Applied Partial Differential Equations with Fourier Series and Boundary Value Problems: International Edition

This book emphasizes the physical interpretation of mathematical solutions and introduces applied mathematics while presenting differential equations. Coverage includes Fourier series, orthogonal functions, boundary value problems, Green’s functions, and transform methods.   This text is ideal for readers interested in science, engineering, and applied mathematics.

Table of Contents:
Normal 0 false false false 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             1.4.1 Prescribed Temperature             1.4.2 Insulated Boundaries 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             2.3.1 Introduction             2.3.2 Separation of Variables             2.3.3 Time-Dependent Equation             2.3.4 Boundary Value Problem             2.3.5 Product Solutions and the Principle of Superposition             2.3.6 Orthogonality of Sines             2.3.7 Formulation, Solution, and Interpretation of an Example             2.3.8 Summary 2.4 Worked Examples with the Heat Equation: Other Boundary Value Problems             2.4.1 Heat Conduction in a Rod with Insulated Ends             2.4.2 Heat Conduction in a Thin Circular Ring             2.4.3 Summary of Boundary Value Problems 2.5 Laplace’s Equation: Solutions and Qualitative Properties             2.5.1 Laplace’s Equation Inside a Rectangle             2.5.2 Laplace's Equation for a Circular Disk             2.5.3 Fluid Flow Past a Circular Cylinder (Lift)             2.5.4 Qualitative Properties of Laplace's Equation   3. Fourier Series 3.1 Introduction 3.2 Statement of Convergence Theorem 3.3 Fourier Cosine and Sine Series             3.3.1 Fourier Sine Series             3.3.2 Fourier Cosine Series             3.3.3 Representing f(x) by Both a Sine and Cosine Series             3.3.4 Even and Odd Parts             3.3.5 Continuous Fourier 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             4.6.1 Snell's Law of Refraction             4.6.2 Intensity (Amplitude) of Reflected and Refracted Waves             4.6.3 Total Internal Reflection   5. Sturm-Liouville Eigenvalue Problems 5.1 Introduction 5.2 Examples             5.2.1 Heat Flow in a Nonuniform Rod             5.2.2 Circularly Symmetric Heat Flow 5.3 Sturm-Liouville Eigenvalue Problems             5.3.1 General Classification             5.3.2 Regular Sturm-Liouville Eigenvalue Problem             5.3.3 Example and Illustration of Theorems 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             6.3.1 Introduction             6.3.2 A Partial Difference Equation             6.3.3 Computations             6.3.4 Fourier-von Neumann Stability Analysis             6.3.5 Separation of Variables for Partial Difference Equations and Analytic Solutions of Ordinary Difference Equations             6.3.6 Matrix Notation             6.3.7 Nonhomogeneous Problems             6.3.8 Other Numerical Schemes             6.3.9 Other Types of Boundary Conditions 6.4 Two-Dimensional Heat Equation 6.5 Wave Equation 6.6 Laplace's Equation 6.7 Finite Element Method             6.7.1 Approximation with Nonorthogonal Functions (Weak Form of the Partial Differential Equation)             6.7.2 The Simplest Triangular Finite Elements   7. Higher Dimensional Partial Differential Equations 7.1 Introduction 7.2 Separation of the Time Variable             7.2.1 Vibrating Membrane: Any Shape             7.2.2 Heat Conduction: Any Region             7.2.3 Summary 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             7.6.1 Rayleigh Quotient             7.6.2 Time-Dependent Heat Equation and Laplace's Equation 7.7 Vibrating Circular Membrane and Bessel Functions             7.7.1 Introduction             7.7.2 Separation of Variables             7.7.3 Eigenvalue Problems (One Dimensional)             7.7.4 Bessel's Differential Equation             7.7.5 Singular Points and Bessel's Differential Equation             7.7.6 Bessel Functions and Their Asymptotic Properties (near z = 0)             7.7.7 Eigenvalue Problem Involving Bessel Functions             7.7.8 Initial Value Problem for a Vibrating Circular Membrane             7.7.9 Circularly Symmetric Case 7.8 More on Bessel Functions             7.8.1 Qualitative Properties of Bessel Functions             7.8.2 Asymptotic Formulas for the