D Müntz spectral method is extended to the 2D space-fractional convection-diffusion equation. Compared to the time-fractional diffusion equation, some new operators such as suitable H¹-projectors are needed to analyze the error of the numerical solution. The main contribution of the present paper consists of an efficient method combining the Crank-Nicolson scheme for the temporal discretization and a new spectral method using the Müntz Jacobi polynomials for the spatial discretization of the 2D space-fractional convection-diffusion equation. A detailed convergence analysis is carried out, and several error estimates are established. Finally, a series of numerical experiments is performed to verify the theoretical claims.