American Journal of Applied Mathematics and Statistics
ISSN (Print): 2328-7306 ISSN (Online): 2328-7292 Website: http://www.sciepub.com/journal/ajams Editor-in-chief: Mohamed Seddeek
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American Journal of Applied Mathematics and Statistics. 2014, 2(5), 336-343
DOI: 10.12691/ajams-2-5-7
Open AccessArticle

Numerical Solution of Singularly Perturbed Differential-Difference Equations with Dual Layer

Lakshmi Sirisha1 and Y.N. Reddy1,

1Department of Mathematics, National Institute of Technology, WARANGAL, INDIA

Pub. Date: October 12, 2014

Cite this paper:
Lakshmi Sirisha and Y.N. Reddy. Numerical Solution of Singularly Perturbed Differential-Difference Equations with Dual Layer. American Journal of Applied Mathematics and Statistics. 2014; 2(5):336-343. doi: 10.12691/ajams-2-5-7

Abstract

In this paper, we discuss the numerical solution of singularly perturbed differential-difference equations exhibiting dual layer behavior. First the second order singularly perturbed differential-difference equation is replaced by an asymptotically equivalent second order singularly perturbed ordinary differential equation. Then, second order stable central difference scheme has been applied to get a three term recurrence relation which is easily solved by Discrete Invariant Imbedding Algorithm. Some numerical examples have been considered to validate the computational efficiency of the proposed numerical scheme. To analyze the effect of the parameters on the solutions, the numerical solutions have also been plotted using graphs. The error bound and convergence of the method have also been established.

Keywords:
singular perturbations differential-difference equations dual layer delay parameter advance parameter

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References:

[1]  R. Bellman, and K. L. Cooke, Differential-Difference Equations. Academic Press, New York, 1963.
 
[2]  R. D. Driver, Ordinary and Delay Differential Equations, , 1977.
 
[3]  K. Denevers and K. Schmit, “An application of the shooting method to boundary value problems for second-order delay equations”, J. Math. Anal. Appl., 36 (1971) 588-597.
 
[4]  G. B. Gustafson and K. Schmitt, “Nonzero solutions of boundary value problems for second order ordinary and delay differential equations”, J. Differential Equations, 12 (1972) 129-147.
 
[5]  R.B. Stein, “A theoretical analysis of neuronal variability”, Biophys. J. 5 (1965) 173-194.
 
[6]  H. C. Tuckwell and W. Richter, “Neuronal inter-spike time distributions and the estimation of neuro-physiological and neuro-anatomical parameters”, J. Theor. Biol., 71 (1978) 167-183.
 
[7]  H. C. Tuckwell and D. K.Cope, “Accuracy of neuronal inter-spike times calculated from a diffusion approximation”, J. Theor. Biol., 83 (1980) 377-387.
 
[8]  W. J. Wilbur and J. Rinzel, “An analysis of Stein's model for stochastic neuronal excitation”, Biol. Cybern., 45 (1982) 107-114.
 
[9]  C.G. Lange R.M. Miura, “Singular perturbation analysis of boundary-value problems for differential difference equations”, SIAM J. Appl. Math., 42 (1982) 502-531.
 
[10]  C.G. Lange R.M. Miura, “Singular perturbation analysis of boundary-value problems for differential-difference equations. V. Small shifts with layer behaviour”, SIAM J. Appl. Math., 54 (1994) 249-272.
 
[11]  C.G. Lange R.M. Miura, “Singular perturbation analysis of boundary-value problems for differential-difference equations. VI. Small shifts with rapid oscillations”, SIAM J. Appl. Math., 54 (1994) 273-283.
 
[12]  M. K. Kadalbajoo and K. K. Sharma, “Numerical analysis of boundary-value problems for singularly perturbed differential-difference equations with small shifts of mixed type”, J. Optim. Theory Appl., 115 (1) (2002) 145-163.
 
[13]  M. K. Kadalbajoo and K. K. Sharma, “Numerical treatment of a mathematical model arising from a model of neuronal variability”, J. Math. Anal. Appl., 307 (2005) 606-627.
 
[14]  L. E. El’sgolts and S. B. Norkin, Introduction to the Theory and Applications of Differential Equations with Deviating Arguments, Academic Press, , 1973.
 
[15]  R. K. Mohanty and N. Jha, “A class of variable mesh spline in compression methods for singularly perturbed two point singular boundary value problems”, Applied Mathematics and Computation, 168 (2005) 704-716.