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1984 | Buch

Introduction Kapitel lesen Erstes Kapitel lesen

Nonlinear Singular Perturbation Phenomena

Theory and Applications

verfasst von: K. W. Chang, F. A. Howes

Verlag: Springer New York

Buchreihe : Applied Mathematical Sciences

Enthalten in: Professional Book Archive

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Über dieses Buch

Our purpose in writing this monograph is twofold. On the one hand, we want to collect in one place many of the recent results on the exist­ ence and asymptotic behavior of solutions of certain classes of singularly perturbed nonlinear boundary value problems. On the other, we hope to raise along the way a number of questions for further study, mostly ques­ tions we ourselves are unable to answer. The presentation involves a study of both scalar and vector boundary value problems for ordinary dif­ ferential equations, by means of the consistent use of differential in­ equality techniques. Our results for scalar boundary value problems obeying some type of maximum principle are fairly complete; however, we have been unable to treat, under any circumstances, problems involving "resonant" behavior. The linear theory for such problems is incredibly complicated already, and at the present time there appears to be little hope for any kind of general nonlinear theory. Our results for vector boundary value problems, even those admitting higher dimensional maximum principles in the form of invariant regions, are also far from complete. We offer them with some trepidation, in the hope that they may stimulate further work in this challenging and important area of differential equa­ tions. The research summarized here has been made possible by the support over the years of the National Science Foundation and the National Science and Engineering Research Council.

