The finite difference method for Caputo-type parabolic equation with fractional Laplacian: One-dimension case

In this paper, we present the finite difference method for Caputo-type parabolic equation with fractional Laplacian, where the time derivative is in the sense of Caputo with order in (0, 1) and the spatial derivative is the fractional Laplacian. The Caputo derivative is evaluated by the L1 approximation, and the fractional Laplacian with respect to the space variable is approximated by the Caffarelli-Silvestre extension. The difference schemes are provided together with the convergence and error estimates. Finally, numerical experiments are displayed to verify the theoretical results.