Mixed boundary value problems in potential theory

Front cover......Page 1
Dedication......Page 2
Title page......Page 3
Date-line......Page 4
Preface......Page 5
CONTENTS......Page 7
1.1. Electrostatic Problems......Page 9
1.2. Steady-State Diffusion Problems......Page 15
1.3. Elastostatic Problems......Page 18
1.4. Hydrodynamic Problems......Page 28
1.5. The Basic Elementary Problems......Page 31
1.6. Generalized Axisymmetric Potential Theory......Page 32
2.1. Integrals involving Bessel Functions......Page 34
2.2. Infinite Series involving Bessel Functions......Page 41
2.3. Some Remarks on Integral Equations......Page 48
2.4. Operators of Fractional Integration......Page 54
2.5. Relations between the Operator of Hankel Transforms and the Erdelyi-Kober Operators......Page 60
2.6. Jacobi Polynomials and Associated Legendre Functions......Page 62
3.1. Weber's Solution for a Disk Charged to Unit Potential......Page 71
3.2. Beltrami's Symmetric Potentials......Page 72
3.3. Use of Oblate Spheroidal Coordinates......Page 74
3.4. Copson's Solution......Page 77
3.5. Elementary Solution of the Dual Integral Equations of Beltrami's Method......Page 82
3.6. Methods based on the Integral Representation of Harmonic Functions......Page 85
4.1. Introduction......Page 88
4.2.1. Peters' Solution......Page 92
4.2.2. Titchmarsh's Solution......Page 94
4.2.3. Noble's Solution......Page 96
4.2.4. Gordon-Copson Solution......Page 99
4.3. Functions derived from Solutions of Dual Integral Equations......Page 101
4.4. Special Cases corresponding to Axisymmetric Problems in Potential Theory......Page 104
4.5. Dual Integral Equations with Trigonometrical Kernels......Page 106
4.6.1. Reduction to a Fredholm Equation......Page 114
4.6.2. Reduction to a System of Algebraic Equations......Page 121
4.6.3. Integral Equation for $\chi_1(x)$......Page 123
4.7. The General Problem......Page 126
4.7.1. Reduction to the Solution of two Integral Equations......Page 127
4.7.2. The Multiplying Factor Method......Page 128
4.7.3. The Integral Representation Method......Page 129
4.7.4. Identification of the Operators......Page 130
4.8. Approximate Solution of Dual Integral Equations......Page 131
4.9. Simultaneous Dual Integral Equations......Page 137
5.1. Introduction......Page 142
5.2. Dual Relations involving Fourier-Bessel Series......Page 143
5.2.1. The Reduction of Problem (a) to the Solution of a System of Algebraic Equations......Page 144
5.2.2. The Reduction of Problem (a) to the Solution of an Integral Equation......Page 147
5.2.3. The Reduction of Problem (b) to the Solution of an Integral Equation......Page 150
5.3. Dual Relations involving Dini Series......Page 152
5.4. Dual Relations involving Trigonometric Series......Page 158
5.4.1. Dual Relations involving Fourier Sine Series......Page 160
5.4.2. Dual Relations involving Sine Series analogous to Dini Series......Page 166
5.4.3. Dual Relations involving Fourier Cosine Series......Page 169
5.4.4. Dual Relations involving Cosine Series analogous to Fourier-Bessel Series......Page 170
5.4.5 Tranter's Formulae......Page 171
5.5. Dual Relations involving Series of Jacobi Polynomials......Page 173
5.6. Dual Relations involving Series of Associated Legendre Functions......Page 181
6.1. The Origin of Triple Relations......Page 186
6.2. Triple Integral Equations of Titchmarsh Type......Page 188
6.3. Triple Equations corresponding to Axisymmetric Potential Functions......Page 192
6.4. Reduction of Triple Integral Equations to Dual Series Relations......Page 195
6.5. Triple Equations involving Series of Legendre Polynomials......Page 198
6.5.1. Solution of Equations of the First Kind......Page 200
6.5.2. Solution of Equations of the Second Kind......Page 204
7.1. Kobayashi Potentials......Page 207
7.2. Dovnorovich's Solution of the First Basic Problem......Page 209
7.3. Galin's Theorem......Page 211
7.4. A Simple Superposition Method......Page 215
7.5. Green's Solution of the First Basic Problem and Related Solutions......Page 217
7.6. Boundary Value Problems concerning a Spherical Cap......Page 223
7.7. Boundary Value Problems concerning two Spherical Caps......Page 227
7.8. The Potential due to a Circular Annulus......Page 233
8.1.1. Derivation of Love's Integral Equation......Page 238
8.1.2. Solution of the Integral Equation......Page 242
8.1.3. Approximate Solutions......Page 246
8.2. Electrified Disk between Earthed Parallel Plates......Page 254
8.3. Electrified Disk within an Earthed Cylinder......Page 261
8.4. Two Coplanar Electrified Disks......Page 267
8.5. Two Parallel Electrified Strips......Page 272
8.6. Field due to a Charged Annular Disk......Page 275
8.7.1. Spherical Cap at Constant Potential......Page 278
8.7.2. Spherical Cap in a Uniform Field......Page 280
Appendix. Table of Relations involving the Erdelyi-Kober Operators and the Modified Operator of Hankel Transforms......Page 282
References......Page 283
Subject Index......Page 287
Authors' Index......Page 289
Index of Symbols......Page 291
Back cover......Page 292