Abstract
An initial-value technique is presented for solving singularly perturbed two-point boundary-value problems for linear and semilinear second-order ordinary differential equations arising in chemical reactor theory. In this technique, the required approximate solution is obtained by combining solutions of two terminal-value problems and one initial-value problem which are obtained from the original boundary-value problem through asymptotic expansion procedures. Error estimates for approximate solutions are obtained. Numerical examples are presented to illustrate the present technique.
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O'malley, R. E., Introduction to Singular Perturbations, Academic Press, New York, New York, 1974.
Baxley, J. V., On Singular Perturbation of Nonlinear Two-Point Boundary-Value Problems, Czechoslovak Mathematical Journal, Vol. 27, pp. 363–377, 1977.
Cohen, D. S., Multiple Solutions of Singular Perturbation Problems, SIAM Journal on Mathematical Analysis, Vol. 3, pp. 72–82, 1972.
Cohen, D. S., Multiple Stable solutions of Nonlinear Boundary-Value Problems Arising in Chemical Reactor Theory, SIAM Journal on Mathematical Analysis, Vol. 20, pp. 1–13, 1973.
Kadalbajoo, M. K., and Reddy, Y. N., Asymptotic and Numerical Analysis of Singular Perturbation Problems: A Survey, Applied Mathematics and Computation, Vol. 30, pp. 223–259, 1989.
Gasparo, M. G., and Macconi, M., Initial-Value Methods for Second-Order Singularly Perturbed Boundary-Value Problems, Journal of Optimization Theory and Applications, Vol. 66, pp. 197–210, 1990.
Kadalbajoo, M. K., and Reddy, Y. N., Initial-Value Technique for a Class of Nonlinear Singular Perturbation Problems, Journal of Optimization Theory and Applications, Vol. 53, pp. 395–406, 1987.
Gasparo, M. G., and Macconi, M., New Initial-Value Method for Singularly Perturbed Boundary-Value Problems, Journal of Optimization Theory and Applications, Vol. 63, pp. 213–224, 1989.
Doolan, E. P., Miller, J. J. H., and Schilders, W. H. A., Uniform Numerical Methods for Problems with Initial and Boundary Layers, Boole Prees, Dublin, Ireland, 1980.
Miller, J. J. H., Optimal Uniform Difference Schemes for Linear Initial-Value Problems, Computers and Mathematics with Applications, Vol. 12B, pp. 1209–1215, 1986.
Farrell, P. A., Uniform and Optimal Schemes for Stiff Initial-Value Problems, Computers and Mathematics with Applications, Vol. 13, pp. 925–936, 1987.
Chang, K. W., and Howes, F. A., Nonlinear Singular Perturbation Phenomena: Theory and Applications, Springer Verlag, New York, New York, 1984.
O'Malley, R. E., Singular Perturbation Methods for Ordinary Differential Equations, Springer Verlag, New York, New York, 1991.
Hemker, P. W., A Numerical Study of Stiff Two-Point Boundary-Value Problems, Mathematics Centre, Amsterdam, Holland, 1977.
Axelsson, O., and Gustafsson, I., A Modified Upwind Scheme for Convective-Transport Equations and the Use of a Conjugate-Gradient Method for the Solution of Nonsymmetric System of Equations, Journal of the Institute of Mathematics and its Applications, Vol. 23, pp. 321–337, 1979.
Dorr, F. W., Parter, S. V., and Shampine, L. F., Application of the Maximum Principle to Singular Perturbation Problems, SIAM Review, Vol. 15, pp. 43–88, 1973.
Jayakumar, J., and Ramanujam, N., Numerical Method for Singular Perturbation Problems Arising in Chemical Reactor Theory, Computers and Mathematics with Applications, Vol. 27, pp. 83–99, 1994.
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Natesan, S., Ramanujam, N. Initial-Value Technique for Singularly Perturbed Boundary-Value Problems for Second-Order Ordinary Differential Equations Arising in Chemical Reactor Theory. Journal of Optimization Theory and Applications 97, 455–470 (1998). https://doi.org/10.1023/A:1022639003366
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DOI: https://doi.org/10.1023/A:1022639003366