Abstract
An initial-value method is given for second-order singularly perturbed boundary-value problems with a boundary layer at one endpoint. The idea is to replace the original two-point boundary value problem by two suitable initial-value problems. The method is very easy to use and to implement. Nontrivial text problems are used to show the feasibility of the given method, its versatility, and its performance in solving linear and nonlinear singularly perturbed problems.
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References
Ascher, U., andWeiss, R.,Collocation for Singular Perturbation Problems, I: First-Order Systems with Constant Coefficients, SIAM Journal on Numerical Analysis, Vol. 20, pp. 537–557, 1983.
Ascher, U., andWeiss, R.,Collocation for Singular Perturbation Problems, II: Linear First-Order Systems without Turning Points, Mathematics of Computation, Vol. 4, pp. 157–187, 1984.
Ascher, U., andWeiss, R.,Collocation for Singular Perturbation Problems, III: Nonlinear Problems without Turning Points, SIAM Journal on Scientific and Statistical Computing, Vol. 5, pp. 811–829, 1984.
Flaherty, J. E., andMathon, W.,Collocation with Polynomial and Tension Splines for Singularly Perturbed Boundary-Value Problems, SIAM Journal on Scientific and Statistical Computing, Vol. 1, pp. 260–289, 1980.
Kreiss, B., andKreiss, H. O.,Numerical Methods for Singular Perturbation Problems, SIAM Journal of Numerical Analysis, Vol. 18, pp. 262–276, 1981.
Roos, H. G.,A Second-Order Monotone Upwind Scheme, Computing, Vol. 36, pp. 57–67, 1986.
De Groen, P. P. N., andHemker, P. W.,Error Bounds for Exponentially Fitted Galerkin Methods to Stiff Two-Points Boundary-Value Problems, Numerical Analysis of Singular Perturbation Problems, Edited by P. W. Hemker and J. J. H. Miller, Academic Press, New York, New York, 1979.
Hemker, P. W.,A Numerical Study of Stiff Two-Point Boundary-Value Problems, Mathematisch Centrum, Amsterdam, Holland, 1977.
Stynes, M., andO'Riordan E.,A Uniformly Accurate Finite-Element Method for a Singular Perturbation Problem in Conservative Form, SIAM Journal on Numerical Analysis, Vol. 23, pp. 369–375, 1986.
Szymczak, W. G., andBabuska, I.,Adaptivity and Error Estimation for the Finite-Element Method Applied to Convection-Diffusion Problems, SIAM Journal on Numerical Analysis, Vol. 21, pp. 910–954, 1984.
Maier, M. R.,An Adaptive Shooting Method for Singularly Perturbed Boundary-Value Problems, SIAM Journal on Scientific and Statistical Computing, Vol. 7, pp. 418–440, 1986.
Kadalbajoo, M. K., andReddy, Y. N.,Numerical Solution of Singular Perturbation Problems by a Terminal Boundary-Value Technique, Journal of Optimization Theory and Applications, Vol. 52, pp. 243–254, 1987.
Roberts, S. M.,A Boundary-Value Technique for Singular Perturbation Problems, Journal of Mathematical Analysis and Applications, Vol. 87, pp. 489–508, 1982.
Roberts, S. M.,The Analytical and Approximate Solution of εy" = yy', Journal of Mathematical Analysis and Applications, Vol. 97, pp. 245–265, 1983.
Roberts, S. M.,Solution of εy" + yy' − y = 0 by a Nonasymptotic Method, Journal of Optimization Theory and Applications, Vol. 44, pp. 303–332, 1984.
Flaherty, J. E., andO'Malley, R. E., Jr.,Numerical Methods for Stiff Systems of Two-Point Boundary-Value Problems, SIAM Journal on Scientific and Statistical Computing, Vol. 5, pp. 865–886, 1984.
Kadalbajoo, M. K., andReddy, 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.
Kevorkian, J., andCole, J. D.,Perturbation Methods in Applied Mathematics, Springer-Verlag, New York, New York, 1981.
O'Malley, R. E., Jr.,Introduction to Singular Perturbations, Academic Press, New York, New York, 1974.
Aiken, R. C., Editor,Stiff Computation, Oxford University Press, Oxford, England, 1985.
Gear, C. W., andShampine, L. F.,A User's View of Solving Stiff Ordinary Differential Equations, SIAM Review, Vol. 21, pp. 1–17, 1979.
Shampine, L. F., andWatts, H. A.,Practical Solution of Ordinary Differential Equations by Runge-Kutta Methods, Sandia National Laboratories, Albuquerque, New Mexico, Report No. SAND-76-0585, 1976.
Petzold, L.,Automatic Selection of Methods for Solving Stiff and Nonstiff Systems of Ordinary Differential Equations, SIAM Journal on Scientific and Statistical Computing, Vol. 4, pp. 136–148, 1983.
Enright, W. H., Jackson, K. R., Norsett, S. P., andThomsen, P. G.,Interpolants for Runge-Kutta Formulas, ACM Transactions on Mathematical Software, Vol. 12, pp. 193–218, 1986.
Shampine, L. F.,Some Practical Runge-Kutta Formulas, Mathematics of Computation, Vol. 46, pp. 135–150, 1986.
Gasparo, M. G., andMacconi, M.,A New Initial-Value Method for Singularly Perturbed Boundary-Value Problems, Istituto Matematico Ulisse Dini, Firenze, Italy, Report No. 5, 1987.
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Communicated by I. Galligani
This work was supported in part by the Consiglio Nazionale delle Ricerche, Contract No. 86.02108.01, and in part by the Ministero della Pubblica Istruzione.
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Gasparo, M.G., Macconi, M. New initial-value method for singularly perturbed boundary-value problems. J Optim Theory Appl 63, 213–224 (1989). https://doi.org/10.1007/BF00939575
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DOI: https://doi.org/10.1007/BF00939575