Differential Equations of Applied Mathematics

CHAPTER 1. FINITE SYSTEMS

1.1. Vectors and Linear Spaces
1.2. Matrices and Linear Transformations
1.3. Ordinary Differential Systems
1.4. Finite Mechanical Systems
1.5. Lagrange's Equations and Hamilton's Principle
1.6. Systems with Constant Coefficients
1.7. The Response Matrix and Distributions
1.8. Two-Point Boundary Conditions

CHAPTER 2. DISTRIBUTIONS AND WA YES

2.1. Equations in Two Independent Variables
2.2. Wave Motion of a String
2.3. Reflection of Waves
2.4. Theory of Distributions
2.5. Applications to the Initial Problem
2.6. The Separation of Variables
2.7. Fourier Series

CHAPTER 3. PARABOLIC EQUATIONS AND FOURIER INTEGRALS

3.1. The Heat Flow Equation
3.2. Heat Flow on a Finite Interval
3.3. Fourier Integral Transforms 
3.4. Diffusion on an Infinite Interval
3.5. Semi-Infinite Intervals
3.6. Fourier Transforms of Distributions
3.7. Finite Difference Calculations

CHAPTER 4. LAPLACE'S EQUATION AND COMPLEX VARIABLES

4.1. Mathematical and Physical Applications
4.2. Boundary Value Problems for Harmonic Functions
4.3. Circular Harmonics 
4.4. Rectangular Harmonics
4.5. Half-Plane Problems
4.6. Complex Integrals
4.7. Fourier and Laplace Transforms
4.8. The Finite Difference Laplace Equation

CHAPTER 5. EQUATIONS OF MOTION 

5.1. Vibrations of a Membrane 
5.2. Lateral Vibration of Rods and Plates 
5.3. Integral Theorems and Vector Calculus 
5.4. Equations of Motion of an Elastic Solid
5.5. Motion of a Fluid 
5.6. Equations of the Electromagnetic Field 
5.7. Equations of Quantum Mechanics 

CHAPTER 6. GENERAL THEORY OF EIGENVALUES AND EIGENFUNCTIONS 

6.1. The Minimum Problem 
6.2. Sequences of Eigenvalues and Eigenfunctions
6.3. Variational Properties of Eigenvalues and Eigenfunctions
6.4. Eigenfunction Expansions
6.5. The Rayleigh-Ritz Approximation Method
6.6. On the Separation of Variables
6.7. Series Expansions and Integral Transforms 

CHAPTER 7. GREEN'S FUNCTIONS 

7.1. Inverses of Differential Operators
7.2. Examples of Green's Functions 
7.3. The Neumann and Robin Functions
7.4. Differential and Integral Equations
7.5. Source Functions for Parabolic Equations 
7.6. Convergence of Series of Distributions

CHAPTER 8. CYLINDRICAL EIGENFUNCTIONS 

8.1. Bessel Functions
8.2. Eigenfunctions for Finite Regions
8.3. The Fourier-Bessel Series
8.4. The Green's Function
8.5. Functions of Large Argument
8.6. Diffraction by a Cylinder
8.7. Modified Bessel Functions
8.8. The Hankel and Weber Formulas

CHAPTER 9. SPHERICAL EIGENFUNCTIONS 

9.1. Legendre Functions
9.2. Eigenfunctions of the Spherical Surface
9.3. Eigenfunctions for the Solid Sphere
9.4. Diffraction by a Sphere: Addition Theorem
9.5. Interior and Exterior Expansions
9.6. Functions of Nonintegral Order

CHAPTER 10. WAVE PROPAGATION IN SPACE

10.1. Characteristic Surfaces
10.2. Source Function for the Wave Equation
10.3. Applications. Huyghens' Premise
10.4. Electromagnetic and Elastic Waves
10.5. Wave Fronts and Rays
10.6. Reflection and Diffraction