An Asymptotic-Fitted Method for Solving Singularly Perturbed Delay Differential Equations

ABSTRACT

In this paper, we presented an asymptotic fitted approach to solve singularly perturbed delay differential equations of second order with left and right boundary. In this approach, the singularly perturbed delay differential equations is modified by approximating the term containing negative shift using Taylor series expansion. After approximating the coefficient of the second derivative of the new equation, we introduced a fitting parameter and determined its value using the theory of singular Perturbation; O’Malley [1]. The three term recurrence relation obtained is solved using Thomas algorithm. The applicability of the method is tested by considering five linear problems (two problems on left layer and one problem on right layer) and two nonlinear problems.

In this paper, we presented an asymptotic fitted approach to solve singularly perturbed delay differential equations of second order with left and right boundary. In this approach, the singularly perturbed delay differential equations is modified by approximating the term containing negative shift using Taylor series expansion. After approximating the coefficient of the second derivative of the new equation, we introduced a fitting parameter and determined its value using the theory of singular Perturbation; O’Malley [1]. The three term recurrence relation obtained is solved using Thomas algorithm. The applicability of the method is tested by considering five linear problems (two problems on left layer and one problem on right layer) and two nonlinear problems.

KEYWORDS

Asymptotic-Fitted Scheme; Delay-Differential Equations; Singular Perturbation; Boundary Layer

Asymptotic-Fitted Scheme; Delay-Differential Equations; Singular Perturbation; Boundary Layer

Cite this paper

A. Andargie and Y. Reddy, "An Asymptotic-Fitted Method for Solving Singularly Perturbed Delay Differential Equations,"*Applied Mathematics*, Vol. 3 No. 8, 2012, pp. 895-902. doi: 10.4236/am.2012.38132.

A. Andargie and Y. Reddy, "An Asymptotic-Fitted Method for Solving Singularly Perturbed Delay Differential Equations,"

References

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[3] H. S. Prasad and Y. N. Reddy, “Numerical Solution of Singularly Perturbed Differential-Difference Equations with Small Shifts of Mixed Type by Differential Quadrature Method,” American Journal of Computational and Applied Mathematics, Vol. 2, No. 1, 2012, pp. 46-52.

[4] M. K. Kadalbajoo and K. K. Sharma, “Numerical Analysis of Singularly Perturbed Delay Differential Equations with Layer Behavior,” Applied Mathematics and Computation, Vol. 157, No. 1, 2004, pp. 11-28. doi:10.1016/j.amc.2003.06.012

[5] M. K. Kadalbajoo and V. P. Ramesh, “Hybrid Method for Numerical Solution of Singularly Perturbed Delay Differential Equations,” Applied Mathematics and Computation, Vol. 187, No. 2, 2007, pp. 797-814. doi:10.1016/j.amc.2006.08.159

[6] M. K. Kadalbajoo and K. K. Sharma, “Numerical Method Based on Finite Difference for Boundary Value Problems for Singularly Perturbed Delay Differential Equations,” Applied Mathematics and Computation, Vol. 197, No. 2, 2008, pp. 692-707. doi:10.1016/j.amc.2007.08.089

[7] M. K. Kadalbajoo and D. Kumar, “Fitted Mesh B-Spline Collocation Method for Singularly Perturbed Differential-Difference Equations with Small Delay,” Applied Mathematics and Computation, Vol. 204, No. 1, 2008, pp. 90-98. doi:10.1016/j.amc.2008.05.140

[8] M. K. Kadalbajoo and D. Kumar, “Computational Method for Singularly Perturbed Nonlinear Differential-Difference Equations with Small Shift,” Applied Mathematical Modeling, Vol. 34, No. 9, 2010, pp. 2584-2596. doi:10.1016/j.apm.2009.11.021

[9] A. Bellen and M. Zennaro, “Numerical Methods for Delay Differential Equations,” Oxford University Press, Oxford, 2003. doi:10.1093/acprof:oso/9780198506546.001.0001

[10] R. D. Driver, “Ordinary and Delay Differential Equations,” Springer-Verlag, New York, 1977. doi:10.1007/978-1-4684-9467-9

[11] R. E. Bellman and K. L. Cooke, “Differential-Difference Equations,” Academy Press, New York, 1963.

[12] L. E. El’sgol’ts, “Qualitative Methods in Mathematical Analyses, Translations of Mathematical Monographs 12,” American Mathematical Society, Providence, 1964.

