Abstract
In this chapter a series of simple examples are considered, some model and some physical, in order to demonstrate the application of various techniques concerning limit process expansions. In general we expect analytic dependence of the exact solution on the small parameter ε, but one of the main tasks in the various problems is to discover the nature of this dependence by working with suitable approximate differential equations. Another problem is to systematize as much as possible the procedures for discovering these expansions.
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References
J. Grassman and B. J. Matkowsky, A variational approach to singularly perturbed boundary value problems for ordinary and partial differential equations with turning points, S.I.A.M. Journal on Applied Mathematics, 32, No. 3, May 1977, pp. 588–597.
H. C. Corben and P. Stehle, Classical Mechanics, John Wiley and Sons, Inc., New York, 1960.
A. Erdelyi, Singular perturbation, Bull Amer. Math. Soc. 68 (1962), 420–424.
E. A. Coddington and N. Levinson, A boundary value problem for a nonlinear differential equation with a small parameter, Proc. Amer. Math. Soc. 3 (1952), 73–81.
W. Wasow, Singular perturbations of boundary value problems for nonlinear differential equations of the second order. Comm. Pure and Applied Math. 9 (1956), 93–113.
A. Erdelyi, On a nonlinear boundary value problem involving a small parameter, J. Aust. Math. Soc. II, Part 4 (1962), 425–439.
J. J. Shepherd, Asymptotic solution of a nonlinear singular perturbation problem. S.I.A.M. J. on Appl. Math. 35 (1978) 176–186.
A. A. Dorodnitsyn, Asymptotic Solution of the van der Pol Equations, Prik. Mat. i Mekh, I1 (1947), 313–328 (in Russian).
M. Urabe, Numerical Study of Periodic Solutions of the van der Pol Equations, in Proc. Int. Symp. on Non-Linear Differential Equations and Non-Linear Mechanics, 184–195. Edited by J. P. LaSalle and S. Lefschetz. New York. Academic Press, 1963.
H. C. Corben and P. Stehle, Classical Mechanics, John Wiley and Sons, Inc., New York, 1960.
P. A. Lagerstrom and J. Kevorkian, Matched conic approximation to the two fixed force-center problem, The Astronomical Journal 68, No. 2, March 1963, pp. 84–92.
P. A. Lagerstrom and J. Kevorkian, Earth-to-moon trajectories in the restricted three-body problem, Journal de Mécanique 2, No. 2, June 1963, pp. 189–218.
P. A. Lagerstrom and J. Kevorkian, Nonplanar earth-to-moon trajectories, AIAA Journal4, No. 1, January 1966, pp. 149–152.
J. V. Breakwell and L. M. Perko, Matched asymptotic expansions, patched conics, and the computation of interplanetary trajectories, Methods in Astrodynamics and Celestial Mechanics 17, eds. R. L. Duncombe and V. G. Szebehely, pp. 159–182, 1966.
P. A. Lagerstrom, Méthodes asymptotiques pour l’étude des équations de Navier—Stokes, Lecture notes, Institut Henri Poincaré, Paris, 1961. Translated by T. J. Tyson, California Institute of Technology, Pasadena, California, 1965.
D. S. Cohen, A. Fokas, and P. A. Lagerstrom, Proof of some asymptotic results for a model equation for low Reynolds number flow, S.LA.M. J. on Appl. Math. 35 (1978), 187–207.
A. D. MacGillivray, On a model equation of Lagerstrom, S.I.A.M. J. on Appl. Math. 34 (1978), 804 812.
W. B. Bush, On the Lagerstrom mathematical model for viscous flow at low Reynolds number, S.I.A.M. J. on Appl. Math. 20 (1971), 279–287.
T. von Karman, and M. Biot, Mathematical Methods in Engineering, McGraw-Hill Book Co., New York, 1940.
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Kevorkian, J., Cole, J.D. (1981). Limit Process Expansions Applied to Ordinary Differential Equations. In: Perturbation Methods in Applied Mathematics. Applied Mathematical Sciences, vol 34. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-4213-8_2
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DOI: https://doi.org/10.1007/978-1-4757-4213-8_2
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