Challa, Lakshmi Sirisha; Reddy, Yanala Narsimha Numerical integration of singularly perturbed delay differential equations using exponential integrating factor. (English) Zbl 1383.65070 Math. Commun. 22, No. 2, 251-264 (2017). Summary: In this paper, we proposed a numerical integration technique with exponential integrating factor for the solution of singularly perturbed differential-difference equations with negative shift, namely the delay differential equation, with layer behaviour. First, the negative shift in the differentiated term is approximated by Taylor’s series, provided the shift is of \(o(\varepsilon)\). Subsequently, the delay differential equation is replaced by an asymptotically equivalent first order neutral type delay differential equation. An exponential integrating factor is introduced into the first order delay equation. Then trapezoidal rule, along with linear interpolation, has been employed to get a three term recurrence relation. The resulting tri-diagonal system is solved by Thomas algorithm. The proposed technique is implemented on model examples, for different values of delay parameter, \(\delta\) and perturbation parameter, \(\varepsilon\). Maximum absolute errors are tabulated and compared to validate the technique. Convergence of the proposed method has also been discussed. Cited in 6 Documents MSC: 65L03 Numerical methods for functional-differential equations 34K26 Singular perturbations of functional-differential equations 34K28 Numerical approximation of solutions of functional-differential equations (MSC2010) 34K40 Neutral functional-differential equations 65L11 Numerical solution of singularly perturbed problems involving ordinary differential equations 65L12 Finite difference and finite volume methods for ordinary differential equations 65L20 Stability and convergence of numerical methods for ordinary differential equations Keywords:singularly perturbed differential-difference equation; negative shift: boundary layer; exponential integrating factor; numerical integration; error bound; numerical example; neutral type delay differential equation; trapezoidal rule; Thomas algorithm; convergence PDFBibTeX XMLCite \textit{L. S. Challa} and \textit{Y. N. Reddy}, Math. Commun. 22, No. 2, 251--264 (2017; Zbl 1383.65070) Full Text: Link References: [1] J. B. Mc Cartin, Exponentially fitting of the delayed recruitment/renewal equation,J. Comput. Appl. Math. 136(2001), 343-356. [2] L. E. El’sgolt’s, S. B. Norkin, Introduction to the Theory and Application of Dif-ferential Equations with Deviating Arguments, Academic Press, New York - London,1973. [3] F. Z. Geng, S. P. Qian, Improved Reproducing Kernel Method for Singularly Per-turbed Differential-Difference Equations with Boundary Layer behavior, Appl. Math.Comput. 252(2015), 58-63. · Zbl 1338.34117 [4] M. K. Kadalbajoo, K. K. Sharma, A numerical analysis of singularly perturbed de-lay differential equations with layer behavior, Appl. Math. Comput. 157(2004), 11-28. · Zbl 1069.65086 [5] M. K. Kadalbajoo, V. P. Ramesh, Hybrid method for numerical solution of Sin-gularly Perturbed Delay Differential Equations, Appl. Math. Comput. 187 (2007),797-814. · Zbl 1120.65088 [6] M. K. Kadalbajoo, K. C. Patidar, K. K. Sharma,ǫ-Uniformly convergent fittedmethods for the numerical solution of the problems arising from singularly perturbedgeneral DDEs, Appl. Math. Comput. 182(2006),119-139. · Zbl 1109.65067 [7] C. G. Lange, R. M. Miura, Singular Perturbation Analysis of Boundary-Value Prob-lems for Differential-Difference Equations. v. small shifts with layer behavior, SIAMJ. Appl. Math. 54(1994), 249-272. · Zbl 0796.34049 [8] C. G. Lange, R. M. Miura, Singular perturbation analysis of boundary-value prob-lems for differential-difference equations. II. Rapid oscillations and resonances, SIAMJ. Appl. Math. 45(1985), 687-707. · Zbl 0623.34050 [9] C. G. Lange, R. M. Miura, Singular Perturbation Analysis of Boundary-Value Prob-lems for Differential-Difference Equations III. Turning Point Problems, SIAM J. Appl.Math. 45(1985), 708-734. · Zbl 0623.34051 [10] J. Mohapatra, S. Nateshan, Uniformly Convergent Numerical Method for Singu-larly Perturbed Differential-Difference Equation Using Grid Equi-distribution, Int. J.Numer. Methods Biomed. Eng. 27(2011),1427-1445. [11] Y. N. Reddy, G. B. S. L. Soujanya, K. Phaneendra, Numerical integration methodfor singularly perturbed delay differential equations, IJASER 10(2012), 249-261. [12] R. S. Varga, Matrix Iterative Analysis, Prentice-Hall, New Jersey, 1962. · Zbl 0133.08602 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.