International Journal of Applied Science and Engineering
Published by Chaoyang University of Technology

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

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


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ABSTRACT


In this paper, we have presented a fitted Galerkin method for singularly perturbed differential equations with layer behaviour.  We have introduced a fitting factor in the Galerkin difference scheme which takes care of the rapid changes occur that in the boundary layer.  This fitting factor is obtained from the theory of singular perturbations.  Thomas algorithm is used to solve the tridiagonal system of the fitted Galerkin method.  The existence and uniqueness of the discrete problem along with stability estimates are discussed. Also we have discussed the convergence of the method.  Maximum absolute errors in numerical results are presented to illustrate the proposed method.


Keywords: Singularly perturbed two-point boundary value problem; boundary layer; Taylor series; Galerkin method; maximum absolute error.


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REFERENCES


  1. [1] Bawa, R. K. and Natesan, S. 2009. An Efficient Hybrid Numerical Scheme for Convection- Dominated Boundary-Value Problems, International Journal ofComputer Mathematics, 86, 2: 261-273.

  2. [2] Bellman, R. 1964. “Perturbation Tech-niques in Mathematics, Physics and Engineering”, Holt, Rinehart, Winston, New York.

  3. [3] Bender, C. M. and Orszag, S. A. 1978. “Advanced Mathematical Methods for Scientists and Engineers”, Mc. Graw-Hill, New York.

  4. [4] Eckhaus, W. 1973. “Matched Asymptotic Expansions and Singular Perturbations”, North Holland Publishing company, Amsterdam, Holland.

  5. [5] Hemker, P. W. and Miller, J. J. H. 1979. “Numerical Analysis of Singular Perturbation Problems”, Academic Press, New York.

  6. [6] Kadalbajoo, M. K. and Reddy, Y. N. 1989. asymptotic and Numerical Analy-sis of singular perturbation problems: A Survey, Applied Mathematics and com-putation, 30: 223-259.

  7. [7] Kadalbajoo, M. K. and Patidar, K. C. 2003. Exponentially Fitted Spline in- Compression for the Numerical Solution of Singular Perturbation Problems, Computers and Mathematics with Ap-plications, 46: 751-767.

  8. [8] Kasi Viswanadham, K. N. S., Muralik-rishna, P., and Prabhakara Rao, C. 2010. Numerical Solution of Fifth Order Boundary Value Problems by Colloca-tion Method with Sixth Order B-Splines, International Journal of Applied Science and Engineering, 8, 2: 119-125.

  9. [9] Kevorkian, J. and Cole, J. D. 1981. “PerturbationMethods in Applied Mathematics”,Springer-Verlag, New York.

  10. [10] Natesan, S. and Ramanujam, N. 1999. Improvement of Numerical Solution of Self- Adjoint Singular Perturbation Problems by Incorporation of Asymp-totic Approximations, Applied Mathe-maticsand Computation, 98: 119-137.

  11. [11] Nayfeh, A. H. 1973. “PerturbationMethods”, Wiley, New York.

  12. [12] O’ Malley, R.E. 1974. “Introduction to Singular Perturbations”, Academic Press, New York.

  13. [13] Ramos, H., Vigo-Aguiar, J., Natesan, S., Garcia-Rubio, R and Queiruga M.A. 2010. Numerical Solution of Nonlinear Singularly Perturbed Problems on Non-uniform meshes by using a Non-Standard Algorithm, Journal of Mathematical Chemistry, 48, 1: 38-54.

  14. [14] Valanarasu, T. and Ramanujam, N. 2003. An Asymptotic Initial Value Method for Singularly Perturbed Boundary Value Problems for Second Order Ordinary Differential Equations, Journal of Opti-mization Theory and Application, 116: 167-182.

  15. [15] Van Dyke, M. 1974. “Perturbation Methods in Fluid Mechanics”, Parabolic Press, Stanford, California.

  16. [16] Vigo-Aguiar, J. and Natesan, S. 2006. An Efficient Numerical Method for Singular Perturbation Problems, Journal of Com-putational and Applied Mathematics, 192: 132-141.


ARTICLE INFORMATION




Accepted: 2011-07-06
Available Online: 2011-09-01


Cite this article:

Soujanya, G.B.S.L., Reddy, Y.N., Phaneendra, K. 2011. A fitted galerkin method for singularly perturbed differential equations with layer behaviour. International Journal of Applied Science and Engineering, 9, 195–206. https://doi.org/10.6703/IJASE.2011.9(3).195