Approximate Solution of the Singular-Perturbation Problem on Chebyshev-Gauss Grid

Mustafa GÜLSU, Yalçn ÖZTÜRK

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DOI: 10.4236/ajcm.2011.14024                

Abstract

Matrix methods, now-a-days, are playing an important role in solving the real life problems governed by ODEs and/or by PDEs. Many differential models of sciences and engineers for which the existing methodologies do not give reliable results, these methods are solving them competitively. In this work, a matrix methods is presented for approximate solution of the second-order singularly-perturbed delay differential equations. The main characteristic of this technique is that it reduces these problems to those of solving a system of algebraic equations, thus greatly simplifying the problem. The error analysis and convergence for the proposed method is introduced. Finally some experiments and their numerical solutions are given.

Keywords

Singular Perturbation Problems, Two-Point Boundary Value Problems, The Shifted Chebyshev Polynomials, Approximation Method, Matrix Method

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	M. K. Kadalbajoo and D. Kumar, “Fitted Mesh B-Spline Collocation Method for Singularly Perturbed Differential- Difference Equations with Small Delay,” Applied Mathe- matics and Computation, Vol. 204, No. 1, 2008, pp. 90- 98. doi:10.1016/j.amc.2008.05.140
[2]	M. K. Kadalbajoo and K. K. Sharma, “A Numerical Me- thod Based on Finite Difference for Boundary Value Pro- blems for Singularly Perturbed Delay Differential Equa- tions,” Applied Mathematics and Computation, Vol. 197, No. 2, 2008, pp. 692-707. doi:10.1016/j.amc.2007.08.089
[3]	M. K. Kadalbajoo and V. P. Ramesh, “Numerical Meth- ods on Shishkin Mesh for Singularly Perturbed Delay Differential Equations with a Grid Adaptation Strategy,” Applied Mathematics and Computation, Vol. 188, No. 2, 2007, pp. 1816-1831. doi:10.1016/j.amc.2006.11.046
[4]	M. K. Kadalbajoo and V. P. Ramesh, “Hybrid Method for Numerical Solution of Singularly Perturbed Delay Dif- ferential Equationsy,” Applied Mathematics and Compu- tation, Vol. 187, No. 2, 2007, pp. 797-814. doi:10.1016/j.amc.2006.08.159
[5]	M. K. Kadalbajoo and K. K. Sharma, “Numerical Analy- sis of Singularly Perturbed Delay Differential Equations with Layer Behavior,” Applied Mathematics and Com- putation, Vol. 157, No. 1, 2004, pp. 11-28. doi:10.1016/j.amc.2003.06.012
[6]	K. C. Patidar and K. K. Sharma, “ε-Uniformly Conver- gent Non-Standard Finite Difference Methods for Singu- larly Perturbed Differential Difference Equations with Small Delay,” Applied Mathematics and Computation, Vol. 175, No. 1, 2006, pp. 864-890. doi:10.1016/j.amc.2005.08.006
[7]	M. H. Adhikari, E. A. Coutsias and J. K. Mclver, “Periodic Solutions of a Singularly Perturbed Delay Differential Equation,” Physica D, Vol. 237, No. 24, 2008, pp. 3307-3321. doi:10.1016/j.physd.2008.07.019
[8]	I. G. Amirsliyeva, F. Erdogan and G. M. Amiraliyev, “A Uniform Numerical Method for Dealing with a Singularly Perturbed Delay Initial Value Problem,” Applied Mathe- matics Letters, Vol. 23, No. 10, 2010, pp. 1221-1225. doi:10.1016/j.aml.2010.06.002
[9]	S. N. Chow and J. M. Paret, “Singularly Perturbed Delay- Differential Equations, Cuopled Nonlinear Oscillators,” North-Holland Publishing Company, Amsterdam, 1983.
[10]	R. E. O’Malley Jr., “Introduction to Singular Perturbation,” Academic Press, New York, 1979.
[11]	M. Glsu, Y. ?ztürk and M. Sezer, “A Newcollocation Method for Solutionof the Mixed Linear Integro-Differ- ential-Difference Equations,” Applied Mathematics and Computation, Vol. 216, No. 7, 2010, pp. 2183-2198. doi:10.1016/j.amc.2010.03.054
[12]	M. Sezer and M. Gulsu, “Polynomial Solution of the Most General Linear Fredholm Integro-Differential-Dif- ference Equation by Means of Taylor Matrix Method,” International Journal of Complex Variables, Vol. 50, No. 5, 2005, pp. 367-382.
[13]	T. J. Rivlin, “Introduction to the Approximation of Func- tions,” Courier Dover Publications, London, 1969.
[14]	[2] P. J. Davis, “Interpolation and Approximation,” Dover Pub- lications, New York, 1963.
[15]	K. Atkinson and W. Han, “Theoretical Numerical Analysis,” 3rd Edition, Springer, 2009.
[16]	J. P. Body, “Chebyshev and Fourier Spectral Methods,” University of Michigan, New York, 2000.
[17]	J. C. Mason and D. C. Handscomb, “Chebyshev Polyno- mials,” Chapman and Hall/CRC, New York, 2003.