1st Edition

Constructive Methods for Linear and Nonlinear Boundary Value Problems for Analytic Functions



ISBN 9781584880578
Published November 29, 1999 by Chapman and Hall/CRC
296 Pages

USD $200.00

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Book Description

Constructive methods developed in the framework of analytic functions effectively extend the use of mathematical constructions, both within different branches of mathematics and to other disciplines. This monograph presents some constructive methods-based primarily on original techniques-for boundary value problems, both linear and nonlinear. From among the many applications to which these methods can apply, the authors focus on interesting problems associated with composite materials with a finite number of inclusions.

How far can one go in the solutions of problems in nonlinear mechanics and physics using the ideas of analytic functions? What is the difference between linear and nonlinear cases from the qualitative point of view? What kinds of additional techniques should one use in investigating nonlinear problems? Constructive Methods for Linear and Nonlinear Boundary Value Problems serves to answer these questions, and presents many results to Westerners for the first time. Among the most interesting of these is the complete solution of the Riemann-Hilbert problem for multiply connected domains.

The results offered in Constructive Methods for Linear and Nonlinear Boundary Value Problems are prepared for direct application. A historical survey along with background material, and an in-depth presentation of practical methods make this a self-contained volume useful to experts in analytic function theory, to non-specialists, and even to non-mathematicians who can apply the methods to their research in mechanics and physics.

Table of Contents

A HISTORICAL SURVEY
NOTATIONS AND AUXILIARY RESULTS
Geometry of Complex Plane
Functional Spaces
Operator Equations in Functional Spaces
Properties of Analytic and Harmonic Functions
Cauchy-Type Integral and Singular Integrals
Schwarz Operator
C-Linear Conjugation Problem
Riemann-Hilbert Boundary Value Problem
Entire Function
Conformal Mappings
R-Linear Problem and its Applications
Notes and Comments
NONLINEAR BOUNDARY VALUE PROBLEMS
Conjugation Problem of Power Type
Problem of Multiplication Type
Entire Functions Methods
General Riemann-Hilbert Problem of Power Type
The Modulus Problem and its Generalization
Linear Fractional Problem
Cherepanov's Mixed Problem
Notes and Comments
METHOD OF FUNCTIONAL EQUATIONS
Dirichlet Problem for a Doubly Connected Domain
A Nonlinear Boundary Value Problem
Linear Functional Equations
Harmonic Measures and Schwarz Operator
Linear Riemann-Hilbert Porblem
Poincaré Series
Mixed Problem for Multiply Connected Domains
Circular Polygons with Zero Angles
Generalized Method of Schwarz and other Methods
Notes and Comments
NONLINEAR PROBLEMS OF MECHANICS
Steady Heat Conduction: Nonlinear Composites
Linearized Problem
Constructive Solution to Integral Equations
Composite Materials with Reactive Inclusions
Steady Heat Conduction on Configurations
An Elastic Problem for Composite Materials
Plane Stokes Flow
Notes and Comments
BIBLIOGRAPHY
INDEX

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Reviews

"The book contains several fresh results and collects material which has been spread in the literature (frequently in Russian, but also from the western schools). With an extensive bibliography of about 300 items, it can serve as a reference text. The presentation is addressed to beginners and experts as well. Since the essential prerequisites are included it should be convenient to use for interested applied scientists with some mathematical background."
-Elias Wegert, in Mathematical Reviews, Issue 2001d


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