Asymptotic numerical method for third-order singularly perturbed convection diffusion delay differential equations
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Asymptotic numerical method for third-order singularly perturbed convection diffusion delay differential equations V. Subburayan1
· R. Mahendran1
Received: 22 June 2018 / Revised: 9 May 2019 / Accepted: 10 June 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract In this paper, an asymptotic numerical method based on a fitted finite difference scheme and the fourth-order Runge–Kutta method with piecewise cubic Hermite interpolation on Shishkin mesh is suggested to solve singularly perturbed boundary value problems for thirdorder ordinary differential equations of convection diffusion type with a delay. An error estimate is derived using the supremum norm and it is of almost first-order convergence. A nonlinear problem is also solved using the Newton’s quasi linearization technique and the present asymptotic numerical method. Numerical results are provided to illustrate the theoretical results. Keywords Third-order differential equations · Convection diffusion equation · Boundary value problem · Singularly perturbed problem · Shishkin mesh · Delay differential equations · Asymptotic numerical methods Mathematics Subject Classification 34K10 · 34K26 · 34K28
1 Introduction Delay differential equations (DDEs) are differential equations in which the unknown function not only evaluated at present but also evaluated at some past values of the independent variable. DDEs arise in many fields of science and technology such as, mathematical physics, control theory, neural network, medicine, biology, population dynamics model, etc. In the recent past years, the problem of finding numerical solutions for higher order singularly perturbed problems without delay are paid more attention, to cite a few (Cañada et al. 2006; Murray 2002; Kuang 1993).
Communicated by Corina Giurgea.
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V. Subburayan [email protected] R. Mahendran [email protected]
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Department of Mathematics, Faculty of Engineering and Technology, SRM Institute of Science and Technology, Kattankulathur , Chennai, Tamilnadu 603 203, India 0123456789().: V,-vol
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V. Subburayan, R. Mahendran
Nowadays, there has been growing interest for solving higher order singularly perturbed delay differential equation, where the highest order derivative is multiplied by small positive parameter ε (0 < ε 1) which contains at least one delay term, called singularly perturbed delay differential equation (SPDDE). This type of differential equation plays a vital role in mathematical modeling of various practical phenomena, such as variational problem in control theory Glizer (2003), predator-prey model Gourley and Kuang (2004), description of human pupil-light reflex Longtin and Milton (1988). It is well-known fact that the solution of the singularly perturbed differential equations with or without delay, generally exhibits boundary layer(s) and interior layer(s). Classical numerical methods for solving such type of problems are known to be inadequate, due to the presence of boundary layer(s) and interior layer(s
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