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.
Author(s): G. F. Roach
Edition: Second
Publisher: Cambridge University Press
Year: 1992
Language: English
Pages: 341
Tags: Математика;Дифференциальные уравнения;
Preface to the First Edition
Preface to the Second Edition
CHAPTER 1 THE CONCEPT OF A GREEN'S FUNCTION
CHAPTER 2 VECTOR SPACES AND LINEAR TRANSFORMATIONS
2.1 Vector Spaces
2.2 Linearly Independent Vectors
2.3 Orthonormal Vectors
2.4 Linear Transformations
CHAPTER 3 SYSTEMS OF FINITE DIMENSION 31
3.1 Matrices and Linear Transformations 31
3.2 Change of Basis 36
3.3 Eigenvalues and Eigenvectors 38
3.4 Symmetric Operators 51
3.5 Bounded Operators 55
3.6 Positive Definite Operators 59
CHAPTER 4 CONTINUOUS FUNCTIONS 61
4.1 Limiting Processes 61
4.2 Continuous Functions 65
CHAPTER 5 INTEGRAL OPERATORS 79
5.1 The Kernel of an Integral Operator 79
5.2 Symmetric Integral Transformations 83
5.3 Separable Kernels 85
5.4 Eigenvalues of a Symmetric Integral Operator 91
5.5 Expansion Theorems for Integral Transformations 99
CHAPTER 6 GENERALIZED FOURIER SERIES AND COMPLETE VECTOR SPACES
6.1 Generalized Fourier Series
6.2 Approximation Theorem
6.3 Complete Vector Spaces
CHAPTER 7 DIFFERENTIAL OPERATORS 141
7.1 Introduction 141
7.2 Inverse Operators and the delta-function 141
7.3 The Domain of a Linear Differential Operator 152
7.4 Adjoint Differential Operators 154
7.5 Self-Adjoint Second-Order Differential Operators 157
7.6 Non-Homogeneous Problems and Symbolic Operators 159
7.7 Green's Functions and Second-Order Differential Operators 163
7.8 rrhe Problem of Eigenfunctions 177
7.9 Green's Functions and the Adjoint Operator 181
7.10 Spectral Representation and Green's Functions 182
CHAPTER 8 INTEGRAL EQUATIONS 187
8.1 Classification of Integral Equations 187
8.2 Method of Successive Approximations 188
8.3 The Fredholm Alternative 195
8.4 Symmetric Integral Equations 206
8.5 Equivalence of Integral and Differential Equations 210
CHAPTER 9 GREEN'S FUNCTIONS IN HIGHER-DIMENSIONAL SPACES 213
9.1 Introduction 213
9.2 Partial Differential Operators and delta-functions 215
9.3 Green's Identities 224
9.4 Fundamental Solutions 227
9.5 Self-Adjoint Elliptic Equations (The Dirichlet Problem) 237
9.6 Self-Adjoint Elliptic Equations (The Neumann Problem) 243
9.7 Parabolic Equations 248
9.8 Hyperbolic Equations 251
9.9 Worked Examples 256
CHAPTER 10 CALCULATION OF PARTICULAR GREEN'S FUNCTIONS 274
10.1 Method of Images 274
10.2 Generalized Green's Function 278
10.3 Mixed Problems 287
CHAPTER 11 APPROXIMATE GREEN'S FUNCTIONS 291
11.1 Introduction 291
11.2 Fundamental Solutions 292
11.3 Generalized Potentials 295
11.4 A Representation Theorem 300
11.5 Choice of Approximate Kernal 302
APPENDIX A SUMMARY OF THE GREEN'S FUNCTION METHOD 304
Al Green's Function Method for Ordinary Differential Equations 304
A2 Green's Function Method for Partial Differential Equations 305
APPENDIX B OPERATORS AND EXPRESSIONS
APPENDIX C THE LEBESGUE INTEGRAL
APPENDIX D DISTRIBUTIONS
BIBLIOGRAPHY
Chapter References
INDEX
