Exponentially Fitted Numerical Method for Singularly Perturbed Differential-Difference Equations

Abstract

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.

Higher-Order Numerical Method for Singularly Perturbed Delay Reaction-Diffusion Problems

“…The efficiency of the method is validated by numerical examples and results for different parameters. The obtained numerical results have been compared with the results obtained by the methods in "Solution of second-order singular perturbed delay differential equation using Trigonometric B-Spline" and "Exponentially Fitted Numerical Method for Singularly Perturbed Differential-Difference Equations", [4,18], (See Tables 1, 2, 3, 7). Numerical confirmation to the contribution of applying the Richardson extrapolation technique is presented in Tables 4, 5, 6.…”

“…Overall, a higher-order numerical method for solving the singularly perturbed delay differential equation is presented. This method is stable, consistent, and produces a more accurate solution than some existing methods for the differential equation under consideration [4,18]. The interested researcher will be formulating the eighth or higher-order convergent numerical methods to obtain a more accurate solution.…”

“…For instance, numerical methods proposed by various authors for solving singularly perturbed delay differential equations of convection-diffusion type are; Parameter-robust numerical method based on defect-correction technique [6], Fitted mesh numerical method [7], Numerical integration method [8], using B-Spline collocation method [9], Fitted Method [10], Terminal boundary-value technique [11], Numerical integration using exponential integrating factor [12], Fitted fourth-order scheme [13] and A non-asymptotic method for general singular perturbation problems [20] so on. Further, more recently, various authors have developed numerical methods for solving singularly perturbed delay reactiondiffusion problems like; Computational method [14][15][16][17], a fourth-order numerical method [18], trigonometric B-spline [4], and Computational method for singularly perturbed delay differential equations of the reaction-diffusion type with negative shift [22].…”

An Exponentially Fitted Numerical Scheme via Domain Decomposition for Solving Singularly Perturbed Differential Equations with Large Negative Shift

“…In [13], a singularly perturbed differentialdifference equation with small negative and positive shifts was solved by using modified Numerov's method. In [14], a singularly perturbed reaction-diffusion problem was treated by developing a fourth-order exponentially fitted numerical scheme on a uniform mesh. Kadalbajoo and Sharma [15] treated singularly perturbed differential equation with small shifts of mixed typed using numerical approach on a uniform mesh.…”

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

“…For example, population ecology, control theory, viscous elasticity, and materials with thermal memory, hybrid optical system, in models for physiological processes, red blood cell system, predator-prey models, and so on as the detailed descriptions given in ( [1] , [2] , [3] , [4] , [5] ). A series of papers developed ( [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] ), and many more to obtain an approximate solution for different classes of singularly perturbed differential-difference equations. A variety of different numerical approaches have been suggested in an attempt to obtain accurate and reliable schemes for the treatment of boundary value problems of singularly perturbed differential-difference equations with a small negative shift in the convection term [9, 12].…”