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

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 Muntz 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 Muntz 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.