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Index
Cover
Title Page
Copyright Page
Dedication
PREFACE
Table of Contents
1: Finite Dimensional Vector Spaces
1.1 Linear Vector Spaces
1.2 Spectral Theory for Matrices
1.3 Geometrical Significance of Eigenvalues
1.4 Fredholm Alternative Theorem
1.5 Least Squares Solutions—Pseudo Inverses
1.6 Numerical Considerations
Further Reading
Problems for Chapter 1
1.7 Appendix: Jordan Canonical Form
2: Function Spaces
2.1 Complete Metric Spaces
2.1.1 Sobolev Spaces
2.2 Approximation in Hilbert Spaces
2.2.1 Fourier Series and Completeness
2.2.2 Orthogonal Polynomials
2.2.3 Trigonometric Series
2.2.4 Discrete Fourier Transforms
2.2.5 Walsh Functions and Walsh Transforms
2.2.6 Finite Elements
2.2.7 Sine Functions
Further Reading
Problems for Chapter 2
3: Integral Equations
3.1 Introduction
3.2 The Fredholm Alternative
3.3 Compact Operators—Hilbert Schmidt Kernels
3.4 Spectral Theory for Compact Operators
3.5 Resolvent and Pseudo-Resolvent Kernels
3.6 Approximate Solutions
3.7 Singular Integral Equations
Further Reading
Problems for Chapter 3
4: Differential Operators
4.1 Distributions and the Delta Function
4.2 Green’s Functions
4.3 Differential Operators
4.3.1 Domain of an Operator
4.3.2 Adjoint of an Operator
4.3.3 The Extended Definition of an Operator
4.3.4 Inhomogeneous Boundary Data
4.3.5 The Fredholm Alternative
4.4 Least Squares Solutions
4.5 Eigenfunction Expansions
4.5.1 Trigonometric Functions
4.5.2 Orthogonal Polynomials
4.5.3 Special Functions
4.5.4 Discretized Operators
Further Reading
Problems for Chapter 4
5: Calculus of Variations
5.1 Euler-Lagrange Equations
5.1.1 Constrained Problems
5.1.2 Several Unknown Functions
5.1.3 Higher Order Derivatives
5.1.4 Variable Endpoints
5.1.5 Several Independent Variables
5.2 Hamilton’s Principle
5.3 Approximate Methods
5.4 Eigenvalue Problems
Further Reading
Problems for Chapter 5
6: Complex Variable Theory
6.1 Complex Valued Functions
6.2 The Calculus of Complex Functions
6.2.1 Differentiation—Analytic Functions
6.2.2 Integration
6.2.3 Cauchy Integral Formula
6.2.4 Taylor and Laurent Series
6.3 Fluid Flow and Conformal Mappings
6.3.1 Laplace’s Equation
6.3.2 Conformal Mappings
6.3.3 Free Boundary Problems—Hodograph Transformation
6.4 Contour Integration
6.5 Special Functions
6.5.1 The Gamma Function
6.5.2 Bessel Functions
6.5.3 Legendre Functions
6.5.4 Sine Functions
Further Reading
Problems for Chapter 6
7: Transform and Spectral Theory
7.1 Spectrum of an Operator
7.2 Fourier Transforms
7.2.1 Transform Pairs
7.2.2 Completeness of Hermite and Laguerre Polynomials
7.2.3 Sine Functions
7.3 Laplace, Mellin and Hankel Transforms
7.4 Z Transforms
7.5 Scattering Theory
Further Reading
Problems for Chapter 7
8: Partial Differential Equations
8.1 Poisson’s Equation
8.1.1 Fundamental solutions
8.1.2 The Method of Images
8.1.3 Transform Methods
8.1.4 Eigenfunctions
8.2 The Wave Equation
8.2.1 Derivations
8.2.2 Fundamental Solutions
8.2.3 Vibrations
8.2.4 Diffraction Patterns
8.3 The Heat Equation
8.3.1 Derivations
8.3.2 Fundamental Solutions and Green’s Functions
8.4 Differential-Difference Equations
Further Reading
Problems for Chapter 8
8.5 Appendix: Thermal Diffusivity
8.6 Appendix: Coordinate Transformation
9: Inverse Scattering Transform
9.1 Inverse Scattering
9.2 Isospectral Flows
9.3 Korteweg-deVries Equation
9.4 The Toda Lattice
Further Reading
Problems for Chapter 9
10: Asymptotic Expansions
10.1 Definitions and Properties
10.2 Integration by Parts
10.3 Laplace’s Method
10.4 Method of Steepest Descents
10.5 Method of Stationary Phase
10.6 REDUCE Procedures
Further Reading
Problems for Chapter 10
11: Regular Perturbation Theory
11.1 The Implicit Function Theorem
11.2 Perturbation of Eigenvalues
11.3 Nonlinear Eigenvalue Problems
11.4 Oscillations and Periodic Solutions
11.4.1 Advance of the Perihelion of Mercury
11.4.2 Van der Pol Oscillator
11.4.3 Knotted Vortex Filaments
11.5 Hopf Bifurcations
11.6 REDUCE
Further Reading
Problems for Chapter 11
12: Singular Perturbation Theory
12.1 Initial Value Problems I
12.1.1 Van der Pol Equation
12.1.2 Adiabatic Invariance
12.1.3 Averaging
12.2 Initial Value Problems II
12.2.1 Michaelis-Menten Enzyme Kinetics
12.2.2 Operational Amplifiers
12.2.3 Slow Selection in Population Genetics
12.3 Boundary Value Problems
12.3.1 Matched Asymptotic Expansions
12.3.2 Flame Fronts
12.3.3 Relaxation Dynamics
Further Reading
Problems for Chapter 12
References
Index
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