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
Hendy A.S. Zaky M.A. De Staelen R.H.
November 2021Elsevier B.V.
Applied Numerical Mathematics
2021#169108 - 121 pp.
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−α+h4), in the case that 0<αJ<1 satisfies [Formula presented], which means that 0.369≤αJ<1, such that αJ is the maximum α-th order in the multi-order fractional operators.
Convergence analysis , Discrete fractional Grönwall-type inequality , Multi-term time fractional diffusion , Non-linear delay , Space fractional derivatives
Text of the article Перейти на текст статьи
Department of Computational Mathematics and Computer Science, Institute of Natural Sciences and Mathematics, Ural Federal University, 19 Mira St., Yekaterinburg, 620002, Russian Federation
Department of Mathematics, Faculty of Science, Benha University, Benha, 13511, Egypt
Department of Applied Mathematics, Physics Division, National Research Centre, Cairo, 12622, Dokki, Egypt
Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan
Beheer en Algemene Directie, C. Heymanslaan 10, Ghent University Hospital, 9000, Belgium
Deans Office of the Faculty of Medicine and Health Sciences, C. Heymanslaan 10, Ghent University, 9000, Belgium
Department of Computational Mathematics and Computer Science
Department of Mathematics
Department of Applied Mathematics
Institute of Mathematics and Mathematical Modeling
Beheer en Algemene Directie
Deans Office of the Faculty of Medicine and Health Sciences
10 лет помогаем публиковать статьи Международный издатель
Книга Публикация научной статьи Волощук 2026 Book Publication of a scientific article 2026