1. Heat Equation

• 1.1 Introduction
• 1.2 Derivation of the Conduction of Heat in a One-Dimensional Rod
• 1.3 Boundary Conditions
• 1.4 Equilibrium Temperature Distribution
• 1.5 Derivation of the Heat Equation in Two or Three Dimensions

2. Method of Separation of Variables

• 2.1 Introduction
• 2.2 Linearity
• 2.3 Heat Equation with Zero Temperatures at Finite Ends
• 2.4 Worked Examples with the Heat Equation: Other Boundary Value Problems
• 2.5 Laplace’s Equation: Solutions and Qualitative Properties

3. Fourier Series

• 3.1 Introduction
• 3.2 Statement of Convergence Theorem
• 3.3 Fourier Cosine and Sine Series
• 3.4 Term-by-Term Differentiation of Fourier Series
• 3.5 Term-By-Term Integration of Fourier Series
• 3.6 Complex Form of Fourier Series

4. Wave Equation: Vibrating Strings and Membranes

• 4.1 Introduction
• 4.2 Derivation of a Vertically Vibrating String
• 4.3 Boundary Conditions
• 4.4 Vibrating String with Fixed Ends
• 4.5 Vibrating Membrane
• 4.6 Reflection and Refraction of Electromagnetic (Light) and Acoustic (Sound) Waves

5. Sturm-Liouville Eigenvalue Problems

• 5.1 Introduction
• 5.2 Examples
• 5.3 Sturm-Liouville Eigenvalue Problems
• 5.4 Worked Example: Heat Flow in a Nonuniform Rod without Sources
• 5.5 Self-Adjoint Operators and Sturm-Liouville Eigenvalue Problems
• 5.6 Rayleigh Quotient
• 5.7 Worked Example: Vibrations of a Nonuniform String
• 5.8 Boundary Conditions of the Third Kind
• 5.9 Large Eigenvalues (Asymptotic Behavior)
• 5.10 Approximation Properties

6. Finite Difference Numerical Methods for Partial Differential Equations

• 6.1 Introduction
• 6.2 Finite Differences and Truncated Taylor Series
• 6.3 Heat Equation
• 6.4 Two-Dimensional Heat Equation
• 6.5 Wave Equation
• 6.6 Laplace’s Equation
• 6.7 Finite Element Method

7. Higher Dimensional Partial Differential Equations

• 7.1 Introduction
• 7.2 Separation of the Time Variable
• 7.3 Vibrating Rectangular Membrane
• 7.4 Statements and Illustrations of Theorems for the Eigenvalue Problem ∇²φ + λφ = 0
• 7.5 Green’s Formula, Self-Adjoint Operators and Multidimensional Eigenvalue Problems
• 7.6 Rayleigh Quotient and Laplace’s Equation
• 7.7 Vibrating Circular Membrane and Bessel Functions
• 7.8 More on Bessel Functions
• 7.9 Laplace’s Equation in a Circular Cylinder
• 7.10 Spherical Problems and Legendre Polynomials

8. Nonhomogeneous Problems

• 8.1 Introduction
• 8.2 Heat Flow with Sources and Nonhomogeneous Boundary Conditions
• 8.3 Method of Eigenfunction Expansion with Homogeneous Boundary Conditions (Differentiating Series of Eigenfunctions)
• 8.4 Method of Eigenfunction Expansion Using Green’s Formula (With or Without Homogeneous Boundary Conditions)
• 8.5 Forced Vibrating Membranes and Resonance
• 8.6 Poisson’s Equation

9. Green’s Functions for Time-Independent Problems

• 9.1 Introduction
• 9.2 One-dimensional Heat Equation
• 9.3 Green’s Functions for Boundary Value Problems for Ordinary Differential Equations
• 9.4 Fredholm Alternative and Generalized Green’s Functions
• 9.5 Green’s Functions for Poisson’s Equation
• 9.6 Perturbed Eigenvalue Problems
• 9.7 Summary

10. Infinite Domain Problems: Fourier Transform Solutions of Partial Differential Equations

• 10.1 Introduction
• 10.2 Heat Equation on an Infinite Domain
• 10.3 Fourier Transform Pair
• 10.4 Fourier Transform and the Heat Equation
• 10.5 Fourier Sine and Cosine Transforms: The Heat Equation on Semi-Infinite Intervals
• 10.6 Worked Examples Using Transforms
• 10.7 Scattering and Inverse Scattering

11. Green’s Functions for Wave and Heat Equations

• 11.1 Introduction
• 11.2 Green’s Functions for the Wave Equation
• 11.3 Green’s Functions for the Heat Equation

12. The Method of Characteristics for Linear and Quasilinear Wave Equations

• 12.1 Introduction
• 12.2 Characteristics for First-Order Wave Equations
• 12.3 Method of Characteristics for the One-Dimensional Wave Equation
• 12.4 Semi-Infinite Strings and Reflections
• 12.5 Method of Characteristics for a Vibrating String of Fixed Length
• 12.6 The Method of Characteristics for Quasilinear Partial Differential Equations
• 12.7 First-Order Nonlinear Partial Differential Equations

13. Laplace Transform Solution of Partial Differential Equations

• 13.1 Introduction
• 13.2 Properties of the Laplace Transform
• 13.3 Green’s Functions for Initial Value Problems for Ordinary Differential Equations
• 13.4 A Signal Problem for the Wave Equation
• 13.5 A Signal Problem for a Vibrating String of Finite Length
• 13.6 The Wave Equation and its Green’s Function
• 13.7 Inversion of Laplace Transforms Using Contour Integrals in the Complex Plane
• 13.8 Solving the Wave Equation Using Laplace Transforms (with Complex Variables)

14. Dispersive Waves: Slow Variations, Stability, Nonlinearity, and Perturbation Methods

• 14.1 Introduction
• 14.2 Dispersive Waves and Group Velocity
• 14.3 Wave Guides
• 14.4 Fiber Optics
• 14.5 Group Velocity II and the Method of Stationary Phase
• 14.7 Wave Envelope Equations (Concentrated Wave Number)
• 14.7.1 Schrödinger Equation
• 14.8 Stability and Instability
• 14.9 Singular Perturbation Methods: Multiple Scales