Fast and high-order difference schemes for the fourth-order fractional sub-diffusion equations with spatially variable coefficient under the first Dirichlet boundary conditions

Abstract In this paper, we focus on the numerical computation for a class of fourth-order fractional sub-diffusion equations with spatially variable coefficient under the first Dirichlet boundary conditions. Two finite difference schemes with second order accuracy are derived by applying L 2 − 1 σ formula and FL 2 − 1 σ formula respectively to approximate the time Caputo derivative. The main novelty is that a novel technique is introduced to deal with the first Dirichlet boundary conditions, which is compatible with the main equation with spatially variable coefficient. The solvability, unconditional stability and convergence of both schemes are proved by using the discrete energy method and mathematical induction. A difference scheme for such problem with two dimensions is also proposed and analyzed. Numerical results show that the suggested schemes have the almost same accuracy and the FL 2 − 1 σ scheme can reduce the storage and computational cost significantly.