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 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 ( τ 2 − α J + h 4 ) , in the case that 0 α J 1 satisfies 3 α J ≥ 3 2 , which means that 0.369 ≤ α J 1 , such that α J is the maximum α-th order in the multi-order fractional operators.

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