American Journal of Numerical Analysis
ISSN (Print): 2372-2118 ISSN (Online): 2372-2126 Website: Editor-in-chief: Emanuele Galligani
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American Journal of Numerical Analysis. 2017, 5(1), 1-10
DOI: 10.12691/ajna-5-1-1
Open AccessArticle

Numerical Solution of Singularly Perturbed Delay Reaction-Diffusion Equations with Layer or Oscillatory Behaviour

Gemechis File1, Gashu Gadisa1, Tesfaye Aga1 and Y. N. Reddy2,

1Department of Mathematics, Jimma University, Jimma, P. O. Box 378, Ethiopia

2Department of Mathematics, National Institute of Technology, Warangal-506 004, India

Pub. Date: February 05, 2017

Cite this paper:
Gemechis File, Gashu Gadisa, Tesfaye Aga and Y. N. Reddy. Numerical Solution of Singularly Perturbed Delay Reaction-Diffusion Equations with Layer or Oscillatory Behaviour. American Journal of Numerical Analysis. 2017; 5(1):1-10. doi: 10.12691/ajna-5-1-1


In this paper, we presented numerical method for solving singularly perturbed delay differential equations with layer or oscillatory behaviour for which a small shift (δ) is in the reaction term. First, the given singularly perturbed delay reaction-diffusion equation is converted into an asymptotically equivalent singularly perturbed two point boundary value problem and then solved by using fourth order finite difference method. The stability and convergence of the method has been investigated. The numerical results have been tabulated and further to examine the effect of delay on the boundary layer and oscillatory behavior of the solution, graphs have been given for different values of δ. Both theoretical and numerical rate of convergence have been established and are observed to be in agreement for the present method. Briefly, the present method improves the findings of some existing numerical methods in the literature.

singularly perturbed problem delay reaction-diffusion equation oscillatory behaviour

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[1]  Gemechis File and Y.N. Reddy, Numerical integration of a class of singularly perturbed delay differential equations with small shift, International Journal of Differential Equation, 572723 (2012), 1-12.
[2]  C. G. Lange and R. M. Miura, Singular perturbation analysis of boundary-value problems for differential-difference equations very small shifts with layer behavior, SIAM J. Appl. Math., 54 (1994), 249-272.
[3]  V.P. Ramesh and M.K. Kadalbajoo, Numerical Algorithm for Singularly Perturbed Delay Differential Equations with Layer and Oscillatory Behavior, Neural, Parallel, and Scientific Computations, 19 (2011), 21-34.
[4]  D. K. Swamy, Quantative analysis of delay differential equations with layer, Advance Research and Innovations in Mechanical, Material Science, Industrial Engineering and Management - ICARMMIEM- (2014), 145-150.
[5]  G. BSL. Soujanya and Y.N. Reddy, Computational method for singularly perturbed delay differential equations with layer or oscillatory behaviour, Appl. Math. Inf. Sci. 10 (2), (2016), 527-536.
[6]  E.R. Doolan, J.J.H. Miller and W.H.A. Schilders, Uniform Numerical Methods for Problems with Initial and Boundary Layers, Boole Press, Dublin, 1980.
[7]  M.K. Kadalbajoo and Y.N. Reddy, Asymptotic and numerical analysis of singular perturbation problems, Applied Mathematics and Computation, 30 (1989), 223-259.
[8]  H.G. Roos, M. Stynes and L. Tobiska, Numerical Methods for Singularly Perturbed Differential Eqns, Convection-Diffusion and Flow Problems, Springer Verlag, Berlin, 1996.
[9]  M.K. Kadalbajoo and V.P. Ramesh, Numerical methods on Shishkin mesh for singularly perturbed delay differential equations with a grid adaptation strategy, Applied Mathematics and Computation, 188 (2007), 1816-1831.
[10]  R. Pratima and K.K. Sharma, Numerical analysis of singularly perturbed delay differential turning point problem, Applied Mathematics and Computation, 218 (2011), 3483-3498.
[11]  L.E. Elsgolt’s and S.B. Norkin, Introduction to the Theory and Applications of Differential Equations with Deviating Arguments, Academic Press, New York, 1973.
[12]  M. K. Kadalbajoo and Y.N. Reddy, A non-asymptotic method for general singular perturbation problems, Journal of Optimization Theory and Applications, 55 (1986), 256 269.
[13]  R.S. Varga, Matrix Iterative Analysis. New Jersey: Prentice-Hall, Englewood Cliffs, 1962.
[14]  D.M. Young, Iterative Solution of Large Linear Systems, Academic Press, Inc. New York, 1971.