FAST COMPACT DIFFERENCE SCHEME FOR THE FOURTH-ORDER TIME MULTI-TERM FRACTIONAL SUB-DIFFUSION EQUATIONS WITH THE FIRST DIRICHLET BOUNDARY

Abstract

<abstract> In this paper, a fast compact difference scheme is proposed for the initial-boundary value problem of fourth-order time multi-term fractional sub-diffusion equations with the first Dirichlet boundary conditions. Using the method of order reduction, the original problem can be converted to an equivalent lower-order system. Then at some super-convergence points, the multi-term Caputo derivatives are fast evaluated based on the sum-of-exponentials (SOE) approximation for the kernel functions appeared in Caputo fractional derivatives. The difficulty caused by the first Dirichlet boundary conditions is carefully handled. The energy method is used to illustrate the unconditional stability and convergence of the proposed fast compact scheme. The convergence accuracy is second-order in time and fourth-order in space if the solution has enough regularity. Compared with the direct scheme without the acceleration in time direction, the CPU time of the current fast scheme is largely reduced, which is shown by numerical examples. </abstract>