# Numerical Integration of Singularly Perturbed Differential-Difference Problem Using Non Polynomial Interpolating Function

## DOI:

https://doi.org/10.26713/jims.v11i2.1151## Keywords:

Singularly perturbed differential-difference problem, Layer behaviour, Numerical integration## Abstract

In this paper, a simple integration of differential-difference problem with singular perturbed nature using non polynomial interpolating function is presented. Firstly, an equivalent first-order problem of the given second order singularly perturbed equation. Resulting first order problem is solved by numerical integration using the non polynomial interpolating function. To analyse the method computationally, several model experiments have been solved and results are compared with upwind method for different values for the advanced, delay and the perturbation parameter. The cause of the small parameters on the layer solutions are presented in graphs.

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## References

R. Bellman and K. L. Cooke, Differential-Difference Equations, Academic Press (1963).

P. P. Chakravarthy and R. N. Rao, A modified Numerov method for solving singularly perturbed differential-difference equations arising in science and engineering, Results in Physics 2 (2012), 100 – 103, DOI: 10.1016/j.rinp.2012.08.001.

P. P. Chakravarthy, S. D. Kumar, R. N. Rao and D. P Ghate, A fitted numerical scheme for second order singularly perturbed delay differential equations via cubic spline in compression, Advances in Differential Equations 2015 (2015), Article ID 300, DOI: 10.1186/s13662-015-0637-x.

M. W. Derstine, H. M. Gibbs, F. A. Hopf and D. L. Kaplan, Bifurcation gap in a hybrid optical system, Phys. Rev. A. 26 (1982), 3720 – 3722, DOI: 10.1103/PhysRevA.26.3720.

L. E. Els'gol'ts and S. B. Norkin, Introduction to the Theory and Application of Differential Equations with Deviating Arguments, Academic Press (1973).

M. K. Kadalbajoo and K. K. Sharma, (varepsilon)-Uniform fitted mesh methods for singular perturbed differential difference equations: mixed type of shifts with layer behavior, Int. J. Comput. Math. 81 (2004), 49 – 62, DOI: 10.1080/00207160310001606052.

M. K. Kadalbajoo and K. K. Sharma, Numerical analysis of boundary-value problems for singularlyperturbed differential-difference equations with small shifts of mixed type, Jour. Optim. Theory & Appli. 115 (2002), 145 – 163, DOI: 10.1023/A:1019681130824.

M. K. Kadalbajoo, K. K. Sharma, Numerical treatment of a mathematical model arising from a model of neuronal variability, J. Math. Anal. Appl. 307 (2005), 606 – 627, DOI: 10.1016/j.jmaa.2005.02.014.

A. S. V. R. Kanth and M. M. Kumar, Numerical treatment for a singularly perturbed convection delayed dominated diffusion equation via tension splines, International Journal of Pure and Applied Mathematics 113 (2017), 110 – 118.

C. G. Lange and R. M. Miura, Singular perturbation analysis of boundary-value problems for differential-difference equations II. Rapid oscillations and resonances, SIAM Journal on Applied Mathematics 45 (1985), 687 – 707, DOI: 10.1137/0145041.

C. G. Lange and R. M. Miura, Singular perturbation analysis of boundary-value problems for differential-difference equations III. Turning point problems, SIAM Journal on Applied Mathematics 45 (1985), 708 – 734, DOI: 10.1137/0145042.

C. G. Lange and R. M. Miura, Singular perturbation analysis of boundary-value problems for differential-difference equations V. Small shifts with layer behaviour, SIAM Journal on Applied Mathematics 54 (1994), 249 – 272, DOI: 10.1137/S0036139992228120.

C. G. Lange and R. M. Miura, Singular perturbation analysis of boundary-value problems for differential-difference equations VI. Small shifts with rapid oscillations, SIAM Journal on Applied Mathematics 54 (1994), 273 – 283, DOI: 10.1137/S0036139992228119.

C. G. Lange and R. M. Miura, Singular perturbation analysis of boundary-value problems for differential-difference equations, SIAM Journal on Applied Mathematics 54 (1982), 502 – 531, DOI: 10.1137/0142036.

A. Longtin and J. Milton, Complex oscillations in the human pupil light reflex with mixed and delayed feedback, Math. Biosci. 90 (1988), 183 – 199, DOI: 10.1016/0025-5564(88)90064-8.

M. C. Mackey and L. Glass, Oscillations and chaos in physiological control systems, Science 197 (1977), 287 – 289, DOI: 10.1126/science.267326.

A. S. V. R. Kanth and P. M. M. Kumar, Numerical method for a class of nonlinear singularly perturbed delay differential equations using parametric cubic spline, International Journal of Nonlinear Sciences and Numerical Simulation 19(3-4) (2018), 1 – 9, DOI: 10.1515/ijnsns-2017-0126.

A. A. Salama and D. G. Al-Amery, Asymptotic-numerical method for singularly perturbed differential difference equations of mixed-type, J. Appl. Math. & Informatics 33 (2015), 485 – 502, DOI: 10.14317/jami.2015.485.

R. B. Stein, A theoretical analysis of neuronal variability, Biophys. J. 5 (1965), 173 – 194, DOI: 10.1016/S0006-3495(65)86709-1.

D. K. Swamy, K. Phaneendra and Y. N. Reddy, Solution of Singularly perturbed differential difference equations with mixed shifts using galerkin method with exponential fitting, Chinese Journal of Mathematics 2016 (2016), ID 1935853, DOI: 10.1155/2016/1935853.

R. K. A. S. Venkata and M. M. K. Palli, A numerical approach for solving singularly perturbed convection delay problems via exponentially fitted spline method, Calcolo 54(3) (2017), 943 – 961, DOI: 10.1007/s10092-017-0215-6.

M. Wazewska-Czyzewska and A. Lasota, Mathematical models of the red cell system, Mat. Sts. 6 (1976), 25 – 40.

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*Journal of Informatics and Mathematical Sciences*,

*11*(2), 195–208. https://doi.org/10.26713/jims.v11i2.1151

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