Inhaltsverzeichnis

Frontmatter
Chapter I. Introduction
Abstract
We are mainly interested in quasilinear and nonlinear boundary value problems, to which some well-known methods, such as the methods of matched asymptotic expansions and two-variable expansions are not immediately applicable. For example, let us consider the following boundary value problem(cf. O’Malley [75], Chapter 5)
$$ \varepsilon y'' = y'^2 , 0 < t < 1, $$
(A)
$$ y(0,\varepsilon ) = 1, y(1,\varepsilon ) = 0. $$
(B)
K. W. Chang, F. A. Howes
Chapter II. A’priori Bounds and Existence Theorems
Abstract
Before discussing in detail the various classes of singularly perturbed boundary value problems, let us give an outline of the principal method of proof that we will use throughout. This method employs the theory of differential inequalities which was developed by M. Nagumo [66] and later refined by Jackson [49]. It enables one to prove the existence of a solution, and at the same time, to estimate this solution in terms of the solutions of appropriate inequalities. Such an approach has been found to be very useful in a number of different applications (see, for example, [5] and [83]). It will be seen that for the general classes of problems which we will study in later chapters, this inequality technique leads elegantly (and easily) to some fairly general results about existence of solutions and their asymptotic behavior. Many results which have been obtained over the years by a variety of methods can now be obtained by this method, which we hope will also very clearly reveal the fundamental asymptotic processes at work.
K. W. Chang, F. A. Howes
Chapter III. Semilinear Singular Perturbation Problems
Abstract
We consider first the semilinear Dirichlet problem
$$ \begin{gathered} \varepsilon y'' = h(t,y), a < t < b, \hfill \\ y(a,\varepsilon ) = A, y(b,\varepsilon ) = B, \hfill \\ \end{gathered} $$
(DP1)
where e is a small positive parameter and prime denotes differentiation with respect to t. Some natural questions to ask regarding this problem are: Does the problem have a solution for all small values of ε? Once the existence of a solution has been established, how does the solution behave as ε + 0+?
K. W. Chang, F. A. Howes
Chapter IV. Quasilinear Singular Perturbation Problems
Abstract
We consider now the singularly perturbed quasilinear Dirichlet problem
$$\begin{gathered} \varepsilon y''{\text{ }} = {\text{ f}}({\text{t}},y)y'{\text{ }} + {\text{ g(t}},y{\text{) }} \equiv {\text{ F}}({\text{t}},y,y'),{\text{ a }} < {\text{ t }} < {\text{ b}}, \hfill \\ y({\text{a}},\varepsilon ){\text{ }} = {\text{A}},{\text{ }}y({\text{b}},\varepsilon ){\text{ }} = {\text{B}}. \hfill \\ \end{gathered} $$
(DP2)
K. W. Chang, F. A. Howes
Chapter V. Quadratic Singular Perturbation Problems
Abstract
In this chapter we investigate the asymptotic behavior of solutions of boundary value problems for the differential equation
$$\varepsilon y''{\text{ = p}}\left( {{\text{t}},{\text{y}}} \right){{\text{y'}}^{\text{2}}}{\text{ + g}}\left( {{\text{t}},{\text{y}}} \right),{\text{a}} < {\text{t}} < {\text{b}}$$
(DE)
The novelty here is the presence of the quadratic term in y’. The more general differential equation
$$\varepsilon y''{\text{ = p}}\left( {{\text{t}},{\text{y}}} \right){{\text{y'}}^{\text{2}}}{\text{ + f}}\left( {{\text{t}},{\text{y}}} \right){\text{y' + g}}\left( {{\text{t}},{\text{y}}} \right)$$
will not be studied, since it can be reduced to the form (DE) in some cases by the familiar device of completing the square. Our decision to study the simpler equation (DE) rather than the more general equation stems from a desire to present representative results for this “quadratic” class of problems without having to deal with extra complexities in notation.
K. W. Chang, F. A. Howes
Chapter VI. Superquadratic Singular Perturbation Problems
Abstract
In previous chapters we have presented fairly comprehensive results for boundary value problems involving the differential equation
$$\varepsilon y''{\text{ = f}}\left( {t,y,y'} \right),{\text{a}} < {\text{t}} < {\text{b}}$$
subject to the fundamental restriction:
$$f(t.y,z) = 0({\left| z \right|^2})as\left| z \right| \to \infty .$$
K. W. Chang, F. A. Howes
Chapter VII. Singularly Perturbed Systems
Abstract
In this chapter we turn our attention to some vector boundary value problems which may be regarded as vector analogs of the scalar problems. However, as the reader will see, our results for vector problems are very incomplete, especially in comparison with the scalar theory. The study of singularly perturbed vector second-order equations is in its infancy, and we hope that this chapter will serve to draw attention to some of the associated difficulties. Indeed, many fundamental questions concerning vector equations or systems have yet to be raised, let alone answered.
K. W. Chang, F. A. Howes
Chapter VII. Examples and Applications
Abstract
Consider the Dirichlet problem \(\eqalign{ & \varepsilon y''{\text{ }} = {\text{ }}{(y - {\text{u}}({\text{t}}))^{2q + 1}},{\text{ }} - 1 < {\text{t}} < 1, \cr & y( - {\text{l}},\varepsilon ){\text{ }} = {\text{A}},{\text{ }}y({\text{l}},\varepsilon ){\text{ B}}, \cr} \) where q is a nonnegative integer. If the function u(t), defined for \( - 1{\text{ }} < {\text{ t }} < {\text{ }}1\),is twice continuously differentiable or has a bounded second derivative, then by Theorem 3.1, for sufficiently small \(\varepsilon > 0\),the Dirichlet problem has a solution \(y{\text{ }} = {\text{ }}y({\text{t}},\varepsilon )\)which satisfies
$$\mathop {\lim }\limits_{\varepsilon \to {0^ + }} {\text{ }}y({\text{t}},\varepsilon ){\text{ }} = {\text{ u}}({\text{t}}){\text{ in }}[ - 1 + \delta ,1 - \delta ],$$
(8.1)
where \(0{\text{ }} < {\text{ }}\delta {\text{ }} < {\text{ }}1\) Moreover, the behavior of the solution \(y({\text{t}},\varepsilon )\) in the boundary layers at t = -1 and/or t = 1 (if u(-1)≠A and/or u(1) ≠ B) can be described by means of the layer functions given in the conclusion of Theorem 3.1.
K. W. Chang, F. A. Howes
Backmatter
Metadaten
Titel
Nonlinear Singular Perturbation Phenomena
verfasst von
K. W. Chang
F. A. Howes
Copyright-Jahr
1984
Verlag
Springer New York
Electronic ISBN
978-1-4612-1114-3
Print ISBN
978-0-387-96066-1
DOI
https://doi.org/10.1007/978-1-4612-1114-3