Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Singularly perturbed boundary value problems with angular limiting solutions

Author: F. A. Howes
Journal: Trans. Amer. Math. Soc. 241 (1978), 155-182
MSC: Primary 34D15
MathSciNet review: 0499510
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A basic result of Haber and Levinson which describes the behavior of solutions of $ \varepsilon y'' = f(t,y,y')$,$ a < t < b$, $ y(a,\varepsilon )$, $ y(b,\varepsilon )$, 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 $ \varepsilon \, > \,0$, 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.

References [Enhancements On Off] (What's this?)

  • [1] Yu. P. Boglaev, The two-point problem for a class of ordinary differential equations with a small parameter coefficient of the derivative, USSR Computational Math. and Math. Phys. 10 (1970), 191-204.
  • [2] N. I. Briš, On boundary value problems for the equation $ \varepsilon y'' = f(x,y,y')$ for small $ \varepsilon $, Dokl. Akad. Nauk SSSR 95 (1954), 429-432.
  • [3] F. W. Dorr, S. V. Parter, and L. F. Shampine, Applications of the maximum principle to singular perturbation problems, SIAM Rev. 15 (1973), 43–88. MR 0320456,
  • [4] S. Haber and N. Levinson, A boundary value problem for a singularly perturbed differential equation, Proc. Amer. Math. Soc. 6 (1955), 866–872. MR 0074634,
  • [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), no. 4, 591–607. Summer Research Conference on Singular Perturbations: Theory and applications (Northern Arizona Univ., Flagstaff, Ariz., 1975). MR 0419917,
  • [7] F. A. Howes, Singularly perturbed nonlinear boundary value problems with turning points. II, SIAM J. Math. Anal. 9 (1978), no. 2, 250–271. MR 0477346,
  • [8] M. Nagumo, Über die Differentialgleichung $ y'' = f(x,y,y')$, Proc. Phys. Math. Soc. Japan 19 (1937), 861-866.
  • [9] R. E. O'Malley, Jr., On singular perturbation problems with interior nonuniformities, J. Math. Mech. 19 (1970), 1103-1112.
  • [10] Klaus Schmitt, Boundary value problems for non-linear second order differential equations, Monatsh. Math. 72 (1968), 347–354. MR 0230969,
  • [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), 509–512 (Russian). MR 0108625
  • [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), 13-84.
  • [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), 823-863.
  • [14] M. I. Višik and L. A. Lyusternik, Initial jump for nonlinear differential equations containing a small parameter, Soviet Math. Dokl. 1 (1960), 749–752. MR 0120427
  • [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.)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 34D15

Retrieve articles in all journals with MSC: 34D15

Additional Information

Article copyright: © Copyright 1978 American Mathematical Society

American Mathematical Society