[13] 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 Journal on Applied Mathematics, Vol. 54, No. 1, 1994, pp. 249-272. doi:10.1137/S0036139992228120

[14] C. G. Lange and R. M. Miura, “Singular Perturbation Analysis of Boundary Value Problems for differential difference equations, (VI). Small Shifts with Rapid Oscillations,” SIAM Journal on Applied Mathematics, Vol. 54, No. 1, 1994, pp. 273-283. doi:10.1137/S0036139992228119

[15] C. G. Lange and R. M. Miura, “Singular Perturbation Analysis of Boundary Value Problems for Differential Difference Equations,” SIAM Journal on Applied Mathematics, Vol. 45, No. 5, 1985, pp. 687-707. doi:10.1137/0145041

[16] A. Andargie and Y. N. Reddy, “An Exponentially Fitted Special Second-Order Finite Difference Method for Solving Singular Perturbation Problems,” Applied Mathematics and Computation, Vol. 190, No. 2, 2007, pp. 1767-1782. doi:10.1016/j.amc.2007.02.051

[17] E. Angel and R. Bellman, “Dynamic Programming and partial Differential Equations,” Academic Press, New York, 1972.

[1] R. E. O’Malley, “Introduction to Singular Perturbations,” Academic Press, New York, 1974.

[2] R. N. Rao and P. Pramod Chakravarthy, “A Fourth Order Finite Difference Method for Singularly Perturbed Differential-Difference Equations,” American Journal of Computational and Applied Mathematics, Vol. 1, No. 1, 2011, pp. 5-10.

[3] H. S. Prasad and Y. N. Reddy, “Numerical Solution of Singularly Perturbed Differential-Difference Equations with Small Shifts of Mixed Type by Differential Quadrature Method,” American Journal of Computational and Applied Mathematics, Vol. 2, No. 1, 2012, pp. 46-52.

[4] M. K. Kadalbajoo and K. K. Sharma, “Numerical Analysis of Singularly Perturbed Delay Differential Equations with Layer Behavior,” Applied Mathematics and Computation, Vol. 157, No. 1, 2004, pp. 11-28. doi:10.1016/j.amc.2003.06.012

[5] M. K. Kadalbajoo and V. P. Ramesh, “Hybrid Method for Numerical Solution of Singularly Perturbed Delay Differential Equations,” Applied Mathematics and Computation, Vol. 187, No. 2, 2007, pp. 797-814. doi:10.1016/j.amc.2006.08.159

[6] M. K. Kadalbajoo and K. K. Sharma, “Numerical Method Based on Finite Difference for Boundary Value Problems for Singularly Perturbed Delay Differential Equations,” Applied Mathematics and Computation, Vol. 197, No. 2, 2008, pp. 692-707. doi:10.1016/j.amc.2007.08.089

[7] M. K. Kadalbajoo and D. Kumar, “Fitted Mesh B-Spline Collocation Method for Singularly Perturbed Differential-Difference Equations with Small Delay,” Applied Mathematics and Computation, Vol. 204, No. 1, 2008, pp. 90-98. doi:10.1016/j.amc.2008.05.140

[8] M. K. Kadalbajoo and D. Kumar, “Computational Method for Singularly Perturbed Nonlinear Differential-Difference Equations with Small Shift,” Applied Mathematical Modeling, Vol. 34, No. 9, 2010, pp. 2584-2596. doi:10.1016/j.apm.2009.11.021

[9] A. Bellen and M. Zennaro, “Numerical Methods for Delay Differential Equations,” Oxford University Press, Oxford, 2003. doi:10.1093/acprof:oso/9780198506546.001.0001

[10] R. D. Driver, “Ordinary and Delay Differential Equations,” Springer-Verlag, New York, 1977. doi:10.1007/978-1-4684-9467-9

[11] R. E. Bellman and K. L. Cooke, “Differential-Difference Equations,” Academy Press, New York, 1963.

[12] L. E. El’sgol’ts, “Qualitative Methods in Mathematical Analyses, Translations of Mathematical Monographs 12,” American Mathematical Society, Providence, 1964.

[13] 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 Journal on Applied Mathematics, Vol. 54, No. 1, 1994, pp. 249-272. doi:10.1137/S0036139992228120

[14] C. G. Lange and R. M. Miura, “Singular Perturbation Analysis of Boundary Value Problems for differential difference equations, (VI). Small Shifts with Rapid Oscillations,” SIAM Journal on Applied Mathematics, Vol. 54, No. 1, 1994, pp. 273-283. doi:10.1137/S0036139992228119

[15] C. G. Lange and R. M. Miura, “Singular Perturbation Analysis of Boundary Value Problems for Differential Difference Equations,” SIAM Journal on Applied Mathematics, Vol. 45, No. 5, 1985, pp. 687-707. doi:10.1137/0145041

[16] A. Andargie and Y. N. Reddy, “An Exponentially Fitted Special Second-Order Finite Difference Method for Solving Singular Perturbation Problems,” Applied Mathematics and Computation, Vol. 190, No. 2, 2007, pp. 1767-1782. doi:10.1016/j.amc.2007.02.051

[17] E. Angel and R. Bellman, “Dynamic Programming and partial Differential Equations,” Academic Press, New York, 1972.