Green's functions / G.F. Roach.

Format:

Book

Author/Creator:

Roach, G. F. (Gary Francis)

Language:

English

Subjects (All):

Green's functions.

Boundary value problems.

Physical Description:

xiii, 325 pages ; 24 cm

Edition:

Second edition.

Place of Publication:

Cambridge [Cambridgeshire] ; New York : Cambridge University Press, 1982.

Summary:

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.

Contents:

Chapter 1 The Concept of a Green's Function 1

Chapter 2 Vector Spaces and Linear Transformations 9

2.1 Vector Spaces 9

2.2 Linearly Independent Vectors 16

2.3 Orthonormal Vectors 20

2.4 Linear Transformations 24

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 112

6.1 Generalized Fourier Series 112

6.2 Approximation Theorem 121

6.3 Complete Vector Spaces 127

Chapter 7 Differential Operators 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 The 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

9.2 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 Functions 278

10.3 Mixed Problems 287

Chapter 11 Approximate Green's Functions 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

A1 Green's Function Method for Ordinary Differential Equations 304

A2 Green's Function Method for Partial Differential Equations 305

Appendix B Operators and Expressions 307

Appendix C The Lebesgue Integral 312

Appendix D Distributions 316.

Notes:

Bibliography: pages 319-321.

Includes index.

ISBN:

0521238900

0521282888

OCLC:

7577996