A general framework for the numerical analysis of high-order finite difference solvers for nonlinear multi-term time-space fractional partial differential equations with time delay

摘要

This paper is devoted to introducing a novel methodology to prove the convergence and stability of a Crank-Nicolson difference approximation for a class of multi-term time-fractional diffusion equations with nonlinear delay and space fractional derivatives in case of sufficient smooth solutions. The temporal fractional derivatives are approximated by a specific form of LI scheme at t_(k+1/2)- A fourth-order difference approximation for the spatial fractional derivatives is employed by using the weighted average of the shifted Gruenwald formulae. This methodology is based on a class of discrete fractional Groenwall inequalities convenient with the quadrature formula used to approximate the Caputo derivative at t_(k+1/2). In the present work, the method of energy inequalities is utilized to show that the used difference scheme is stable and converges to the exact solution with order O (τ~(2-αJ) +h~4), in the case that 0 < αJ < 1 satisfies 3~(αJ) ≥3/2, which means that 0.369 ≤ αJ < 1, such that αJ is the maximum α-th order in the multi-order fractional operators.