Absolutely stable difference scheme for a general class of singular perturbation problems

Advances in Difference Equations

article

El-Zahar, Essam R.¹  Alotaibi, A. M.³  Ebaid, Abdelhalim³  Baleanu, Dumitru⁴  Tenreiro Machado, José⁷  Hamed, Y. S.⁸ 

[1] Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University;Department of Basic Engineering Science, Faculty of Engineering, Menoufia University;Department of Mathematics, Faculty of Science, University of Tabuk;Department of Mathematics, Cankaya University;Institute of Space Sciences;Department of Medical Research, China Medical University Hospital, China Medical University;Institute of Engineering;Department of Mathematics and Statistics, Faculty of Science, Taif University;Department of Physics and Engineering Mathematics, Faculty of Electronic Engineering, Menoufia University

关键词: Singular perturbation problems;    Finite difference schemes;    Absolutely stable;    Boundary and interior layers;   

DOI  :  10.1186/s13662-020-02862-z

学科分类：航空航天科学

来源: SpringerOpen

【 摘 要 】

This paper presents an absolutely stable noniterative difference scheme for solving a general class of singular perturbation problems having left, right, internal, or twin boundary layers. The original two-point second-order singular perturbation problem is approximated by a first-order delay differential equation with a variable deviating argument. This delay differential equation is transformed into a three-term difference equation that can be solved using the Thomas algorithm. The uniqueness and stability analysis are discussed, showing that the method is absolutely stable. An optimal estimate for the deviating argument is obtained to take advantage of the second-order accuracy of the central finite difference method in addition to the absolute stability property. Several problems having left, right, interior, or twin boundary layers are considered to validate and illustrate the method. The numerical results confirm that the deviating argument can stabilize the unstable discretized differential equation and that the new approach is effective in solving the considered class of singular perturbation problems.

【 授权许可】

CC BY