International Journal of Applied Science and Engineering
Published by Chaoyang University of Technology

Gemechis File* and Y. N. Reddy

Department of Mathematics, National Ins


 

Download Citation: |
Download PDF


ABSTRACT


In this paper, a computational method is presented for solving singularly perturbed delay differential equations with negative shift whose solution has boundary layer. First, the second order singularly perturbed delay differential equation is replaced by an asymptotically equivalent first order delay differential equation. Then, Trapezoidal integration formula and linear interpolation are employed to get three term recurrence relation which is solved easily by Discrete Invariant Imbedding Algorithm. The method is demonstrated by implementing several model examples by taking various values for the delay parameter and the perturbation parameter.


Keywords: Singular perturbations; delay differential equations; delay parameter; boundary layer; perturbation parameter; trapezoidal rule.


Share this article with your colleagues

 


REFERENCES


[1] Angel, E. and Bellman, R. 1972. “Dynamic Programming and Partial differential equations”. Academic Press. New York.

[2] Elsgolt's, L. E. and Norkin, S. B. 1973. “Introduction to the Theory and Applications of Differential Equations with Deviating Arguments”. Academic Press. New York.

[3] Kadalbajoo, M. K. and Reddy Y. N. 1986. A non-asymptotic method for general linear singular perturbation problems. Journal of Optimization Theory and Applications, 55: 256-269.

[4] Kadalbajoo, M. K. and Sharma, K. K. 2008. A numerical method based on finite difference for Boundary value problems for singularly perturbed delay differential equations. Applied Mathematics and Computation, 197: 692-707.

[5] Kadalbajoo, M. K., Patidar, K. C., and Sharma, K. K. 2006. e-Uniformly convergent fitted methods for the numerical solution of the problems arising from singularly perturbed general DDEs. Applied Mathematics and Computation, 182: 119- 139.

[6] Kadalbajoo, M. K. and Sharma, K. K. 2004. Numerical Analysis of Boundary Value Problems for Singularly Perturbed Differential-Difference Equations: Small Shifts of Mixed Type with Rapid Oscillations. Comm. Numer. Meth. Eng. 20: 167-182.

[7] Kadalbajoo, M. K. and Sharma, K. K. 2004. Numerical Analysis of Singularly Perturbed Delay Differential Equations with Layer Behavior. Appl.Math. Comput, 157, 1: 11-28.

[8] Kuang, Y. 1993. “Delay Differential Equations with Applications in Population Dynamics”.  Academic Press.

[9] Lange, C. G. and Miura, R. M. 1994. Singular Perturbation Analysis of Boundary-Value Problems for Differential Difference Equations. V. Small Shifts with Layer Behavior. SIAM Journal on Applied Mathematics, 54: 249-272.

[10] Lange, C. G. and Miura, R. M. 1994. Singular Perturbation Analysis of Boundary-Value Problems for Differential Difference Equations. VI. Small Shifts with Rapid Oscillations. SIAM Journal on Applied Mathematics, 54: 273-283.

[11] Longtin, A. and Milton, J. 1988. Complex oscillations in the human pupil light reflex with mixed and delayed feedback. Mathematical Biosciences, 90: 183-199.

[12] Patidar, K. C. and Sharma, K. K. 2006. -Uniformly Convergent Non-Standard Finite Difference Methods for Singularly Perturbed Differential Difference Equations with Small Delay. Applied Mathematics and Computation, 175: 864-890.

[13] Phaneendra, K., Reddy, Y. N., and Soujanya, GBSL. 2011. A Seventh Order Numerical Method for Singular Perturbed Differential-Difference Equations with Negative Shift. Nonlinear Analysis: Modeling and Control, 16, 2: 206-219.

[14] Rao, R. N. and Chakravarthy, P. P. 2011. A Fourth Order Finite Difference Method for Singularly Perturbed Differential-difference Equations. American Journal of Computational and Applied Mathematics, 1, 1: 5-10.

[15] Yadaw, A. S. and Kadalbajoo, M. K. 2009. Parameter-Uniform Ritz-Galerkin Finite Element Method for Singularly Perturbed Delay Differential Equations with Delay in Convection Term. International Journal of Pure and Applied Mathematics, 57, 4: 459-474.


ARTICLE INFORMATION


Received: 2012-04-27
Revised: 2012-11-05
Accepted: 2012-11-13
Available Online: 2013-03-01


Cite this article:

File, G., Reddy, Y.N. 2013. Computational method for solving singularly perturbed delay differential Equations with negative shift. International Journal of Applied Science and Engineering, 11, 101–113. https://doi.org/10.6703/IJASE.2013.11(1).101


We use cookies on this website to improve your user experience. By using this site you agree to its use of cookies.