A transformed $ L1 $ Legendre-Galerkin spectral method for time fractional Fokker-Planck equations

Abstract

<abstract><p>The numerical solutions of time $ \alpha $-order $ (\alpha \in (0, 1)) $ Caputo fractional Fokker-Planck equations is considered. The constructed method is consist of the transformed $ L1 $ ($ TL1 $) scheme in the temporal direction and the Legendre-Galerkin spectral method in the spatial direction. It has been shown that the $ TL1 $ Legendre-Galerkin spectral method in $ L^2 $-norm is exponential order convergent in space and ($ 2-\alpha $)-th order convergent in time. Several numerical examples are given to verify the obtained theoretical results.</p></abstract>