DOMAIN DECOMPOSITION METHOD FOR SINGULARLY PERTURBED DIFFERENTIAL DIFFERENCE EQUATIONS WITH LAYER BEHAVIOR

Abstract

In this paper, a domain decomposition method has been presented for solving singularly perturbed differential difference equations with delay as well as advances whose solution exhibits boundary layer behavior. By introducing a terminal point, the original problem is divided into inner and outer region problems. An implicit terminal boundary condition at the terminal point has been determined. The outer region problem with the implicit boundary condition is solved and produces an explicit boundary condition for the inner region problem. Then, the modified inner region problem (using the stretching transformation) is solved as a two-point boundary value problem. Fourth order stable central difference method has been used to solve both the inner and outer region problems. The proposed method is iterative on the terminal point. To demonstrate the applicability of the method, some numerical examples have been solved for different values of the perturbation parameter, delay and advance parameters. The stability and convergence of the scheme has also investigated

Keywords

• Singular perturbation, Differential difference equations, Finite Differences, Terminal Boundary Condition, Boundary layer

References

1. Stein, R. B., A theoretical analysis of neuronal variability, Biophys. J. 5, 173–194, 1965.
2. Stein, R. B., Some models of neuronal variability, Biophys. J. 7, 37–67, 1967.
3. Tuckwell, H. C., Firing rates of motor neurons with strong random synaptic excitation, Biol. Cybernet., 24, 147–152, 1976.
4. Tuckwell, H. C., Introduction to Theoretical Neurobiology, vol. 1, Cambridge University Press, Cambridge, 1988.
5. Tuckwell, H. C., Introduction to Theoretical Neurobiology, vol. 2, Cambridge University Press, Cambridge, 1988.
6. Tuckwell, H. C. and Wan, F.Y.M., Time to first spike in stochastic Hodgkin–Huxley systems, Physica A, 351, 427–438, 2005.
7. Lange, C.G. and Miura, R. M., Singular perturbation analysis of boundary-value problems for differential-difference equations, SIAM J. Appl. Math. 42, 502–531, 1982.
8. Derstine, M.W., Gibbs, F.A.H.H.M. and Kaplan, D. L., Bifurcation gap in a hybrid optical system, Phys. Rev. A, 26, 3720–3722, 1982.

9. Longtin, A. and Milton, J., Complex oscillations in the human pupil light reflex with mixed and delayed feedback, Math. Biosci., 90, 183–199, 1988.
10. Wazewska-Czyzewska, M. and Lasota, A., Mathematical models of the red cell system, Mat. Stos., 6, 25–40, 1976.
11. Mackey, M. C. and Glass, L., Oscillations and chaos in physiological control systems, Science, 197, 287–289, 1977.
12. Lange, C.G. and Miura, R. M., Particular solutions of forced generalized airy equations, J. Math. Anal. Appl., 109, 303–310, 1985.
13. Lange, C.G. and Miura, R. M., Singular perturbation analysis of boundary-value problems for differential-difference equations. II: Rapid oscillations and resonances, SIAM J. Appl. Math. 45, 687–707, 1985.
14. Lange, C.G. and Miura, R. M., Singular perturbation analysis of boundary-value problems for differential-difference equations. III: Turning point problems, SIAM J. Appl. Math. 45, 708–734, 1985.
15. Lange, C.G. and Miura, R. M., Singular perturbation analysis of boundary-value problems for differential-difference equations. IV: A nonlinear example with layer behavior, Stud. Appl. Math. 84, 231–273, 1991.
16. Lange, C.G. and Miura, R. M., Singular perturbation analysis of boundary-value problems for differential-difference equations. V: Small shifts with layer behavior, SIAM J. Appl. Math. 54, 249–272, 1994.
17. Lange, C.G. and Miura, R. M., Singular perturbation analysis of boundary-value problems for differential-difference equations. VI: Small shifts with rapid oscillations, SIAM J. Appl. Math. 54, 273–283, 1994.
18. Sharma, K. K., Numerical Analysis for Boundary Value Problems for Singularly Perturbed Differential-Difference Equations with Delay as well as Advance, Ph.D. Thesis, Department of Mathematics, Indian Institute of Technology Kanpur, 2003.
19. Patidar, K.C. and Sharma, K. K., Uniformly convergent non-standard finite difference methods for singularly perturbed differential-difference equations with delay and advance, Int. J. Numer. Methods Eng. 66, 272–296, 2006.
20. Kadalbajoo, M. K., Sharma, K.K. and Patidar, K.C., e-Uniformly convergent fitted methods for the numerical solution of the problems arising from singularly perturbed general DDEs, Appl. Math. Comput. 182, 119–139, 2006.
21. Kumar, D. and Kadalbajoo, M. K., A parameter-uniform numerical method for timedependent singularly perturbed differential-difference equations, Applied Mathematical Modelling, 35, 2805–2819, 2011.
22. Elsgolt’s, L. E. and Norkin, S. B., Introduction to the Theory and Applications of Differential Equations with Deviating Arguments, Academic Press, New York, 1973.
23. Choo, J. Y., and Schultz, D. H.. Stable higher order methods for differential equations with small coefficients for the second order terms, Journal of Computers and Mathematics with Applications, 25(1993) 105-123.
24. Hsiao G. C, and Jordan K. E., Solutions to the difference equations of singular perturbation problems, Academic Press, New York, 1979.
25. Lorenz J., Combinations of initial and boundary value method for a class of singular perturbation problems, Academic Press, New York, 1979.
26. Keller, H. B, Numerical Methods for Two point boundary value problems, Blaisdell Publishing Company, Waltham, 1968.
27. Greenspan, D. and Casulli, V., Numerical analysis for Applied Mathematics, Science and Engineering, Addison-Wesley publishing Co., Inc., 1988.