A space-time Galerkin Müntz spectral method for the time fractional Fokker–Planck equation

Abstract

In this paper, we propose a space-time Galerkin spectral method for the time fractional Fokker–Planck equation. This approach is based on combining temporal Müntz Jacobi polynomials spectral method with spatial Legendre polynomials spectral method. Based on the well-posedness and regularity for the re-scaled problem of a linear model problem which reflects the main difficulty for solving the equivalent equation (i.e. the time fractional convection-diffusion equation): the singularity of the solution in time, we explain in detail why we use the Müntz polynomials to approximate in time. The well-posedness and stability of the discrete scheme as well as its continuous problem are established. Moreover, the error estimation of the space-time approach is derived. We find that the proposed method can attain spectral accuracy regardless of whether the solution of the original equation is smooth or non-smooth. Numerical experiments substantiate the theoretical results.

Keywords:

• Time fractional Fokker–Planck equation
• smooth and non-smooth solutions
• well-posedness
• Müntz Jacobi polynomials
• space-time Galerkin spectral method
• error estimate

Mathematics Subject Classifications:

• 35R11
• 65M70
• 65M60

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research is supported by the National Natural Science Foundation of China [No. 12071403], the Research Foundation of Education Department of Hunan Province of China [No. 21A0108] and the Postdoctoral Fellowship Program of China Postdoctoral Science Foundation [No. GZC20230213].