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.