Constructive Methods for Linear and Nonlinear Boundary Value Problems for Analytic Functions: 1st Edition (Hardback) book cover

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

1st Edition

By v Mityushev, S V Rogosin

Chapman and Hall/CRC

296 pages

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Hardback: 9781584880578
pub: 1999-11-29
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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.

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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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

About the Series

Monographs and Surveys in Pure and Applied Mathematics

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Subject Categories

BISAC Subject Codes/Headings:
MAT003000
MATHEMATICS / Applied
MAT007000
MATHEMATICS / Differential Equations