Green's Functions

By: G. F. Roach

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Paperback | 26 July 1982 | Edition Number 2

At a Glance

Paperback
340 Pages

Edition Type
Revised

Dimensions(cm)
22.86 x 15.24 x 1.93

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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.

Industry Reviews

'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

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You Can Find This Book In

Non-Fiction Mathematics Calculus & Mathematical Analysis Differential Calculus & Equations Applied Mathematics Integral Calculus & Equations

Preface to the First Edition	p. xi
Preface to the Second Edition	p. xiv
The Concept of a Green's Function	p. 1
Vector Spaces and Linear Transformations	p. 9
Vector Spaces	p. 9
Linearly Independent Vectors	p. 16
Orthonormal Vectors	p. 20
Linear Transformations	p. 24
Systems of Finite Dimension	p. 31
Matrices and Linear Transformations	p. 31
Change of Basis	p. 36
Eigenvalues and Eigenvectors	p. 38
Symmetric Operators	p. 51
Bounded Operators	p. 55
Positive Definite Operators	p. 59
Continuous Functions	p. 61
Limiting Processes	p. 61
Continuous Functions	p. 65
Integral Operators	p. 79
The Kernel of an Integral Operator	p. 79
Symmetric Integral Transformations	p. 83
Separable Kernels	p. 85
Eigenvalues of a Symmetric Integral Operator	p. 91
Expansion Theorems for Integral Transformations	p. 99
Generalized Fourier Series and Complete Vector Spaces	p. 112
Generalized Fourier Series	p. 112
Approximation Theorem	p. 121
Complete Vector Spaces	p. 127
Differential Operators	p. 141
Introduction	p. 141
Inverse Operators and the [delta]-function	p. 141
The Domain of a Linear Differential Operator	p. 152
Adjoint Differential Operators	p. 154
Self-Adjoint Second-Order Differential Operators	p. 157
Non-Homogeneous Problems and Symbolic Operators	p. 159
Green's Functions and Second-Order Differential Operators	p. 163
The Problem of Eigenfunctions	p. 177
Green's Functions and the Adjoint Operator	p. 181
Spectral Representation and Green's Functions	p. 182
Integral Equations	p. 187
Classification of Integral Equations	p. 187
Method of Successive Approximations	p. 188
The Fredholm Alternative	p. 195
Symmetric Integral Equations	p. 206
Equivalence of Integral and Differential Equations	p. 210
Green's Functions in Higher-Dimensional Spaces	p. 213
Introduction	p. 213
Partial Differential Operators and [delta]-functions	p. 215
Green's Identities	p. 224
Fundamental Solutions	p. 227
Self-Adjoint Elliptic Equations (The Dirichlet Problem)	p. 237
Self-Adjoint Elliptic Equations (The Neumann Problem)	p. 243
Parabolic Equations	p. 248
Hyperbolic Equations	p. 251
Worked Examples	p. 256
Calculation of Particular Green's Functions	p. 274
Method of Images	p. 274
Generalized Green's Functions	p. 278
Mixed Problems	p. 287
Approximate Green's Functions	p. 291
Introduction	p. 291
Fundamental Solutions	p. 292
Generalized Potentials	p. 295
A Representation Theorem	p. 300
Choice of Approximate Kernal	p. 302
Summary of the Green's Function Method	p. 304
Green's Function Method for Ordinary Differential Equations	p. 304
Green's Function Method for Partial Differential Equations	p. 305
Operators and Expressions	p. 307
The Lebesgue Integral	p. 312
Distributions	p. 316
Bibliography	p. 319
Chapter References	p. 321
Index	p. 323
Table of Contents provided by Syndetics. All Rights Reserved.

Green's Functions

At a Glance

Paperback

Industry Reviews

Shipping

How to return your order

Defective items

You Can Find This Book In

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