A fast numerical solution of the 3D nonlinear tempered fractional integrodifferential equation. Issue 2 (17th October 2022)

• Record Type: Journal Article
• Authors: Qiao, Leijie; Qiu, Wenlin; Tang, Bo
• Is Part Of: Numerical methods for partial differential equations. Volume 39:Issue 2(2023)
• Page Start: 1333
• Page End: 1354
• Publication Date: 2022-10-17
• Subjects: ADI Galerkin finite element method -- nonlinear tempered fractional integrodifferential equation -- stability and convergence; Differential equations, Partial -- Numerical solutions -- Periodicals; 515.353
• DOI: 10.1002/num.22936
• Languages: English
• ISSNs: 0749-159X

Abstract

Abstract: In this paper, we investigate the numerical solution of the three‐dimensional (3D) nonlinear tempered fractional integrodifferential equation which is subject to the initial and boundary conditions. The backward Euler (BE) method in association with the first‐order convolution quadrature rule is employed to discretize this equation for time, and the Galerkin finite element method is applied for space, which is combined with an alternating direction implicit (ADI) algorithm, in order to reduce the computational cost for solving the three‐dimensional nonlocal problem. Then a fully discrete BE ADI Galerkin finite element scheme can be obtained by linearizing the non‐linear term. Thereafter we prove a positive‐type lemma, from which the stability and convergence of the proposed numerical scheme are derived based on the energy method. Numerical experiments are performed to verify the effectiveness of the proposed approach.