Elementary Applied Partial Differential Equations

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Cover Title page PREFACE CHAPTER 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 Appendix 1.5 Review of Gradient and a Derivation of Fourier's Law of Heat Conduction CHAPTER 2 METHOD OF SEPARATION OF VARIABLES 2.1 Introduction 2.2 Linearity 2.3 Heat Equation with Zero Temperatures at Finite Ends Appendix 2.3 Orthogonality of Functions 2.4 Worked Examples with the Heat Equation (Other Boundary Value Problems) 2.5 Laplace's Equation: Solutions and Qualitative Properties CHAPTER 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 CHAPTER 4 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 CHAPTER 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 Appendix 5.5 Matrix Eigenvalue Problem and Orthogonality of Eigenvectors 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 CHAPTER 6 PARTIAL DIFFERENTIAL EQUATIONS WITH AT LEAST THREE INDEPENDENT VARIABLES 6.1 Introduction 6.2 Separation of the Time Variable 6.3 Vibrating Rectangular Membrane Appendix 6.3 Outline of Alternative Method to Separate Variables 6.4 Statements and Illustrations of Theorems for the Eigenvalue Problem ∇²φ+λφ = 0 6.5 Self-Adjoint Operators and Multidimensional Eigenvalue Problems Appendix 6.5 Gram-Schmidt Method 6.6 Rayleigh Quotient 6.7 Vibrating Circular Membrane and Bessel Functions 6.8 More on Bessel Functions 6.9 Laplace's Equation in a Circular Cylinder CHAPTER 7 NONHOMOGENEOUS PROBLEMS 7.1 Introduction 7.2 Heat Flow with Sources and Nonhomogeneous Boundary Conditions 7.3 Method of Eigenfunction Expansion with Homogeneous Boundary Conditions (Differentiating Series of Eigenfunctions) 7.4 Method of Eigenfunction Expansion Using Green's Formula (with or without Homogeneous Boundary Conditions) 7.5 Forced Vibrating Membranes and Resonance 7.6 Poisson's Equation CHAPTER 8 GREEN'S FUNCTIONS FOR TIME-INDEPENDENT PROBLEMS 8.1 Introduction 8.2 One-Dimensional Heat Equation 8.3 Green's Functions for Boundary Value Problems for Ordinary Differentiai Equations Appendix 8.3 Establishing Green's Formula with Dirac Delta Functions 8.4 Fredholm Alternative and Modified Green's Functions 8.5 Green's Functions for Poisson's Equation 8.6 Perturbed Eigenvalue Problems 8.7 Summary CHAPTER 9 INFINITE DOMAIN PROBLEMS - FOURIER TRANSFORM SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS 9.1 Introduction 9.2 Heat Equation on an Infinite Domain 9.3 Complex Form of Fourier Series 9.4 Fourier Transform Pair Appendix 9.4 Derivation of the Inverse Fourier Transform of a Gaussian 9.5 Fourier Transform and the Heat Equation 9.6 Fourier Sine and Cosine Transforms - The Heat Equation on Semi-infinite Intervals 9.7 Worked Examples Using Transforms CHAPTER 10 GREEN'S FUNCTIONS FOR TIME-DEPENDENT PROBLEMS 10.1 Introduction 10.2 Green's Functions for the Wave Equation 10.3 Green's Functions for the Heat Equation CHAPTER 11 THE METHOD OF CHARACTERISTICS FOR LlNEAR AND QUASI-LiNEAR WAVE EQUATIONS 11.1 Introduction 11.2 Characteristics for First-Order Wave Equations 11.3 Method of Characteristics for the One-Dimensional Wave Equation 11.4 Semi-infinite Strings and Reflections 11.5 Method of Characteristics for a Vibrating String of Fixed Length 11.6 The Method of Characteristics for Quasi-Linear Partial Differentiai Equations CHAPTER 12 A BRIEF INTRODUCTION TO LAPLACE TRANSFORM. SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS 12.1 Introduction 12.2 Elementary Properties of the Laplace Transform 12.3 Green's Functions for Initial Value Problems for Ordinary Differentiai Equations 12.4 An Elementary Signal Problem for the Wave Equation 12.5 A Signal Problem for a Vibrating String of Finite Length 12.6 The Wave Equation and Its Green's Function 12.7 Inversion of Laplace Transforms Using Contour Integrais in the Complex Plane 12.8 Solving the Wave Equation Using Laplace Transforms (with Complex Variables) CHAPTER 13 AN ELEMENTARY DISCUSSION OF FINITE DIFFERENCE. NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS 13.1 Introduction 13.2 Finite Differences and Truncated Taylor Series 13.3 Heat Equation 13.4 Two-Dimensional Heat Equation 13.5 Wave Equation 13.6 Laplace's Equation SELECTED ANSWERS TO STARRED EXERCISES BIBLIOGRAPHY INDEX (missing)