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Green's Functions - G. F. Roach

Green's Functions

Paperback

Published: 26th July 1982
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Green's functions are an important tool used in solving boundary value problems associated with ordinary and partial differential equations.

This self-contained and systematic introduction to Green's functions has been written with applications in mind. The material is presented in an unsophisticated and rather more practical manner than usual. Consequently advanced undergraduates and beginning postgraduate students in mathematics and the applied sciences will find this account particularly attractive. Many exercises and examples have been supplied throughout to reinforce comprehension and to increase familiarity with the technique.

'It is well planned and splendidly motivated and the computational material is covered without obscuring the conceptual development.' Bulletin of the Institute of Mathematics and its Applications 'The exposition is generally clear and well motivated mathematically... Generally this book should fill the need of those who want an introduction to the theory of Green's Functions but lack the mathematical background to understand more advanced accounts.' Mathematical Reviews

Preface to the First Editionp. xi
Preface to the Second Editionp. xiv
The Concept of a Green's Functionp. 1
Vector Spaces and Linear Transformationsp. 9
Vector Spacesp. 9
Linearly Independent Vectorsp. 16
Orthonormal Vectorsp. 20
Linear Transformationsp. 24
Systems of Finite Dimensionp. 31
Matrices and Linear Transformationsp. 31
Change of Basisp. 36
Eigenvalues and Eigenvectorsp. 38
Symmetric Operatorsp. 51
Bounded Operatorsp. 55
Positive Definite Operatorsp. 59
Continuous Functionsp. 61
Limiting Processesp. 61
Continuous Functionsp. 65
Integral Operatorsp. 79
The Kernel of an Integral Operatorp. 79
Symmetric Integral Transformationsp. 83
Separable Kernelsp. 85
Eigenvalues of a Symmetric Integral Operatorp. 91
Expansion Theorems for Integral Transformationsp. 99
Generalized Fourier Series and Complete Vector Spacesp. 112
Generalized Fourier Seriesp. 112
Approximation Theoremp. 121
Complete Vector Spacesp. 127
Differential Operatorsp. 141
Introductionp. 141
Inverse Operators and the [delta]-functionp. 141
The Domain of a Linear Differential Operatorp. 152
Adjoint Differential Operatorsp. 154
Self-Adjoint Second-Order Differential Operatorsp. 157
Non-Homogeneous Problems and Symbolic Operatorsp. 159
Green's Functions and Second-Order Differential Operatorsp. 163
The Problem of Eigenfunctionsp. 177
Green's Functions and the Adjoint Operatorp. 181
Spectral Representation and Green's Functionsp. 182
Integral Equationsp. 187
Classification of Integral Equationsp. 187
Method of Successive Approximationsp. 188
The Fredholm Alternativep. 195
Symmetric Integral Equationsp. 206
Equivalence of Integral and Differential Equationsp. 210
Green's Functions in Higher-Dimensional Spacesp. 213
Introductionp. 213
Partial Differential Operators and [delta]-functionsp. 215
Green's Identitiesp. 224
Fundamental Solutionsp. 227
Self-Adjoint Elliptic Equations (The Dirichlet Problem)p. 237
Self-Adjoint Elliptic Equations (The Neumann Problem)p. 243
Parabolic Equationsp. 248
Hyperbolic Equationsp. 251
Worked Examplesp. 256
Calculation of Particular Green's Functionsp. 274
Method of Imagesp. 274
Generalized Green's Functionsp. 278
Mixed Problemsp. 287
Approximate Green's Functionsp. 291
Introductionp. 291
Fundamental Solutionsp. 292
Generalized Potentialsp. 295
A Representation Theoremp. 300
Choice of Approximate Kernalp. 302
Summary of the Green's Function Methodp. 304
Green's Function Method for Ordinary Differential Equationsp. 304
Green's Function Method for Partial Differential Equationsp. 305
Operators and Expressionsp. 307
The Lebesgue Integralp. 312
Distributionsp. 316
Bibliographyp. 319
Chapter Referencesp. 321
Indexp. 323
Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9780521282888
ISBN-10: 0521282888
Audience: Professional
Format: Paperback
Language: English
Number Of Pages: 340
Published: 26th July 1982
Publisher: CAMBRIDGE UNIV PR
Country of Publication: GB
Dimensions (cm): 22.96 x 15.42  x 2.13
Weight (kg): 0.55
Edition Number: 2
Edition Type: Revised