ASYMPTOTIC-NUMERICAL METHOD FOR SINGULARLY PERTURBED DIFFERENTIAL DIFFERENCE EQUATIONS OF MIXED-TYPE

- Journal title : Journal of applied mathematics & informatics
- Volume 33, Issue 5_6, 2015, pp.485-502
- Publisher : The Korean Society of Computational and Applied Mathematics
- DOI : 10.14317/jami.2015.485

Title & Authors

ASYMPTOTIC-NUMERICAL METHOD FOR SINGULARLY PERTURBED DIFFERENTIAL DIFFERENCE EQUATIONS OF MIXED-TYPE

SALAMA, A.A.; AL-AMERY, D.G.;

SALAMA, A.A.; AL-AMERY, D.G.;

Abstract

A computational method for solving singularly perturbed boundary value problem of differential equation with shift arguments of mixed type is presented. When shift arguments are sufficiently small (o(ε)), most of the existing method in the literature used Taylor`s expansion to approximate the shift term. This procedure may lead to a bad approximation when the delay argument is of O(ε). The main idea for this work is to deal with constant shift arguments, which are independent of ε. In the present method, we construct the formally asymptotic solution of the problem using the method of composite expansion. The reduced problem is solved numerically by using operator compact implicit method, and the second problem is solved analytically. Error estimate is derived by using the maximum norm. Numerical examples are provided to support the theoretical results and to show the efficiency of the proposed method.

Keywords

Singularly perturbed;Differential difference equations;Boundary layer;Asymptotic expansion;Compact method;

Language

English

References

1.

G.M. Amiraliyev and E. Cimen, Numerical method for a singularly perturbed convectiondiffusion problem with delay, Appl. Math. Comput. 216 (2010), 2351-2359.

2.

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.

3.

R.D. Driver, Ordinary and delay differential equations, Springer, New York, 1977.

4.

V.Y. Glizer, Asymptotic analysis and solution of a finite-horizon H_{∞} control problem for singularly-perturbed linear systems with small state delay, J. Optim. Theory Appl. 117 (2003), 295-325.

5.

M.K. Kadalbajoo, K.C. Patidar and K.K. Sharma, ϵ-uniformly convergent fitted methods for the numerical solution of the problems arising from singularly perturbed general DDEs, Appl. Math. Comput. 182 (2006), 119-139.

6.

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 (2002), 145-163.

7.

M.K. Kadalbajoo and K.K. Sharma, ϵ-uniform fitted mesh method for singularly perturbed differential difference equations: mixed type of shifts with layer behavior, Int. J. Comput. Math. 81 (2004), 49-62.

8.

C.G. Lange and R.M. Miura, Singular perturbation analysis of boundary value problems of differential-difference equations, SIAM J. Appl. Math. 42 (1982), 502-531.

9.

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 J. Appl. Math. 54 (1994), 249-272.

10.

A. Longtin and J. Milton, Complex oscillations in the human pupil light reflex with mixed and delayed feedback, Math. Biosci. 90 (1988), 183-199.

11.

M.C. Mackey and L. Glass, Oscillations and chaos in physiological control systems, Science 197 (1977), 287-289.

12.

J.J.H. Miller, E. O’Riordan and G.I. Shishkin, Fitted numerical methods for singular perturbation problems, World Scientific, Singapore, 1996.

13.

J. Mohapatra and S. Natesan, Uniformly convergent numerical method for singularly perturbed differential-difference equation using grid equidistribution, Int. J. Numer. Meth. Biomed. Engng. 27 (2011), 1427-1445.

14.

A.H. Nayfeh, Introduction to perturbation methods, John Wiley and Sons, New York, 1981.

15.

R.E.Jr. O’Malley, Singular-perturbation methods for ordinary differential equations, Springer Verlag, New York, 1990.

16.

A.A. Salama, The operator compact implicit methods for the numerical treatment of ordinary differential and integro-differential equations, Ph.D. Thesis, Assiut University, Assiut, 1992.

17.

A.A. Salama and D.G. Al-Amery, High-order method for singularly perturbed differentialdifference equations with small shifts, Int. J. Pure Appl. Math. 8 (2013), 273-295.

18.

V. Subburayan and N. Ramanujam, An initial value technique for singularly perturbed convection-diffusion problems with a negative shift, J. Optim Theory Appl. 158 (2013), 234-250.

19.

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