Introduction to the General Theory of Singular Perturbations

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Introduction to the General Theory of Singular Perturbations

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     1992 by the American Mathematical Society
     ISBN 0-8218-4569-1
     QA871.L813 1992 515'.35-dc20
     LCCN 92-26927

Contents

Prefaces
     Preface to the English Edition
     Preface
     Author's Preface

CHAPTER 1  Introduction. General Survey

     §1. On perturbations

     §2. The basic idea of classical perturbation theory

     §3. Singular perturbations

     §4. Basic concepts. Terminology

     §5. The Schlesinger-Birkhoff theorem

     §6. The Schlesinger-Birkhoff theorem and asymptotic integration

     §7. Further development of the theory of singular perturbations

     §8. Comparison of two types of asymptotic expansions

     §9. Some notation and auxiliary concepts


Part I  Asymptotic Integration of Various Problems for Ordinary Differential Equations

     CHAPTER 2  The Method of Regularization of Singular Perturbations

          §1. The formalism of the regularization method

               1. Formulation of the problem

               2. Regularization of singularities

               3. Formal construction of a series for the solution

          §2. The space of resonance-free solutions

               1. The structure of the space

               2. Properties of the basic operator in the space of resonance-free solutions

          §3. The theory of resonance-free solutions

               1. The adjoint operator

               2. Normal solvability of the basic operator

               3. Uniqueness of the solution

          §4. Formal regularized series

               1. Determination of the coefficients of the series of perturbation theory

               2. Uniqueness and other properties of the regularized series.

          §5. Estimation of the remainder term of the asymptotic series for the fundamental matrix

               1. Formal construction of the fundamental matrix

               2. The asymptotic character of the serie

          §6. Estimation of the remainder term of the asymptotic series for the solution of the Cauchy problem

               1. Auxiliary notation and a lemma

               2. Estimation of the remainder term

          §7. Convergence of regularized series in the usual sense

               1. Systems with a diagonal matrix of coefficients

               2. Examples.

               3. Ordinary convergence of the asymptotic series

               4. Convergence in a finite-dimensional Hilbert space

               5. An example

          §8. The method of regularization in the case of null points of the spectrum

               1. Formulation of the problem

               2. The formalism of the regularization method

               3. Construction of the adjoint operator in the space of resonance-free solutions.

               4. Questions of solvability in the space of resonance-free solutions.

               5. A limit theorem

     CHAPTER 3  Asymptotic Integration of a Boundary Value Problem

          §1. Special features of boundary value problems

               1. Characteristic features of boundary value problems

               2. Formulation of the problem

               3. Stability of the boundary value problem

          §2. Construction of an algorithm for asymptotic integration of a boundary value problem for general systems

               1. The formalism of the method of regularization for a boundary value problem

               2. Solvability theorems in the space of resonance-free solutions

               3. Solvability of the iteration problems

               4. Formal asymptotic solution of the original problem

          §3. Construction of the Green function

               1. Reduction of the system to quasi-diagonal form

               2. Construction of two fundamental matrices of special form

               3. Construction of a fundamental matrix of a singularly perturbed system with special boundary conditions

               4. Construction of the matrix [

               5. Construction of the matrix Green function

               6. A remark on the construction of the Green function for a more general system.

          §4. Estimation of the remainder term

               1. The problem for the remainder term.

               2. An estimate theorem

     CHAPTER 4  Asymptotic Integration of Linear Integro-Differential Equations

          §1. Special features of the regularization of singularities in the presence of integrals of the desired solutions in the oscillatory case

               1. Formulation of the problem in the simplest case.

               2. Partial regularization of the problems

          §2. Complete regularization and asymptotic integration

               1. Regularization and the formalism of the method

               2. Solvability of iteration problems

               3. Estimation of the remainder term

          §3. The Cauchy problem for integro-differential systems

               1. Formulation of the problem and regularization of singularities

               2. Determination of the coefficients of the formal asymptotic series.

               3. Estimation of the remainder term.

               4. An example

          §4. Integro-differential systems of Fredholm type

               1. Auxiliary propositions

               2. Formulation of the problem and regularization of the operation of differentiation.

               3. Regularization of the integral term and of the problem for determining the elements of the asymptotic solution

               4. Solvability of the iteration problem

               5. Estimation of the remainder term

     CHAPTER 5  Some Problems with Rapidly Oscillating Coefficients

          § 1. Construction of the asymptotic series and conditions for the solvability of the iteration problems

