## Description

**Solutions – Applied Partial Differential Equations with Fourier Series and Boundary Value Problems (Classic Version), 5th edition by Richard Haberman**

**Table of Contents**

### 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
*∇*^{2}*φ*+*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

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