Barbu / Morosanu Singularly Perturbed Boundary-Value Problems

1. Auflage 2007
ISBN: 978-3-7643-8331-2
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 156, 231 Seiten

Reihe: International Series of Numerical Mathematics

This book offers a detailed asymptotic analysis of some important classes of singularly perturbed boundary value problems which are mathematical models for phenomena in biology, chemistry, and engineering. The authors are particularly interested in nonlinear problems, which have gone little-examined so far in literature dedicated to singular perturbations. The treatment presented here combines successful results from functional analysis, singular perturbation theory, partial differential equations, and evolution equations.

Autoren/Hrsg.

Barbu, Luminita

Morosanu, Gheorghe

Inhaltsverzeichnis

1;Contents;7
2;Preface;11
3;Part I Preliminaries;14
3.1;Regular and Singular Perturbations;15
3.2;Evolution Equations in Hilbert Spaces;28
3.2.1;Function spaces and distributions;28
3.2.2;Monotone operators;32
3.2.3;Evolution equations;38
4;Part II Singularly Perturbed Hyperbolic Problems;46
4.1;Presentation of the Problems;47
4.2;Hyperbolic Systems with Algebraic Boundary Conditions;52
4.2.1;4.1 A zeroth order asymptotic expansion;53
4.2.2;4.2 Existence, uniqueness and regularity of the solutions of problems Pe and P0;55
4.2.3;4.3 Estimates for the remainder components;68
4.3;Hyperbolic Systems with Dynamic Boundary Conditions;74
4.3.1;5.1 A .rst order asymptotic expansion for the solution of problem;75
4.3.2;5.2 A zeroth order asymptotic expansion for the solution of problem;92
4.3.3;5.3 A zeroth order asymptotic expansion for the solution of problem;105
4.3.4;5.4 A zeroth order asymptotic expansion for the solution of problem;114
5;Part III Singularly Perturbed Coupled Boundary Value Problems;120
5.1;The Stationary Case;125
5.1.1;7.1 Asymptotic analysis of problem;127
5.1.2;7.2 Asymptotic analysis of problem;138
5.1.3;7.3 Asymptotic analysis of problem;145
5.2;The Evolutionary Case;151
5.2.1;8.1 A first order asymptotic expansion for the solution of problem;153
5.2.2;8.2 A first order asymptotic expansion for the solution of problem;169
5.2.3;8.3 A zeroth order asymptotic expansion for the solution of problem;181
6;Part IV Elliptic and Hyperbolic Regularizations of Parabolic Problems;186
6.1;Presentation of the Problems;187
6.2;The Linear Case;190
6.2.1;10.1 Asymptotic analysis of problem;192
6.2.2;10.2 Asymptotic analysis of problem;200
6.2.3;10.3 Asymptotic analysis of problem;204
6.2.4;10.4 An Example;209
6.3;The Nonlinear Case;214
6.3.1;11.1 Asymptotic analysis of problem;216
6.3.2;11.2 Asymptotic analysis of problem;221
6.3.3;11.3 Asymptotic analysis of problem;226
7;Bibliography;232

Leseproben

Chapter 5 Hyperbolic Systems with Dynamic Boundary Conditions(p. 61-62)

In this chapter we investigate the first four problems presented in Chapter 3. All these problems include dynamic boundary conditions (which involve the derivatives of v(0, t), v(1, t), (BC.2) also include integrals of these functions). Note that all the four problems are singularly perturbed of the boundary layer type with respect to the sup norm.

The chapter consists of four sections, each of them addressing one of the four problems.

As a first step in our treatment, we construct a formal asymptotic expansion for each of the four problems, by employing the method presented in Chapter 1. For three of the four problems we construct zeroth order asymptotic expansions. In the case of problem (LS), (IC), (BC.1), we construct a first order expansion of the solution in order to offer an example of a higher order asymptotic expansion. It should be pointed out that first or even higher order asymptotic expansions can be constructed for all the problems considered in this chapter but additional assumptions on the data should be required and much more laborious computations are needed.

Once a formal asymptotic expansion is determined, we will continue with its validation. More precisely, as a second step in our analysis, we will formulate and prove results concerning the existence, uniqueness, and higher regularity of the terms which occur in each of the previously determined asymptotic expansions. As in the previous chapter, we need higher regularity to show that our asymptotic expansions are well defined and to derive estimates for the remainder components. Our investigations here are mainly based on classic methods in the theory of evolution equations in Hilbert spaces associated with monotone operators as well as on linear semigroup theory. It should be pointed out that each of the four problems requires a different framework and separate analysis. All the operators associated with the corresponding reduced (unperturbed) problems are subdifferentials, except for the reduced problem in Subsection 5.4.2. When the PDE system under investigation is nonlinear (see Subsections 5.2.2 and 5.3.2), the treatment becomes much more complex.