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 t(k+1/2). A fourth-order difference approximation for the spatial fractional derivatives is employed by using the weighted average of the shifted Grunwald formulae. This methodology is based on a class of discrete fractional Gronwall 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 (tau(2-alpha J) +h(4)), in the case that 0 < alpha(J) < 1 satisfies 3(alpha J) >= 3/2 , which means that 0.369 <= alpha(J) < 1, such that alpha(J) is the maximum alpha-th order in the multi-order fractional operators. (C) 2021 IMACS. Published by Elsevier B.V. All rights reserved.