K. Phaneendra*, Y. N. Reddy, and GBSL. Soujanya

Department of Mathematics, National Institute of Technology, Warangal, India

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ABSTRACT

In this paper, a class of third order singularly perturbed boundary value problems with suitable boundary conditions is considered. The third order boundary value problem is transformed to asymptotically equivalent second order boundary value problem.  This problem is solved efficiently by using fitted Numerov method.  Linear and non-linear examples are solved to illustrate the method and relative errors with L2-norms are presented to support the method.


Keywords: Fitted numerov method; boundary value problem; tridiagonal system; fitting factor; relative error.


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REFERENCES

  1. [1] O'Malley, R.E. 1990. “Singular Perturbation Methods for Ordinary Differential Equations”. Springer-Verlag, New York.

  2. [2] Nayfeh, A.H. 1981. “Introduction to Perturbation Methods”. John Wiley and Sons, New York.

  3. [3] Ross, H.G., Stynes, M., and Tobiska, L. 1996. “Numerical Methods for Singularl Perturbed Differential Equations-Convection-Diffusion and Flow Problems”. Springer-Verlag.

  4. [4] Howes, F.A., 1983. The asymptotic solution of a class of third-order boundary value problem arising in the theory of thin film flow, SIAM J. Appl. Math., 43, 5: 993-1004.

  5. [5] Meyer R. E. and Parter, S. V.  1980. “Some old and new results on singularly Perturbed boundary value problems, in Singular Perturbations and Asymptotics”. Academic Press, New York, 41-85.

  6. [6] Jayakumar J. and Ramanujam, N. 1993. A computational method for solving singular perturbation problems, Applied Mathematics and Computation, 55, 31-48.

  7. [7] Doolan, E.P., Miller, J.J.H., and Schilders, W.H.A. 1980. “Uniform Numerical Methods for Problems with Initial and Boundary Layers”. Boole Press, Dublin.

  8. [8] Ramesh Babu, A. and Ramanujam, N. 2007. An asymptotic finite element method for singularly perturbed third and fourth order ordinary differential equations with discontinuous source term, Applied Mathematics and Computation, 191, 2: 372-380.

  9. [9] Bender, C.M. and Orszag, S.A. 1978. “Advanced mathematical methods for scientists and Engineers”, Mc. Graw-Hill, New York.

  10. [10] Phaneendra, K., Pramod Chakravarthy P., and Reddy, Y.N. 20101. A Fitted Numerov Method for Singular Perturbation Problems Exhibiting Twin Layers, Applied Mathematics & Information Sciences– An International Journal, Dixie W Publishing Corporation, U.S. A., 4, 3, 341-352, 2010.


ARTICLE INFORMATION

Received: 2011-09-06
Revised: 2012-02-01
Accepted: 2012-02-26
Available Online: 2012-09-01


Cite this article:

Phaneendra, K., Reddy, Y.N., Soujanya, G.B.S.L. 2012. Asymptotic-numerical method for third-order singular perturbation problems. International Journal of Applied Science and Engineering, 10, 241–248. https://doi.org/10.6703/IJASE.2012.10(3).241

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