In this paper, a numerical technique is presented to approximate the solution of a singular perturbed delay differential equation. The continual emerge of singular perturbed delay differential equations in a mathematical model of real life applications trigger the researchers for the numerical treatment of these equations. The numerical technique is based on trigonometric cubic B-spline functions in which derivatives are approximated as a linear sum of basis functions. The obtained matrix system is solved by using the Thomas Algorithm. The convergence of the employed proposal is scrutinized and computational work is carried out on four examples to test the capability of the proposed scheme. The approximated solution is compared with the existing technique and to present the behavior of the obtained solution graphs are plotted.

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Solution of Second Order Singular Perturbed Delay Differential Equation Using Trigonometric B-Spline

In this paper, existing available literature on numerical solution of singularly perturbed and singular perturbed delay differential equations is reviewed. These equation mathematical modeling of variational problems in control theory,physical and biological phenomena like optically bistable devices, description of human pupil reflex and a variety of models for physiological processes or diseases. The purpose investigate the classof singular type problems that are solved till now using different numerical methods. The objective is to motivate the researchers to develop new methods for calculating efficient solution of such problems. There is literature to solve these typesof equations both numerically and analytically. This paper limits its coverage to the work done by numerous researchers between 2002 and 2017. l. [5] presented a singular perturbed differential equations wide range of applications in real life.In this survey we have given very brief review for the work done on singular perturbed differential equations and we have given detail review of singular perturbed delay diff The paper is organized as follow: In section two, singular perturbed differential equations are introduced and applications of these equations survey of numerical solution on singular perturbed differential equation is given. In section four, singular perturbed delay differential equations are introduced and applications of these equations are given. In section five, methods used for solving singular perturbed delay differential equations are presented. Finally the paper is concluded with further scope of work in this area followed by references. Introduction to singularly perturbed differential equations A singularly perturbed differential equation differential equation in which the highest multiplied by a small parameter known as perturbation parameter. The solution of singularly perturbed differential equations (SPDE) varies rapidly in the regions called as layers which may be apparent in the solution or its derivatives often appear at the boundary of the domain.The study of solution of these equations is of great significance due to the formation of sharp boundary or interior layers when the perturbation parameter approaches to zero. In real life, many problems in control theory, elasticity, fluid mechanics, biosciences are modeled by singular perturbed differential equations, for instant, in red cell system [6], variational problems in control 6505, Impact Factor: SJIF: 5.995

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A review on singular perturbed delay differential equations

Novel approach to solve singularly perturbed boundary value problems with negative shift parameter.

This paper presents a uniform convergent numerical method for solving singularly perturbed delay reaction-diffusion equations. The stability and convergence analysis are investigated. Numerical results are tabulated and the effect of the layer on the solution is examined. In a nutshell, the present method improves the findings of some existing numerical methods reported in the literature.
Keywords: Singularly perturbed, Time delay, Reaction-diffusion equation, Layer

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Numerical treatment of singularly perturbed delay reaction-diffusion equations

This paper presents a numerical method to solve singularly perturbed differential-difference equations The solution of this problem exhibits layer or oscillatory behavior depending on the sign of the sum of the coefficients in reaction terms A fourth-order exponentially fitted numerical scheme on uniform mesh is developed The stability and convergence of the proposed method have been established The effect of delay parameter (small shift) on the boundary layer(s) has also been analyzed and depicted in graphs The applicability of the proposed scheme is validated by implementing it on four model examples Maximum absolute errors in comparison with the other numerical experiments are tabulated to illustrate the proposed method

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Exponentially Fitted Numerical Method for Singularly Perturbed Differential-Difference Equations

In this paper, we presented numerical method for solving singularly perturbed delay differential equations with layer or oscillatory behaviour for which a small shift (δ) is in the reaction term. First, the given singularly perturbed delay reaction-diffusion equation is converted into an asymptotically equivalent singularly perturbed two point boundary value problem and then solved by using fourth order finite difference method. The stability and convergence of the method has been investigated. The numerical results have been tabulated and further to examine the effect of delay on the boundary layer and oscillatory behavior of the solution, graphs have been given for different values of δ. Both theoretical and numerical rate of convergence have been established and are observed to be in agreement for the present method. Briefly, the present method improves the findings of some existing numerical methods in the literature.

Numerical Solution of Singularly Perturbed Delay Reaction-Diffusion Equations with Layer or Oscillatory Behaviour

This paper presents a fitted fourth order numerical scheme for solving singularly perturbedconvection-diffusion equations. The obtained scheme is transformed into a three-term recurrence relation and solved by Thomas algorithm. The stability and convergence of thepresent method have been investigated. The numerical results are presented by tables and graphs. The present method helps us to get good results and also to know the behavior ofthe solution in the boundary layer for perturbation parameter e is less than mesh size h.Moreover, the present method improves the findings of some existing numerical methods reported in the literature.

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Fitted Fourth Order Scheme for Singularly Perturbed Delay Convection-Diffusion Equations

This paper presents a numerical method to solve singularly perturbed delay differential equations. The solution of this problem exhibits layer or oscillatory behaviour depending on the sign of the sum of coefficients in reaction terms. A fourth order finite difference scheme on a uniform mesh is developed. The stability and convergence of the proposed method have been established. The effect of delay parameter (small shift) on the boundary layer(s) has also been analyzed and depicted in graphs. The applicability of the proposed scheme is validated by implementing it on four model examples. To show the accuracy of the method, the results are presented in terms of maximum absolute errors.

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Gashu Gadisa

Papers

Numerical Solution of Singularly Perturbed Delay Reaction-Diffusion Equations with Layer or Oscillatory Behaviour

Fitted Fourth Order Scheme for Singularly Perturbed Delay Convection-Diffusion Equations

Fourth Order Numerical Method for Singularly Perturbed Delay Differential Equations