               1. Formalism of the method

               2. The space of solutions

               3. The adjoint operator

               4. Construction of new recurrent problems

               5. Solvability theorems

          §2. Justification of asymptotic convergence

               1. Estimation of the remainder term.

               2. Remark

          §3. Solution of the problem of parametric amplification

               1. An example

               2. Solution of the auxiliary system

     CHAPTER 6  Problems with an Unstable Spectrum

          §1. The only point of the spectrum has a zero of arbitrary order

               1. On the problem in the simplest formulation

               2. Regularization of the problem

               3. Asymptotic integration

               4. Passage to the limit.

          §2. One of the two points of the spectrum has a zero of first order

               1. Special features of the problem.

               2. Choice of regularizing functions and regularization

               3. Special features of solving the iteration problems

               4. The main theorem

          §3. The inhomogeneous problem with a turning point

               1. Preliminary facts regarding the problem

               2. Formulation of the problem

               3. Regularization of the problem

               4. Special features of the asymptotic integration of problems with turning points.

               5. Solvability of the iteration problems

               6. Estimation of the remainder term

               7. Proof of Lemma 18

          §4. The structure of the fundamental matrix of solutions of singularly perturbed equations with a regular singular point

               1. The fundamental system of solution

               2. Obtaining formal solutions

               3. Asymptotic convergence of the series

               4. The fundamental system in the case of two algebraic singularities

     CHAPTER 7  Singularly Perturbed Problems for Nonlinear Equations

          §1. Weakly nonlinear singularly perturbed problems in the resonance case

               1. Formal solutions of weakly nonlinear problems

               2. Questions of solvability in the space of resonance-free solutions

               3. The asymptotic character of solutions

               4. Examples.

          §2. Regularized asymptotic solutions of strongly nonlinear singularly perturbed problems

               1. Regularization of strongly nonlinear problems

               2. Some function classes and their properties

               3. Theorems on the solvability of the iteration problems

               4. The asymptotic character of formal solutions

          §3. Connection of the regularization method with the averaging method

               1. Regularized asymptotic solutions

               2. Asymptotic solutions obtained by the averaging method

               3. Global solvability of the truncated equations


Part II  Singularly Perturbed Partial Differential Equations

     CHAPTER 8  Asymptotic Integration of Linear Parabolic Equations

          §1. A parabolic singularly perturbed problem with one viscous boundary

               1. Few words about the Fourier method

               2. Formulation of the problem and basic assumptions

          §2. The scheme of the regularization method in the selfadjoint case

               1. Regularization and the iteration problems

               2. The space of resonance-free solutions

               3. Solvability of the iteration problems

               4. Asymptotic convergence of the series.

          §3. Connection with the Fourier method and boundary layer theory

               1. Remarks

               2. Example

               3. Remarks on the adiabatic approximation in quantum mechanics

          §4. Asymptotic integration of a parabolic equation with two viscous boundaries

               1. Formulation of the problem for the linearized one-dimensional Navier-Stokes equation

               2. Regularization of singularities by "viscosity

               3. The iteration problems. The space of resonance-free solutions

               4. Theorems on normal and unique solvability

               5. Construction of the series of perturbation theo

               6. Estimation of the remainder term

          §5. Unsolved problems

               1. Problems without spectrum

               2. Problems with two intersecting viscous boundaries

               3. Multidimensional problems

     CHAPTER 9  Application of the Regularization Method to Some Elliptic Problems in a Cylindrical Domain

          §1. Formalism of the method for an elliptic problem

               1. Formulation of the problem

               2. Regularization and obtaining iteration problems

          §2. Asymptotic well-posedness and convergence of the method

               1. Unique solvability of the iteration problems

               2. A theorem on asymptotic convergence of the series.

               3. The leading term of the asymptotics

     CHAPTER 10  Asymptotic Integrationof Some Singularly Perturbed Evolution Equations

          §1. Asymptotic integration of singularly perturbed problems in Hilbert space in the case of discrete spectrum of the operator

               1. Formulation of the problem and regularization of singularities by a parameter.

               2. Construction of a formal asymptotic solution of the regularized problem

               3. A theorem on estimation of the remainder term

               4. An example

          §2. Generalization of the regularization method to the case of continuous spectrum of the limit operator

               1. Formulation of the problem and basic conditions

               2. Regularization and the space of resonance-free solutions H

               3. Uniqueness of the asymptotic series

               4. Example

          §3. An example of a problem with continuous spectrum and a spectral measure depending on a parameter

               1. Regularization of the problem and the space of solutions

               2. Construction of the regularized series

               3. Conclusion

References

Subject Index

Titles in This Series

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