Numerical Simulation for a Differential Difference Equation With an Interior Layer

Authors

DOI:

https://doi.org/10.26713/cma.v14i1.2047

Keywords:

Differential-difference equation, Numerical integration, Layer behaviour, Convergence

Abstract

This paper addresses the solution of a differential-difference type equation having an interior layer behaviour. An approach is suggested to solve this equation using a numerical integration scheme and linear interpolation. Taylor expansions are utilized to handle the shift arguments. In order to solve the discretized equation, the tridiagonal solver is applied. The approach is analyzed for convergence. Numerical examples are demonstrated to validate the scheme. Maximum errors in the solution, in contrast to the other methods, are organized to explain the approach. The layer profile in the solutions of the examples is illustrated in graphs.

Downloads

Download data is not yet available.

References

C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill Book Company, New York (1978), URL: http://www.lmm.jussieu.fr/~lagree/COURS/M2MHP/Bender-Orszag-chap9-11.pdf.

M. Bestehorn and E. V. Grigorieva, Formation and propagation of localized states in extended systems, Annalen der Physik 516(7-8) (2004), 423 – 431, DOI: 10.1002/andp.200451607-806.

E. P. Doolan, J. J. H. Miller and W. H. A. Schilders, Uniform Numerical Methods for Problems with Initial and Boundary Layers, Boole Press, Dublin (1980).

R. D. Driver, Ordinary and Delay Differential Equations, Applied Mathematical Sciences, Vol. 20, Springer, New York (1977), DOI: 10.1007/978-1-4684-9467-9.

L. E. El’sgol’ts and S. B. Norkin, Introduction to the Theory and Application of Differential Equations with Deviating Arguments, Mathematics in Science and Engineering Book series, Vol. 105, Academic Press (1973), URL: https://www.sciencedirect.com/bookseries/mathematics-in-science-and-engineering/vol/105/suppl/C.

M. K. Kadalbajoo and K. K. Sharma, An exponentially fitted finite difference scheme for solving boundary-value problems for singularly-perturbed differential-difference equations: small shifts of mixed type with layer behavior, Journal of Computational Analysis and Applications 8(2) (2006), 151 – 171, URL: http://www.eudoxuspress.com/JoCAAAv8-06.pdf.

M. K. Kadalbajoo and K. K. Sharma, Numerical treatment of a mathematical model arising from a model of neuronal variability, Journal of Mathematical Analysis and Applications 307(2) (2005), 606 – 627, DOI: 10.1016/j.jmaa.2005.02.014.

R. B. Kellogg and A. Tsan, Analysis of some difference approximations for a singular perturbation problem without turning points, Mathematics of Computation 32 (1978), 1025 – 1039, DOI: 10.1090/S0025-5718-1978-0483484-9.

P. Kokotovic, H. K. Khalil and J. O’Reilly, ´ Singular Perturbation Methods in Control: Analysis and Design, Classics in Applied Mathematics Book Series, Academic Press, New York (1986), DOI: 10.1137/1.9781611971118.

Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, New York (1993).

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, URL: https://www.jstor.org/stable/2101624.

C. G. Lange and R. M. Miura, Singular perturbation analysis of boundary-value problems for differential-difference equations V. Small shifts with layer behavior, SIAM Journal on Applied Mathematics 54 (1994), 249 – 272, URL: https://www.jstor.org/stable/2102177.

M. Wazewska-Czyzewska and A. Lasota, Mathematical problems of the dynamics of the red blood cells, Matematyka Stosowana-Matematyka dla Społecze´nstwa (Mathematica Applicanda) 4(6) (1976), 23 – 40, DOI: 10.14708/ma.v4i6.1173.

A. Martin and S. Raun, Predator-prey models with delay and prey harvesting, Journal of Mathematical Biology 43(3) (2001), 247 – 267, DOI: 10.1007/s002850100095.

R. E. Mickens, Nonstandard Finite Difference Models of Differential Equations, World Scientific, Singapore (1994).

J. J. H. Miller, R. E. O’Riordan and G. I. Shishkin, Fitted Numerical Methods for Singular Perturbation Problems, World Scientific, Singapore (1996), DOI: 10.1142/8410.

R. E. O’Malley Jr., Introduction to Singular Perturbations, Applied Mathematics and Mechanics Book series, Vol. 14, Academic Press, New York (1974), URL: https://www.sciencedirect.com/bookseries/applied-mathematics-and-mechanics/vol/14/suppl/C.

K. C. Patidar and K. K. Sharma, Uniformly convergent non-standard finite difference methods for singularly perturbed differential-difference equations with delay and advance, International Journal for Numerical Methods in Engineering 66(2) (2006), 272 – 296, DOI: 10.1002/nme.1555.

P. Rai and K. K. Sharma, Parameter uniform numerical method for singularly perturbed differential-difference equations with interior layers, International Journal of Computer Mathematics 88(16) (2011), 3416 – 3435, DOI: 10.1080/00207160.2011.591387.

A. A. Salama and D. G. Al-Amery, Asymptotic-numerical method for singularly perturbed differential difference equations of mixed-type, Journal of Applied Mathematics & Informatics 33(5-6) (2015), 485 – 502, DOI: 10.14317/jami.2015.485.

L. Sirisha, K. Phaneendra and Y. N. Reddy, Mixed finite difference method for singularly perturbed differential difference equations with mixed shifts via domain decomposition, Ain Shams Engineering Journal 9(4) (2018), 647 – 654, DOI: 10.1016/j.asej.2016.03.009.

R. B. Stein, Some models of neuronal variability, Biophysical Journal 7(1) (1967), 37 – 68, DOI: 10.1016/S0006-3495(67)86574-3.

D. K. Swamy, K. Phaneendra and Y. N. Reddy, Accurate numerical method for singularly perturbed differential-difference equations with mixed shifts, Khayyam Journal of Mathematics 4 (2018), 110 – 122, DOI: 10.22034/KJM.2018.57949.

M. M. Woldaregay and G. F. Duressa, Exponentially fitted tension spline method for singularly perturbed differential difference equations, Iranian Journal of Numerical Analysis and Optimization 11 (2021), 261 – 282, DOI: 10.22067/ijnao.2021.68227.1009.

Downloads

Published

09-05-2023
CITATION

How to Cite

Amala, P., Lalu, M., & Phaneendra, K. (2023). Numerical Simulation for a Differential Difference Equation With an Interior Layer. Communications in Mathematics and Applications, 14(1), 189–202. https://doi.org/10.26713/cma.v14i1.2047

Issue

Vol. 14 No. 1 (2023)

Section

Research Article

License

Creative Commons Licence 4.0 by  

Authors who publish with this journal agree to the following terms:

  • Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a CCAL that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
  • Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
  • Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work.