A time-space spectral method for the time-space fractional Fokker-Planck equation and its inverse problem

In the paper, we consider a time-space spectral method to get the numerical solution of time-space fractional Fokker-Planck initial-boundary value problem. The temporal discretization is constructed by Jacobi polynomials and the spatial discretization is composed by Legendre polynomials. Moreover, we present the stability and convergence analysis strictly. The main advantages of the provided method are spectrally accurate in time and space and high computational efficiency. In addition, we introduce the inverse problem based on the spectral form with high-order accuracy of the direct problem for the first time, the Levenberg-Marquardt (L-M) method is proposed to estimate the two fractional derivatives α and 2β. Some numerical results presented are consistent with the theoretical analysis.