Kernel Functions and Differential Equations (eBook)

Front Cover 1
Kernel Functions and Elliptic Differential Equations in Mathematical Physics 4
Copyright Page 5
Contents 12
Preface 8
Note to the Reader 10
Part A: Boundary Value Problems for Partld Dlfferentlal EquaUonr of Elllptlc Type 16
CHAPTER I: THEORY OF HEAT CONDUCTION 16
1. The differential equation of heat transfer 16
2. The special case of steady flow 18
3. Point sources and fundamental singularities 22
4. Fundamental solutions 25
5. Discontinuities 30
6. Dirichlet's principle 32
7. A modified heat equation 35
8. Formal considerations 37
CHAPTER II: FLUID DYNAMICS 40
1. The fundamental equations 40
2. Stationary irrotational flow 43
3. Incompressible, irrotational, stationary fluid flow 45
4. Sources and sinks 47
5. Fundamental solutions and flow patterns 50
6. Infinite domains 53
7. Virtual mass 58
8. Dirichlet identities 61
9. Virtual mass as an extremum 70
10. Variational formulas 73
11. Free boundaries, discontinuity surfaces, and virtual mass 79
12. Two-dimensional flows 86
13. Boundary value problems and fundamental solutions in plane flows 91
14. Obstacles in a plane flow 97
15. The variation of the Green's function 108
16. Lavrentieff's extremum problem for lift 113
17. Free boundaries in plane flow 116
18. The complex kernel function and its applications 124
19. Axially symmetric motion of an incompressible fluid 134
20. Associated partial differential equations 139
21. Two-dimensional compressible fluid flow 143
22. Construction of solutions of the differential equations in the hodograph plane 148
23. The hodograph method and boundary value problems in the physical plane 156
24. Singularities in the hodograph plane and their application 162
CHAPTER III: ELECTRO- AND MAGNETOSTATICS 170
1. The general equations of the electrostatic field 170
2. Conductor potentials and induction coefficients 172
3. The energy of the electrostatic field 175
4. The magnetostatic field 180
5. The conductor potential spherical harmonics
6. Polarization of a conducting surface 203
7. Forces and moments in a conductor surface 208
8. Orthogonal harmonic functions 213
CHAPTER IV: ELASTICITY 221
1. Strain tensor fields 221
2. Stress tensor fields 224
3. Stress-strain relations 226
4. The energy integral and boundary value problems 230
5. Betti's formula and the fundamental singularity 234
6. Fundamental solutions 237
7. Construction of the fundamental solutions in terms of orthonormal solution fields 241
8. Particular solutions 244
9. The theory of thin plates 247
10. Fundamental solutions of the biharmonic equation 251
11. Plane strain 261
Part B: Kernel Function Methods in the Theory of Boundary Value Problems 273
CHAPTER I : PROPERTIES OF SOLUTIONS 273
1. Introduction 273
2. Notation and definitions 274
3. Properties OI the solutions of (1.1) 276
4. Existence theorems for certain solutions 277
5. Fundamental singularities and fundamental solutions 283
6. Dirichlet integrals and fundamental functions 288
CHAPTER II: THE KERNEL FUNCTIONS AND THEIR PROPERTIES 290
1. Kernel functions 290
2. Orthonormal systems and construction of the kernel 295
3. Integral operators and the construction of complete sets of solutions 301
4. Dirichlet identities 312
5. Choice of the fundamental singularity 316
6. The regularity of l (P,Q) 318
7. An integral equation for the kernel 324
8. The eigenvalues of the kernel I (P,Q) 330
9. Series developments and integral equations 341
CHAPTER III: VARIATIONAL AND COMPARISON THEORY 350
1. Relations between kernels of different domains 350
2. Eigenfunctions of the y-transformation 361
3. Doubly orthogonal systems 366
4. An application of the operators to the theory of orthogonal functions 370
5. Comparison formulas for Green’s and Neumann’s functions 372
6. Variational theory 375
CHAPTER IV: EXISTENCE THEORY 386
1. Boundary value problem and orthogonal projection 386
CHAPTER V: DEPENDENCE OF KERNELS ON BOUNDARY CONDITIONS AND THE DIFFERENTIAL EQUATION 396
1. The positiveness of the fundamental functions 396
2. Stability of fundamental functions 399
3. Stability of eigenvalues 403
4. Construction of a fundamental singularity 407
CHAPTER VI : GENERALIZATIONS 410
1. Different types of metrics 410
2. Further generalizations 417
List of Symbols Used in Part B 419
Bibliography Books 423
Articles 427
Author Index 436
Subject Index 440