Boundary value problems of mathematical physics. Volume 1 / Ivar Stakgold.

Format:

Book

Author/Creator:

Stakgold, Ivar.

Series:

Classics in applied mathematics ; 29.

Classics in applied mathematics ; 29

Language:

English

Subjects (All):

Boundary value problems.

Mathematical physics.

Physical Description:

xii, 340 pages : illustrations ; 23 cm.

Place of Publication:

Philadelphia : Society for Industrial and Applied Mathematics, [2000]

Summary:

For more than 30 years, this two-volume set has helped prepare graduate students to use partial differential equations and integral equations to handle significant problems arising in applied mathematics, engineering, and the physical sciences. Originally published in 1967, this graduate-level introduction is devoted to the mathematics needed for the modern approach to boundary value problems using Green's functions and using eigenvalue expansions.

Contents:

1. The Green's Function

1.1 The String Subject to Transverse Loading 1

1.2 The Dirac Delta Function 18

1.3 The Theory of Distributions 28

1.4 Preliminary Results on Linear Equations of the Second Order 58

1.5 Boundary Value Problems 64

1.6 Alternative Theorems and the Modified Green's Function 79

2. Introduction to Linear Spaces

2.1 Functions and Transformations 92

2.2 Linear Spaces 96

2.3 Metric Spaces, Normed Linear Spaces, and Inner Product Spaces 99

2.4 Properties of a Separable Hilbert Space 116

2.5 Functionals 135

2.6 Transformations 139

2.7 Linear Transformations on E[superscript (c)][subscript n] 146

2.8 The Inverse of a Linear Transformation in Hilbert Space 165

2.9 The Spectrum of an Operator 180

2.10 Completely Continuous Operators 184

2.11 Extremal Properties of Bounded Operators 187

3. Linear Integral Equations

3.2 The Neumann Series (Method of Successive Approximations) 206

3.3 The Spectrum of a Self-adjoint Hilbert-Schmidt Operator 212

3.4 The Solution of the Inhomogeneous Equation with a Symmetric Hilbert-Schmidt Kernel 220

3.5 Extremal Principles 223

3.6 Approximations Based on Extremal Principles 226

3.7 Questions Relating to Continuity and Uniform Convergence

The Bilinear Series for the Kernel and the Iterated Kernels 236

3.8 Approximate Methods for the Solution of Integral Equations 241

3.9 Nonsymmetric Hilbert-Schmidt Operators 250

4. Spectral Theory of Second-Order Differential Operators

4.2 The Regular Boundary Value Problem 268

4.3 Introductory Examples of Singular Problems 283

4.4 The General Singular Problem 295

A.1 Static and Dynamic Problems for Strings and Membranes 323

A.2 Static and Dynamic Problems for Beams and Plates 326

A.3 The Equation of Heat Conduction 327

B.1 Bessel Functions 329

B.2 Wronskian Relationships 330

B.3 The Modified Bessel Function 331

B.4 The Behavior of Cylinder Functions at Zero and at Infinity 332

5. Distributions and Generalized Solutions

5.2 Test Functions 3

5.3 Distributions 4

5.4 Convergence of Distributions 10

5.5 Additional Properties of Distributions 17

5.6 Fourier Transforms 23

5.7 Partial Differential Equations for Distributions 39

5.8 Fundamental Solutions 48

5.9 Classification of Partial Differential Equations 72

6. Potential Theory

6.2 Interior Dirichlet Problem for the Unit Circle 90

6.3 Some Properties of Harmonic Functions 99

6.4 Surface Layers 110

6.5 Integral Equations of Potential Theory 122

6.6 Green's Function for the Negative Laplacian 130

6.7 Methods for Determining the Green's Function 146

6.8 Some Physical Applications of Potential Theory 171

7. Equations of Evolution

7.2 Causal Green's Function for Heat Conduction 197

7.3 Methods for Finding the Causal Green's Function 204

7.4 Uniqueness and Continuous Dependence on the Data 222

7.5 Miscellaneous Topics Related to the Heat Equation 227

7.6 Preliminary Considerations for the Undamped Wave Equation 243

7.7 Causal Green's Function for the Wave Equation 246

7.8 Problems in One Space Dimension 249

7.9 Problems in More than One Dimension 253

7.10 Wave Equation with External Damping 257

7.11 Monochromatic Excitation and the Principle of Limiting Absorption 259

7.12 Green's Function for the Helmholtz Operator and Applications 265

7.13 Half-Plane Excited by a Line Source or a Plane Wave 281

7.14 Representation of Solutions of the Helmholtz Equation in Exterior Domains 294

7.15 Scattering Problem 299

7.16 Wiener-Hopf Method 311

8. Variational and Related Methods

8.2 Best Approximation in a Subspace 335

8.3 Maximum Theorem 337

8.4 Ritz-Rayleigh Method 340

8.5 Complementary Variational Principles 344

8.6 Capacity Problem 350

8.7 Natural Boundary Conditions 352

8.8 Indefinite and Nonsymmetric Operators 355

8.9 Other Methods for Upper Bounds to Functionals Associated with Positive Operators 358

8.10 Method of Least Squares 361

8.11 Extremal Principles for Eigenvalue Problems on Euclidean n Space 369

8.12 Eigenvalue Problems in Hilbert Space 372

8.13 Lower Bounds to Eigenvalues 381

Appendix A. Spherical Harmonics 393

Appendix B. Asymptotic Expansions 399.

Notes:

Originally published: New York : Macmillan, 1967-1968, in series: Macmillan series in advanced mathematics and theoretical physics.

Includes bibliographical references and indexes.

Other Format:

Online version: Stakgold, Ivar. Boundary value problems of mathematical physics.

ISBN:

0898714567

9780898714562

OCLC:

42960653

Online:

Publisher description