Applied Partial Differential Equations with Fourier Series and Boundary Value Problems (Classic Version)

About our author Richard Haberman is Professor of Mathematics at Southern Methodist University, having previously taught at The Ohio State University, Rutgers University, and the University of California at San Diego. He received S.B. and Ph.D. degrees in applied mathematics from the Massachusetts Institute of Technology. He has supervised six Ph.D. students at SMU. His research has been funded by NSF and AFOSR. His research in applied mathematics has been published in prestigious international journals and include research on nonlinear wave motion (shocks, solitons, dispersive waves, caustics), nonlinear dynamical systems (bifurcations, homoclinic transitions, chaos), singular perturbation methods (partial differential equations, matched asymptotic expansions, boundary layers) and mathematical models (fluid dynamics, fiber optics). He is a member of the Society for Industrial and Applied Mathematics and the American Mathematical Society. He has taught a wide range of undergraduate and graduate mathematics. He has published undergraduate texts on Mathematical Models (Mechanical Vibrations, Population Dynamics, and Traffic Flow) and Ordinary Differential Equations.

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