Graded mesh discretization for coupled system of nonlinear multi-term time-space fractional diffusion equations

• Published in: Engineering with computers Vol. 38; no. 2; pp. 1351 - 1363
• Main Authors: Hendy, Ahmed S., Zaky, Mahmoud A.
• Format: Journal Article
• Language: English
• Published: London Springer London 01.04.2022; Springer Nature B.V
• Subjects: Applied mathematics; Approximation; Boundary conditions; Boundary value problems; CAE) and Design; Calculus of Variations and Optimal Control; Optimization; Classical Mechanics; Computational mathematics; Computer Science; Computer-Aided Engineering (CAD; Computers; Control; Convergence; Discretization; Finite difference method; Math. Applications in Chemistry; Mathematical and Computational Engineering; Methods; Numerical analysis; Original Article; Partial differential equations; Singularities; Spectral methods; Systems Theory; Viscoelasticity; Galerkin–Legendre spectral approximation; Multi-term time-space fractional diffusion; Nonsmooth solution; 1-type schemes; Discrete fractional Grönwall inequality
• ISSN: 0177-0667; 1435-5663
• DOI: 10.1007/s00366-020-01095-8

In this paper, we develop an efficient finite difference/spectral method to solve a coupled system of nonlinear multi-term time-space fractional diffusion equations. In general, the solutions of such equations typically exhibit a weak singularity at the initial time. Based on the L 1 formula on nonuniform meshes for time stepping and the Legendre–Galerkin spectral method for space discretization, a fully discrete numerical scheme is constructed. Taking into account the initial weak singularity of the solution, the convergence of the method is proved. The optimal error estimate is obtained by providing a generalized discrete form of the fractional Grönwall inequality which enables us to overcome the difficulties caused by the sum of Caputo time-fractional derivatives and and the positivity of the reaction term over the nonuniform time mesh. The error estimate reveals how to select an appropriate mesh parameter to obtain the temporal optimal convergence. Furthermore, numerical experiments are presented to confirm the theoretical claims.