Direct and Indirect Boundary Integral Equation Methods: 1st Edition (Hardback) book cover

Direct and Indirect Boundary Integral Equation Methods

1st Edition

By Christian Constanda

Chapman and Hall/CRC

216 pages

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Hardback: 9780849306396
pub: 1999-09-28
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Description

The computational power currently available means that practitioners can find extremely accurate approximations to the solutions of more and more sophisticated mathematical models-providing they know the right analytical techniques. In relatively simple terms, this book describes a class of techniques that fulfill this need by providing closed-form solutions to many boundary value problems that arise in science and engineering.

Boundary integral equation methods (BIEM's) have certain advantages over other procedures for solving such problems: BIEM's are powerful, applicable to a wide variety of situations, elegant, and ideal for numerical treatment. Certain fundamental constructs in BIEM's are also essential ingredients in boundary element methods, often used by scientists and engineers.

However, BIEM's are also sometimes more difficult to use in plane cases than in their three-dimensional counterparts. Consequently, the full, detailed BIEM treatment of two-dimensional problems has been largely neglected in the literature-even when it is more than marginally different from that applied to the corresponding three-dimensional versions.

This volume discusses three typical cases where such differences are clear: the Laplace equation (one unknown function), plane strain (two unknown functions), and the bending of plates with transverse shear deformation (three unknown functions). The author considers each of these with Dirichlet, Neumann, and Robin boundary conditions. He subjects each to a thorough investigation-with respect to the existence and uniqueness of regular solutions-through several BIEM's. He proposes suitable generalizations of the concept of logarithmic capacity for plane strain and bending of plates, then uses these to identify contours where non-uniqueness may occur. In the final section, the author compares and contrasts the various solution representations, links them by means of boundary operators, and evaluates them for their suitability for numeric computation.

Reviews

"The text is written clearly and the proofs are given in detail."

M. Aron, Proceedings of the Edinburgh Mathematical Society, Vol. 44, 445-448, 2001

"…the book offers a comprehensive treatment of the subject matter and constitutes a very useful source of information for mathematicians and other scientists interested in boundary integral equation methods.

M. Aron, Proceedings of the Edinburgh Mathematical Society, Vol. 44, 445-448, 2001

Table of Contents

Introduction

THE LAPLACE EQUATION

Notation and Prerequisites

The Fundamental Boundary Value Problems

Green's Formulae

Uniqueness Theorems

The Harmonic Potentials

A Classification of Boundary Integral Equation Methods

The Classical Indirect Method

The Alternative Indirect Method

The Modified Indirect Method

The Refined Indirect Method

The Direct Method

The Substitute Direct Method

PLANE STRAIN

Notation and Prerequisites

The Fundamental Boundary Value Problems

The Betti and Somigliana Formulae

Uniqueness Theorems

The Elastic Potentials

Properties of the Boundary Operators

The Classical Indirect Method

The Alternative Indirect Method

The Modified Indirect Method

The Refined Indirect Method

The Direct Method

The Substitute Direct Method

BENDING OF ELASTIC PLATES

Notation and Prerequisites

The Fundamental Boundary Value Problems

The Betti and Somigliana Formulae

Uniqueness Theorems

The Plate Potentials

Properties of the Boundary Operators

Boundary Integral Equation Methods

WHICH METHOD?

Notation and Prerequisites

Connections between the Indirect Methods

Connections between the Direct and Indirect Methods

Overall View and Conclusions

APPENDIX

About the Series

Monographs and Surveys in Pure and Applied Mathematics

Learn more…

Subject Categories

BISAC Subject Codes/Headings:
MAT000000
MATHEMATICS / General
MAT003000
MATHEMATICS / Applied
MAT007000
MATHEMATICS / Differential Equations