TABLE OF CONTENTS
Part I. BV Problems---an Intuitive Introduction
S1. Diffusion
- Transient heat flow in a block
- A numerical model
- An analytic model: the equation of diffusion
S2. Steady State Dirichlet Problems
- SSP 1. Square regions
- SSP 2. Steady state within a sector
- SSP 3. An infinite strip
- SSP 4. Annular regions
- Solutions via Brownian motion
S3. Flux
- The heat equation revisited
- Potential (the wave equation)
- Conservation theorems
- Classical uniqueness via
- Green's identity
- the maximum principle
- Nondimensional variables
- The method of characteristics
S4. Steady State Neumann Problems
- SSP 5. A rod
- SSP 6. The rod revisited
- SSP 7. Closed-loop ground-coupled heat pumps
- SSP 8. Forces crushing a capacitor
- SSP 9. Steady state within a square
- SSP 10. Steady state within a quarter disk
- SSP 11. A mixed steady problem within a hemisphere
- The method of conformal mapping
S5. Transient Problems: an introduction to separation of variables
- BVP 1. A rod with specified end temperatures
- BVP 2. Heat lost from a slab to ambient
- BVP 3. the Fourier ring problem
- BVP 4. The plucked string
- BVP 5. A circular drum
- BVP 6. Quenching a ball
- SSP 1. Steady state in square regions (revisited)
- A critique of the method
Part II. Function Spaces
S6. Norms
- Error between signals
- L^{\infty}, L^{1}, and L^{2}
- Vector space properties
- Convergence in norm
- Completeness
S7. Hilbert Space
- Definition
- Details
- Examples
- Orthogonal complements
- Gram-Schmidt process
S8. Recasting BVPs into Operator Format
- Differential operators
- Eigenvalues
- Time rate-of-change of shape
- Strong reformulation
S9. Hermitian Operators
- Dense domains
- Closed graph
- Adjoints
- Orthogonality of eigenfunctions
- Negative definite operators
- Lyapunov stability
S10. Resolvents
- Spectrum
- Examples
- Hilbert-Schmidt operators
- Compact operators
- The spectral theorem
- A summary
- Rellich's theorem
S11. Separation of Variables
- A synopsis
- The theorem on separation of variables
- A check list for separation to occur
- Tricks of the trade
Part III. Expansions in Series of Orthogonal Functions
S12. Trigonometric Expansions
- Sine, cosine, and Fourier series
- Pointwise convergence
- Signal processing
S13. Rectangular Problems
- BVP 7. Quenching a block
- BVP 8. Deep earth temperatures
- BVP 9. Current within a flat conductor
- BVP 10. A trapped quantum particle
- BVP 11. Quantum tunneling
- BVP 12. Transverse vibrations of the Bernoulli-Euler beam
- BVP 13. Rectangular waveguides
S14. Bessel Functions
- Power series representation
- Standard formulae
- Integral representation
- Asymptotics
- Orthogonality and completeness
- Other Bessel functions
S15. Cylindrical Problems
- BVP 14. Quenching a solid cylinder
- BVP 5. A circular drum (revisited)
- SSP 7. The closed-loop ground-coupled heat pump (revisited)
- BVP 15. Current within a round conductor
S16. Orthogonal Polynomials
- Weighted inner products
- Completeness
- Polynomial eigenfunctions
- Choosing weights
- Recurrence formulae
- Norm formulae
- Rodrigues's formula
- Non-polynomial eigenfunctions
S17. Spherical Problems
- SSP 12. A spherical capacitor
- BVP 6. Quenching a ball (revisited)
- BVP 16. A spherical bell
- BVP 17. The hydrogen atom
Part IV. An Introduction to Selected Other Topics
S18. Sturm-Liouville Problems
- Statement of the problem
- The correct inner product
- Compact resolvents
- The fundamental theorem of Sturm-Liouville problems
- Examples
S19. Choosing Inner Products
- Examples
- Background: adjoints and biorthogonal series
- The correct inner product
S20. The Use of a Symbolic Manipulator
- Special functions
- Fourier series
- Fourier-Bessel series
- Steady Dirichlet problems
- Eigenmodes
- Animation of time-varying solutions
S21. The Mikusinski Operational Calculus
- Background
- Convolution quotients
- Generalized integration and differentiation
- Generalized solutions
- BVP 18. A semi-infinite slab
- BVP 19. Spilled pollutants
- SSP 7. The closed-loop ground-coupled heat pump (revisited)
S22. Fourier Integrals
- SSP 2. Steady state within a sector (revisited)
- SSP 13. A doubly infinite solid rod
- BVP 18. A semi-infinite slab (reprise)
- Justification of the method
- Cauchy principle value
- Application to BVP 18
- Another origin of the method
- The Fourier transform
S23. The Galerkin Numerical Method
- Truncation of series solutions
- Outline of the Galerkin method
- SSP 14. Steady flow within a square tube
- SSP 7. The closed-loop ground-coupled heat pump (reprise)
- BVP 20. Normal vibrational modes of a triangular brace
S24. Sobolev Methods
- The Dirichlet inner product
- The dominance of the Dirichlet inner product
- The Sobolev space W^{1,2}_{0}(\O)
- Rellich's compact embedding theorem
- Weak restatement of problems
- BVP4. The plucked string (revisited)
Bibliography
Appendix A. Measure and Integration
- Lebesgue measure
- The Lebesgue integral
- Lebesgue dominated convergence theorem
- Fubini's theorem
Appendix B. Quantum Mechanics
- Classical mechanics of a single particle
- Measurement
- Classical distributed vibrations
- Quantum mechanics of a single particle
- Poisson bracket
- Consequences of the commutation relation
- Arguments for the commutation relation
- The uncertainty principle