Effective difference methods for solving the variable coefficient fourth-order fractional sub-diffusion equations

This paper is concerned with the numerical approximations for the variable coefficient fourth-order fractional sub-diffusion equations subject to the second Dirichlet boundary conditions. We construct two effective difference schemes with second order accuracy in time by applying the second order approximation to the time Caputo derivative and the sum-of-exponentials approximation. By combining the discrete energy method and the mathematical induction method, the proposed methods proved to be unconditional stable and convergent. In order to overcome the possible singularity of the solution near the initial stage, a difference scheme based on non-uniform mesh is also given. Some numerical experiments are carried out to support our theoretical results. The results indicate that the our two main schemes has the almost same accuracy and the fast scheme can reduce the storage and computational cost significantly.