Singularly perturbed boundary value problems with angular limiting solutions
Author:
F. A. Howes
Journal:
Trans. Amer. Math. Soc. 241 (1978), 155182
MSC:
Primary 34D15
MathSciNet review:
0499510
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Abstract: A basic result of Haber and Levinson which describes the behavior of solutions of ,, , , prescribed, in the presence of a reduced solution with corners is modified to treat related classes of problems. Under various stability assumptions, solutions are shown to remain, for small , in a o(l)neighborhood of an angular reduced solution with the possible exception of narrow layers near the boundaries in some cases. Each aspect of the theory developed here is illustrated by several examples.
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P. Habets and M. Laloy, Étude de problèmes aux limites par la methode des sur et sous solutions, Lecture Notes, Catholic Univ. of Louvain, 1974.
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of simultaneous differential equations involving a small parameter in the
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W. R. Wasow, Asymptotic expansions for ordinary differential equations, ``MIR,'' Moscow, 1968. (Russian edition of the book originally published by Wiley, New York, 1965.)
 [1]
 Yu. P. Boglaev, The twopoint problem for a class of ordinary differential equations with a small parameter coefficient of the derivative, USSR Computational Math. and Math. Phys. 10 (1970), 191204.
 [2]
 N. I. Briš, On boundary value problems for the equation for small , Dokl. Akad. Nauk SSSR 95 (1954), 429432.
 [3]
 F. W. Dorr, S. V. Parter and L. F. Shampine, Applications of the maximum principle to singular perturbation problems, SIAM Review 15 (1973), 4388. MR 0320456 (47:8995)
 [4]
 S. Haber and N. Levinson, A boundary value problem for a singularly perturbed differential equation, Proc. Amer. Math. Soc. 6 (1955), 866872. MR 0074634 (17:618e)
 [5]
 P. Habets and M. Laloy, Étude de problèmes aux limites par la methode des sur et sous solutions, Lecture Notes, Catholic Univ. of Louvain, 1974.
 [6]
 F. A. Howes, A class of boundary value problems whose solutions possess angular limiting behavior, Rocky Mountain J. Math., 6 (1976), 591607. MR 0419917 (54:7934)
 [7]
 , Singularly perturbed nonlinear boundary value problems with turning points. II, SIAM J. Math. Anal. (to appear). MR 0477346 (57:16877b)
 [8]
 M. Nagumo, Über die Differentialgleichung , Proc. Phys. Math. Soc. Japan 19 (1937), 861866.
 [9]
 R. E. O'Malley, Jr., On singular perturbation problems with interior nonuniformities, J. Math. Mech. 19 (1970), 11031112.
 [10]
 K. Schmitt, Boundary value problems for nonlinear second order differential equations, Monaish. Math. 72 (1968), 347354. MR 0230969 (37:6526)
 [11]
 A. B. Vasil'eva, Uniform approximation to the solution of a set of simultaneous differential equations involving a small parameter in the derivative and its application to boundary value problems, Dokl. Akad. Nauk SSSR 124 (1959), 509512. MR 0108625 (21:7341)
 [12]
 , Asymptotic behavior of solutions to certain problems involving nonlinear differential equations containing a small parameter multiplying the highest derivatives, Russian Math. Surveys 18 (1963), 1384.
 [13]
 , Asymptotic methods in the theory of ordinary differential equations containing small parameters in front of the higher derivatives, USSR Computational Math. and Math. Phys. 3 (1963), 823863.
 [14]
 M. I. Vishik and L. A. Lyusternik, Initial jump for nonlinear differential equations containing a small parameter, Soviet Math. Dokl. 1 (1960), 749752. MR 0120427 (22:11181)
 [15]
 W. R. Wasow, Asymptotic expansions for ordinary differential equations, ``MIR,'' Moscow, 1968. (Russian edition of the book originally published by Wiley, New York, 1965.)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197804995103
PII:
S 00029947(1978)04995103
Article copyright:
© Copyright 1978
American Mathematical Society
