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 Introduction1.2 Derivation of the Conduction of Heat in a One-Dimensional Rod1.3 Boundary Conditions1.4 Equilibrium Temperature Distribution1.5 Derivation of the Heat Equation in Two or Three Dimensions2. Method of Separation of Variables 2.1 Introduction2.2 Linearity2.3 Heat Equation with Zero Temperatures at Finite Ends2.4 Worked Examples with the Heat Equation: Other Boundary Value Problems2.5 Laplace's Equation: Solutions and Qualitative Properties3. Fourier Series 3.1 Introduction3.2 Statement of Convergence Theorem3.3 Fourier Cosine and Sine Series3.4 Term-by-Term Differentiation of Fourier Series3.5 Term-By-Term Integration of Fourier Series3.6 Complex Form of Fourier Series4. Wave Equation: Vibrating Strings and Membranes 4.1 Introduction4.2 Derivation of a Vertically Vibrating String4.3 Boundary Conditions4.4 Vibrating String with Fixed Ends4.5 Vibrating Membrane4.6 Reflection and Refraction of Electromagnetic (Light) and Acoustic (Sound) Waves5. Sturm-Liouville Eigenvalue Problems 5.1 Introduction5.2 Examples5.3 Sturm-Liouville Eigenvalue Problems5.4 Worked Example: Heat Flow in a Nonuniform Rod without Sources5.5 Self-Adjoint Operators and Sturm-Liouville Eigenvalue Problems5.6 Rayleigh Quotient5.7 Worked Example: Vibrations of a Nonuniform String5.8 Boundary Conditions of the Third Kind5.9 Large Eigenvalues (Asymptotic Behavior)5.10 Approximation Properties6. Finite Difference Numerical Methods for Partial Differential Equations 6.1 Introduction6.2 Finite Differences and Truncated Taylor Series6.3 Heat Equation6.4 Two-Dimensional Heat Equation6.5 Wave Equation6.6 Laplace's Equation6.7 Finite Element Method7. Higher Dimensional Partial Differential Equations 7.1 Introduction7.2 Separation of the Time Variable7.3 Vibrating Rectangular Membrane7.4 Statements and Illustrations of Theorems for the Eigenvalue Problem ∇2φ + λφ = 07.5 Green's Formula, Self-Adjoint Operators and Multidimensional Eigenvalue Problems7.6 Rayleigh Quotient and Laplace's Equation7.7 Vibrating Circular Membrane and Bessel Functions7.8 More on Bessel Functions7.9 Laplace's Equation in a Circular Cylinder7.10 Spherical Problems and Legendre Polynomials8. Nonhomogeneous Problems 8.1 Introduction8.2 Heat Flow with Sources and Nonhomogeneous Boundary Conditions8.3 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 Resonance8.6 Poisson's Equation9. Green's Functions for Time-Independent Problems 9.1 Introduction9.2 One-dimensional Heat Equation9.3 Green's Functions for Boundary Value Problems for Ordinary Differential Equations9.4 Fredholm Alternative and Generalized Green's Functions9.5 Green's Functions for Poisson's Equation9.6 Perturbed Eigenvalue Problems9.7 Summary10. Infinite Domain Problems: Fourier Transform Solutions of Partial Differential Equations 10.1 Introduction10.2 Heat Equation on an Infinite Domain10.3 Fourier Transform Pair10.4 Fourier Transform and the Heat Equation10.5 Fourier Sine and Cosine Transforms: The Heat Equation on Semi-Infinite Intervals10.6 Worked Examples Using Transforms10.7 Scattering and Inverse Scattering11. Green's Functions for Wave and Heat Equations 11.1 Introduction11.2 Green's Functions for the Wave Equation11.3 Green's Functions for the Heat Equation12. The Method of Characteristics for Linear and Quasilinear Wave Equations 12.1 Introduction12.2 Characteristics for First-Order Wave Equations12.3 Method of Characteristics for the One-Dimensional Wave Equation12.4 Semi-Infinite Strings and Reflections12.5 Method of Characteristics for a Vibrating String of Fixed Length12.6 The Method of Characteristics for Quasilinear Partial Differential Equations12.7 First-Order Nonlinear Partial Differential Equations13. Laplace Transform Solution of Partial Differential Equations 13.1 Introduction13.2 Properties of the Laplace Transform13.3 Green's Functions for Initial Value Problems for Ordinary Differential Equations13.4 A Signal Problem for the Wave Equation13.5 A Signal Problem for a Vibrating String of Finite Length13.6 The Wave Equation and its Green's Function13.7 Inversion of Laplace Transforms Using Contour Integrals in the Complex Plane13.8 Solving the Wave Equation Using Laplace Transforms (with Complex Variables)14. Dispersive Waves: Slow Variations, Stability, Nonlinearity, and Perturbation Methods 14.1 Introduction14.2 Dispersive Waves and Group Velocity14.3 Wave Guides14.4 Fiber Optics14.5 Group Velocity II and the Method of Stationary Phase14.7 Wave Envelope Equations (Concentrated Wave Number)14.7.1 Schrödinger Equation14.8 Stability and Instability14.9 Singular Perturbation Methods: Multiple Scales