Chebyshev Differential Quadrature for Numerical Solutions of Third- and Fourth-Order Singular Perturbation Problems
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RESEARCH ARTICLE
Chebyshev Differential Quadrature for Numerical Solutions of Third- and Fourth-Order Singular Perturbation Problems Gu¨lsemay Yig˘it1,2
•
Mustafa Bayram3
Received: 18 November 2017 / Revised: 2 March 2019 / Accepted: 15 March 2019 Ó The National Academy of Sciences, India 2019
Abstract In this paper, linear and nonlinear singularly perturbed problems are studied by a numerical approach based on polynomial differential quadrature. The weighting coefficient matrix is acquired using Chebyshev polynomials. Different classes of perturbation problems are considered as test problems to show the accuracy of method. Then, the quadrature results are compared with analytical solutions of well-known existing solutions. Keywords Singular perturbation The differential quadrature Chebyshev polynomials
1 Introduction The analytical and numerical methods to solve singular perturbation problems have been widely used in many fields of fluid dynamics, reaction–diffusion processes, particle physics and combustion processes [1]. These types of problems are represented by ordinary differential equations including which is assumed to be a small parameter. Solutions of the problems have non-uniform behavior when the parameter ! 0 [2, 3]. & Gu¨lsemay Yig˘it [email protected] Mustafa Bayram [email protected] 1
Department of Basic Sciences, School of Engineering and Natural Sciences, Altınbas¸ University, Istanbul, Turkey
2
Graduate School of Science and Engineering, Yıldız Technical University, Istanbul, Turkey
3
Faculty of Engineering and Natural Sciences, Biruni University, Istanbul, Turkey
In this study, perturbation problems are analyzed by using polynomial-based differential quadrature method. These problems are classified by setting ¼ 0. If the order of the higher-order problem is reduced by one, the problem becomes convection–diffusion type, and if the order is reduced by two, it becomes reaction–diffusion type [4]. Then, to show the efficiency of the method, different types of problems have been solved and solutions are compared with exact or asymptotic solutions depending on the small parameter . Many studies have been carried out on singular perturbation including both theoretical and numerical methods. The existence of third-order singularly perturbation problem is considered based on asymptotic methods [5], and lower and upper solutions method [6]. Boundary value technique is also one of the popular numerical methods to solve such problems. To obtain numerical results, model equation is transformed into a system of a first differential equation and a second-order singularly perturbed equation [7, 8]. Such problems with discontinuous source term are also considered by asymptotic finite element method [9]. Cui and Geng [10] studied third-order singularly perturbed problem using an analytical method with asymptotic expansion. Phaneendra et al. [11] solved such kind of problems by using fitted Numerov method by transforming third-order boundary value problem to the second-order.
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