An Initial Value Method for Singularly Perturbed System of Reaction-Diffusion Type Delay Differential Equations

Abstract

In this paper an asymptotic numerical method named as Initial Value Method (IVM) is suggested to solve the singularly perturbed weakly coupled system of reaction-diffusion type second order ordinary differential equations with negative shift (delay) terms. In this method, the original problem of solving the second order system of equations is reduced to solving eight first order singularly perturbed differential equations without delay and one system of difference equations. These singularly perturbed problems are solved by the second order hybrid finite difference scheme. An error estimate for this method is derived by using supremum norm and it is of almost second order. Numerical results are provided to illustrate the theoretical results.

Third order singularly perturbed delay differential equation of reaction diffusion type with integral boundary condition

“…Therefore, it is necessary to improve suitable numerical methods which are uniformly convergent to solve the problem. Some authors have worked on singularly perturbed differential equations with delay using uniformly convergent numerical methods [1][2][3][4][5][6].…”

Solving Linear Second-Order Singularly Perturbed Differential Difference Equations via Initial Value Method

“…However, much of this feedback require a finite time to sense information and react to it. A popular way to describe this process is to formulate delay differential equations or differential difference equations (DDEs) where the evolution of a dependent variable is a function of time which depends on not only current time but also earlier time [3]. A singularly perturbed differential difference equation is a differential equation in which the highest derivative is multiplied by a small parameter and which involves at least one shift term.…”