Eigenvalues             7.8.3 Zeros of Bessel Functions and Nodal Curves             7.8.4 Series Representation of Bessel Functions 7.9 Laplace’s Equation in a Circular Cylinder             7.9.1 Introduction             7.9.2 Separation of Variables             7.9.3 Zero Temperature on the Lateral Sides and on the Bottom or Top             7.9.4 Zero Temperature on the Top and Bottom             7.9.5 Modified Bessel Functions 7.10 Spherical Problems and Legendre Polynomials             7.10.1 Introduction             7.10.2 Separation of Variables and One-Dimensional Eigenvalue Problems             7.10.3 Associated Legendre Functions and Legendre Polynomials             7.10.4 Radial Eigenvalue Problems             7.10.5 Product Solutions, Modes of Vibration, and the Initial Value Problem             7.10.6 Laplace's Equation Inside a Spherical Cavity   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             9.3.1 One-Dimensional Steady-State Heat Equation             9.3.2 The Method of Variation of Parameters             9.3.3 The Method of Eigenfunction Expansion for Green's Functions             9.3.4 The Dirac Delta Function and Its Relationship to Green's Functions             9.3.5 Nonhomogeneous Boundary Conditions             9.3.6 Summary 9.4 Fredholm Alternative and Generalized Green’s Functions             9.4.1 Introduction             9.4.2 Fredholm Alternative             9.4.3 Generalized Green's Functions 9.5 Green’s Functions for Poisson’s Equation             9.5.1 Introduction             9.5.2 Multidimensional Dirac Delta Function and Green's Functions             9.5.3 Green's Functions by the Method of Eigenfunction Expansion and the Fredholm Alternative             9.5.4 Direct Solution of Green's Functions (One-Dimensional Eigenfunctions)             9.5.5 Using Green's Functions for Problems with Nonhomogeneous Boundary Conditions             9.5.6 Infinite Space Green's Functions             9.5.7 Green's Functions for Bounded Domains Using Infinite Space Green's Functions             9.5.8 Green's Functions for a Semi-Infinite Plane (y > 0) Using Infinite Space Green's Functions: The Method of Images             9.5.9 Green's Functions for a Circle: The Method of Images 9.6 Perturbed Eigenvalue Problems             9.6.1 Introduction             9.6.2 Mathematical Example             9.6.3 Vibrating Nearly Circular Membrane 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             10.3.1 Motivation from Fourier Series Identity             10.3.2 Fourier Transform             10.3.3 Inverse Fourier Transform of a Gaussian 10.4 Fourier Transform and the Heat Equation             10.4.1 Heat Equation             10.4.2 Fourier Transforming the Heat Equation: Transforms of Derivatives             10.4.3 Convolution Theorem             10.4.4 Summary of Properties of the Fourier Transform 10.5 Fourier Sine and Cosine Transforms: The Heat Equation on Semi-Infinite Intervals             10.5.1 Introduction             10.5.2 Heat Equation on a Semi-Infinite Interval I             10.5.3 Fourier Sine and Cosine Transforms             10.5.4 Transforms of Derivatives             10.5.5 Heat Equation on a Semi-Infinite Interval II             10.5.6 Tables of Fourier Sine and Cosine Transforms 10.6 Worked Examples Using Transforms             10.6.1 One-Dimensional Wave Equation on an Infinite Interval             10.6.2 Laplace's Equation in a Semi-Infinite Strip             10.6.3 Laplace's Equation in a Half-Plane             10.6.4 Laplace's Equation in a Quarter-Plane             10.6.5 Heat Equation in a Plane (Two-Dimensional Fourier Transforms)             10.6.6 Table of Double-Fourier 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             11.2.1 Introduction             11.2.2 Green's Formula             11.2.3 Reciprocity             11.2.4 Using the Green's Function             11.2.5 Green's Function for the Wave Equation             11.2.6 Alternate Differential Equation for the Green's Function             11.2.7 Infinite Space Green's Function for the One-Dimensional Wave Equation and d’Alembert's Solution             11.2.8 Infinite Space Green's Function for the Three-Dimensional Wave Equation (Huygens’ Principle)             11.2.9 Two-Dimensional Infinite Space Green's Function             11.2.10 Summary 11.3 Green’s Functions for the Heat Equation             