**AUTHORS:**Rakesh Ranjan, H. S. Prasad

**Download as PDF**

**KEYWORDS:**
Singular perturbation problems, Bounary value problems, Stability and convergence, Numerical Integration

**REFERENCES:**

[1] E. Angel, R. Bellman, Dynamic Programming and Partial Differential Equations, Academic Press– New York 1972.

[2] C. M. Bender, S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill– New York 1978.

[3] P. P. Chakravarthy, S. D. Kumar, R. N. Rao, An exponentially fitted finite difference scheme for a class of singularly perturbed delay differential equations with large delays, Ain Shams Engineering Journal, xxx, 2015, pp. xxx-xxx.

[4] L. E. El’sgol’ts, S. B. Norkin, Introduction to the Theory and Application of Differential Equations with Deviating Arguments, Academic Press– New York 1973.

[5] R. R. Gold, Magnetohydrodynamic pipe flow, Part I. Journal of Fluid Mechanics, 13, 1962, pp. 505-512.

[6] P. W. Hemker, J. J. H. Miller, Numerical Analysis of Singular Perturbation Problems, Academic Press– New York 1979.

[7] M. K. Kadalbajoo, Y. N. Reddy, Asymptotic, numerical analysis of singular perturbation problems: a survey, Applied Mathematics and Computation, 30, 1989, pp. 223-259.

[8] M. K. Kadalbajoo, Y. N. Reddy, An Approximate Method for Solving a Class of Singular Perturbation Problems, Journal of Mathematical for smaller values of N the tendency of maximum absolute error is becoming uniform is fast with respect to larger values of N. Analysis and Applications, 133, 1988, pp. 306- 323.

[9] M. K. Kadalbajoo, Y.-N. Reddy, Numerical Treatment of Singularly Perturbed Two Point Boundary Value Problems, Applied Mathematics and Computation, 21, 1987, pp. 93-110.

[10] M. K. Kadalbajoo, V. K. Aggarwal, Fitted mesh B-spline collocation method for solving self-adjoint singularly perturbed boundary value problems, Applied Mathematics and Computation, 161, 2005, pp. 973987.

[11] M. K. Kadalbajoo, K. K. Sharma, Parameteruniform fitted mesh method for singularly perturbed delay differential equations with layer behavior, Electronic Transactions on Numerical Analysis, 23, 2006, pp. 180-201.

[12] M. K. Kadalbajoo, P. Arora, B-spline collocation method for the singular-perturbation problem using artificial viscosity, Computers and Mathematics with Applications, 57, 2009, pp. 650-663.

[13] M. K. Kadalbajoo, V. Gupta, A brief survey on numerical methods for solving singularly perturbed problems, Applied Mathematics and Computation, 217, 2010, pp. 3641-3716.

[14] J. Kevorkian, J. D. Cole, Perturbation Methods in Applied Mathematics, Springer– New York 1981.

[15] M. Kumar, A. K. Singh, Singular Perturbation Problems in Nonlinear Elliptic Partial Differential Equations: A Survey, International Journal of Nonlinear Science, 17, 3, 2014, pp. 195-214.

[16] J. J. H. Miller, E. O’Riordan, G. I. Shishkin, Fitted Numerical Methods for Singular Perturbation Problems, Error Estimates in the Maximum Norm for Linear Problems in One and Two Dimensions, World Scientific– Singapore 1996.

[17] J. J. H. Miller, Singular Perturbation Problems in Chemical Physics Analytic and Computational Methods, John Wiley– New York 1997.

[18] A. H. Nayfeh, Perturbation Methods, Wiley– New York 1979.

[19] R. E. O’Malley, Introduction to Singular Perturbations, Academic Press– New York 1974.

[20] R. E. O’Malley, Singular Perturbation Methods for Ordinary Differential Equations, Springer– New York 1991.

[21] H. G. Roos, M. Stynes, L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations, Springer– Berlin 1996.

[22] H. J. Reinhardt, Singular Perturbations of difference methods for linear ordinary differential equations, Applicable Analysis, 10, 1980, pp. 53-70.

[23] V. Shanthi, N. Ramanujam, S. Natesan, Fitted mesh method for singularly perturbed reactionconvection-diffusion problems with boundary and interior layers, J. Appl. Math. and Computing, 22, 1, 2006, pp. 49-65.

[24] V. Subburayan, N. Ramanujam, Asymptotic Initial Value Technique for singularly perturbed convectiondiffusion delay problems with boundary and weak interior layers, Applied Mathematics Letters, 25, 2012, pp. 2272-2278