Kernel Functions and Differential Equations
Kernel Functions and Differential Equations
Academic Press
Published on 31. August 2009
431 pages
978-0-08-087312-1 (ISBN)
Description
Kernel Functions and Differential Equations
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Content
- Front Cover
- Kernel Functions and Elliptic Differential Equations in Mathematical Physics
- Copyright Page
- Contents
- Preface
- Note to the Reader
- Part A: Boundary Value Problems for Partld Dlfferentlal EquaUonr of Elllptlc Type
- CHAPTER I: THEORY OF HEAT CONDUCTION
- 1. The differential equation of heat transfer
- 2. The special case of steady flow
- 3. Point sources and fundamental singularities
- 4. Fundamental solutions
- 5. Discontinuities
- 6. Dirichlet's principle
- 7. A modified heat equation
- 8. Formal considerations
- CHAPTER II: FLUID DYNAMICS
- 1. The fundamental equations
- 2. Stationary irrotational flow
- 3. Incompressible, irrotational, stationary fluid flow
- 4. Sources and sinks
- 5. Fundamental solutions and flow patterns
- 6. Infinite domains
- 7. Virtual mass
- 8. Dirichlet identities
- 9. Virtual mass as an extremum
- 10. Variational formulas
- 11. Free boundaries, discontinuity surfaces, and virtual mass
- 12. Two-dimensional flows
- 13. Boundary value problems and fundamental solutions in plane flows
- 14. Obstacles in a plane flow
- 15. The variation of the Green's function
- 16. Lavrentieff's extremum problem for lift
- 17. Free boundaries in plane flow
- 18. The complex kernel function and its applications
- 19. Axially symmetric motion of an incompressible fluid
- 20. Associated partial differential equations
- 21. Two-dimensional compressible fluid flow
- 22. Construction of solutions of the differential equations in the hodograph plane
- 23. The hodograph method and boundary value problems in the physical plane
- 24. Singularities in the hodograph plane and their application
- CHAPTER III: ELECTRO- AND MAGNETOSTATICS
- 1. The general equations of the electrostatic field
- 2. Conductor potentials and induction coefficients
- 3. The energy of the electrostatic field
- 4. The magnetostatic field
- 5. The conductor potential
- spherical harmonics
- 6. Polarization of a conducting surface
- 7. Forces and moments in a conductor surface
- 8. Orthogonal harmonic functions
- CHAPTER IV: ELASTICITY
- 1. Strain tensor fields
- 2. Stress tensor fields
- 3. Stress-strain relations
- 4. The energy integral and boundary value problems
- 5. Betti's formula and the fundamental singularity
- 6. Fundamental solutions
- 7. Construction of the fundamental solutions in terms of orthonormal solution fields
- 8. Particular solutions
- 9. The theory of thin plates
- 10. Fundamental solutions of the biharmonic equation
- 11. Plane strain
- Part B: Kernel Function Methods in the Theory of Boundary Value Problems
- CHAPTER I : PROPERTIES OF SOLUTIONS
- 1. Introduction
- 2. Notation and definitions
- 3. Properties OI the solutions of (1.1)
- 4. Existence theorems for certain solutions
- 5. Fundamental singularities and fundamental solutions
- 6. Dirichlet integrals and fundamental functions
- CHAPTER II: THE KERNEL FUNCTIONS AND THEIR PROPERTIES
- 1. Kernel functions
- 2. Orthonormal systems and construction of the kernel
- 3. Integral operators and the construction of complete sets of solutions
- 4. Dirichlet identities
- 5. Choice of the fundamental singularity
- 6. The regularity of l (P,Q)
- 7. An integral equation for the kernel
- 8. The eigenvalues of the kernel I (P,Q)
- 9. Series developments and integral equations
- CHAPTER III: VARIATIONAL AND COMPARISON THEORY
- 1. Relations between kernels of different domains
- 2. Eigenfunctions of the y-transformation
- 3. Doubly orthogonal systems
- 4. An application of the operators to the theory of orthogonal functions
- 5. Comparison formulas for Green's and Neumann's functions
- 6. Variational theory
- CHAPTER IV: EXISTENCE THEORY
- 1. Boundary value problem and orthogonal projection
- CHAPTER V: DEPENDENCE OF KERNELS ON BOUNDARY CONDITIONS AND THE DIFFERENTIAL EQUATION
- 1. The positiveness of the fundamental functions
- 2. Stability of fundamental functions
- 3. Stability of eigenvalues
- 4. Construction of a fundamental singularity
- CHAPTER VI : GENERALIZATIONS
- 1. Different types of metrics
- 2. Further generalizations
- List of Symbols Used in Part B
- Bibliography Books
- Articles
- Author Index
- Subject Index
