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

Abstract

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 L1 scheme at tk+1/2. A fourth-order difference approximation for the spatial fractional derivatives is employed by using the weighted average of the shifted Grünwald formulae. This methodology is based on a class of discrete fractional Grönwall inequalities convenient with the quadrature formula used to approximate the Caputo derivative at tk+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(τ2−αJ+h4), in the case that 0<αJ<1 satisfies 3αJ≥32, which means that 0.369≤αJ<1, such that αJ is the maximum α-th order in the multi-order fractional operators.