11.3.1 Introduction             11.3.2 Non-Self-Adjoint Nature of the Heat Equation             11.3.3 Green's Formula             11.3.4 Adjoint Green's Function             11.3.5 Reciprocity             11.3.6 Representation of the Solution Using Green's Functions             11.3.7 Alternate Differential Equation for the Green's Function             11.3.8 Infinite Space Green's Function for the Diffusion Equation             11.3.9 Green's Function for the Heat Equation (Semi-Infinite Domain)             11.3.10 Green's Function for the Heat Equation (on a Finite Region)   12. The Method of Characteristics for Linear and Quasilinear Wave Equations 12.1 Introduction 12.2 Characteristics for First-Order Wave Equations             12.2.1 Introduction             12.2.2 Method of Characteristics for First-Order Partial Differential Equations 12.3 Method of Characteristics for the One-Dimensional Wave Equation             12.3.1 General Solution             12.3.2 Initial Value Problem (Infinite Domain)             12.3.3 D’alembert’s Solution 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             12.6.1 Method of Characteristics             12.6.2 Traffic Flow             12.6.3 Method of Characteristics (Q=0)             12.6.4 Shock Waves             12.6.5 Quasilinear Example 12.7 First-Order Nonlinear Partial Differential Equations             12.7.1 Eikonal Equation Derived from the Wave Equation             12.7.2 Solving the Eikonal Equation in Uniform Media and Reflected Waves             12.7.3 First-Order Nonlinear Partial Differential Equations   13. Laplace Transform Solution of Partial Differential Equations 13.1 Introduction 13.2 Properties of the Laplace Transform             13.2.1 Introduction             13.2.2 Singularities of the Laplace Transform             13.2.3 Transforms of Derivatives             13.2.4 Convolution Theorem 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             14.2.1 Traveling Waves and the Dispersion Relation             14.2.2 Group Velocity I 14.3 Wave Guides             14.3.1 Response to Concentrated Periodic Sources with Frequency ωf             14.3.2 Green's Function If Mode Propagates             14.3.3 Green's Function If Mode Does Not Propagate             14.3.4 Design Considerations 14.4 Fiber Optics 14.5 Group Velocity II and the Method of Stationary Phase             14.5.1 Method of Stationary Phase             14.5.2 Application to Linear Dispersive Waves 14.6 Slowly Varying Dispersive Waves (Group Velocity and Caustics)             14.6.1 Approximate Solutions of Dispersive Partial Differential Equations             14.6.2 Formation of a Caustic 14.7 Wave Envelope Equations (Concentrated Wave Number)             14.7.1 Schrödinger Equation             14.7.2 Linearized Korteweg-de Vries Equation             14.7.3 Nonlinear Dispersive Waves: Korteweg-deVries Equation             14.7.4 Solitons and Inverse Scattering             14.7.5 Nonlinear Schrödinger Equation 14.8 Stability and Instability             14.8.1 Brief Ordinary Differential Equations and Bifurcation Theory             14.8.2 Elementary Example of a Stable Equilibrium for a Partial Differential Equation             14.8.3 Typical Unstable Equilibrium for a Partial Differential Equation and Pattern Formation             14.8.4 Ill posed Problems             14.8.5 Slightly Unstable Dispersive Waves and the Linearized Complex Ginzburg-Landau Equation             14.8.6 Nonlinear Complex Ginzburg-Landau Equation             14.8.7 Long Wave Instabilities             14.8.8 Pattern Formation for Reaction-Diffusion Equations and the Turing Instability 14.9 Singular Perturbation Methods: Multiple Scales             14.9.1 Ordinary Differential Equation: Weakly Nonlinearly Damped Oscillator             14.9.2 Ordinary Differential Equation: Slowly Varying Oscillator             14.9.3 Slightly Unstable Partial Differential Equation on Fixed Spatial Domain             14.9.4 Slowly Varying Medium for the Wave Equation             14.9.5 Slowly Varying Linear Dispersive Waves (Including Weak Nonlinear Effects) 14.10 Singular Perturbation Methods: Boundary Layers Method of Matched Asymptotic Expansions             14.10.1 Boundary Layer in an Ordinary Differential Equation             14.10.2 Diffusion of a Pollutant Dominated by Convection   Bibliography Answers to Starred Exercises Index