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Numerical Solution of Singular Perturbation problems via deviating Argument through the Numerical methods

Author Affiliations

  • 1Kakatiya Institute of Technolog and Science, warangal- 506015, INDIA
  • 2Kakatiya Institute of Technolog and Science, warangal- 506015, INDIA
  • 3Kakatiya Institute of Technolog and Science, warangal- 506015, INDIA

Res. J. Mathematical & Statistical Sci., Volume 2, Issue (9), Pages 9-19, September,12 (2014)


An attempt is made in this article to obtain the numerical solution of singularly perturbed two point boundary value problems. To achieve this singular perturbation problem is reduced to first order differential equation by taking a small deviating argument. The Simpsons 3/8 rule is employed to get the equation in y (xi). Hermite interpolation is used to obtain the value of y at the intermediate points of the boundary, finally yielding to a tridiagonal system of equations. The discrete invariant imbedding method is used to obtain the solution of system of equations. four linear singular perturbation problems of which two are with constant coefficients and two are with variable coefficients are solved to test the applicability and competence of the proposed method. The numerical results obtained by the proposed method are compared with the exact solution and also with the results obtained using Simpsons 1/3 rule. It is observed that the numerical results are very near to the exact solution.


  1. Bellman R., Perturbation Techniques in Mathematics,Physics and Engineering, Holt Rinehart and Winston,(1964)
  2. Bender C.M. and Orszag S.A., Advanced MathematicalMethods for Scientist and Engineers, McGraw-Hill, NewYork, (1978)
  3. O, Introduction to Singular Perturbations, Academic Press, New York, (1974)
  4. Eckhaus W., Asymptotic Analysis of SingularPerturbations, North Holland, New York, (1979)
  5. Cole J.D. and Kevorkian J., Perturbation Methods inApplied Mathematics, Springer, New York, (1979)
  6. Nayfeh A.H., Perturbation Methods, Wiley, New York,(1973)
  7. Vandyke M., Perturbation Methods in Fluid Mechanics, Parabolic, Stanford, Calif, (1975)
  8. Bush A.B., Perturbation Methods for Engineers andScientists, CRC Press, Bocaraton, (1992)
  9. Holmes M.H., Introduction to Perturbation Methods, Springer Verlag, Heidelbers, (1995)
  10. Murdock J.A., Perturbation Theory and Methods, JohnWiley and Sons, New York, (1991)
  11. Niijima K., On a three point difference scheme for asingular perturbation problem without a first derivativeterm, I, II. Mem. Numer. Math, 7, 1-27 (1980)
  12. Miller J.J.H., On a convergence, uniformly in ε, of finitedifference schemes for a two point boundary singularperturbation problems, Proc. Conf. Math. Ins. CatholicUniv. Nijmegen, 1978, Academic Press, New York, 467-474, (1979)
  13. Kadalbajoo M.K. and Patidar K.C., Spline approximationmethod for solving self-adjoint singular perturbationproblems on non-uniform grids, J. Comput. Anal. Appl., 5,425-451, (2003)
  14. El, Introduction to the Theoryand Application of Differential Equations with DeviatingArguments [Book], - New York: Academic Press, (1973)
  15. Reddy Y.N., Numerical Treatment of Singularly Perturbedtwo point boundary value problems, Ph.D. Thesis, IIT,Kanpur, India, (1986)
  16. Reddy Y.N. and Pramod Chakravarthy P., Anexponentially fitted finite difference method for singularperturbation problems, Appl. Math. Comput. vol. 154, 83101, (2004)
  17. Kadalbajoo M.K. and Vikas Gupta, A brief survey onnumerical methods for solving singularly perturbedproblems, Applied Mathematics and Computation, 217,3641-3716, (2010)
  18. Rashidinia, J, Ghasemi, M, and Mahmoodi, Z, Splineapproach to the solution of a singularly-perturbedboundaryvalue problems, Appl. Math. Comput, 189, 7278, (2007)
  19. Reddy Y.N. and Anantha Reddy K., Numerical integrationmethod for general singularly perturbed two pointboundary value problems, Applied Mathematics andComputation, 133, 351-373, (2002)
  20. Reinhardt H.J., Singular perturbations of differencemethod for linear ordinary differential equations, [Journal]// Applicable Anal., 10, 53-70, (1980)