Finite difference scheme for singularly perturbed reaction diffusion problem of partial delay differential equation with nonlocal boundary condition

Advances in Difference Equations

article

Elango, Sekar¹  Tamilselvan, Ayyadurai²  Vadivel, R.³  Gunasekaran, Nallappan⁴  Zhu, Haitao⁵  Cao, Jinde⁵  Li, Xiaodi⁷ 

[1] Department of Mathematics, SASTRA Deemed to be University;Department of Mathematics, Bharathidasan University;Department of Mathematics, Faculty of Science and Technology, Phuket Rajabhat University;Department of Mathematical Sciences, Shibaura Institute of Technology;School of Mathematics, Southeast University;Yonsei Frontier Lab, Yonsei University;School of Mathematics and Statistics, Shandong Normal University

关键词: Parabolic delay differential equations;    Singular perturbation problem;    Integral boundary condition;    Shishkin mesh;    Finite difference scheme;    Boundary layers;   

DOI  :  10.1186/s13662-021-03296-x

学科分类：航空航天科学

来源: SpringerOpen

【 摘 要 】

This paper investigates singularly perturbed parabolic partial differential equations with delay in space, and the right end plane is an integral boundary condition on a rectangular domain. A small parameter is multiplied in the higher order derivative, which gives boundary layers, and due to the delay term, one more layer occurs on the rectangle domain. A numerical method comprising the standard finite difference scheme on a rectangular piecewise uniform mesh (Shishkin mesh) of$N_{r} \times N_{t}$ elements condensing in the boundary layers is suggested, and it is proved to be parameter-uniform. Also, the order of convergence is proved to be almost two in space variable and almost one in time variable. Numerical examples are proposed to validate the theory.

【 授权许可】

CC BY