Book:Tyn Myint-U/Linear Partial Differential Equations for Scientists and Engineers/Fourth Edition

Subject Matter

 * Partial Differential Equations

Contents

 * Preface to the Fourth Edition
 * Preface to the Third Edition


 * 1 Introduction
 * 1.1 Brief Historical Comments
 * 1.2 Basic Concepts and De?nitions
 * 1.3 Mathematical Problems
 * 1.4 Linear Operators
 * 1.5 Superposition Principle
 * 1.6 Exercises


 * 2 First-Order, Quasi-Linear Equations and Method of Characteristics
 * 2.1 Introduction
 * 2.2 Classification of First-Order Equations
 * 2.3 Construction of a First-Order Equation
 * 2.4 Geometrical Interpretation of a First-Order Equation
 * 2.5 Method of Characteristics and General Solutions
 * 2.6 Canonical Forms of First-Order Linear Equations
 * 2.7 Method of Separation of Variables
 * 2.8 Exercises


 * 3 Mathematical Models
 * 3.1 Classical Equations
 * 3.2 The Vibrating String
 * 3.3 The Vibrating Membrane
 * 3.4 Waves in an Elastic Medium
 * 3.5 Conduction of Heat in Solids
 * 3.6 The Gravitational Potential
 * 3.7 Conservation Laws and The Burgers Equation
 * 3.8 The Schrödinger and the Korteweg–de Vries Equations
 * 3.9 Exercises


 * 4 Classification of Second-Order Linear Equations
 * 4.1 Second-Order Equations in Two Independent Variables
 * 4.2 Canonical Forms
 * 4.3 Equations with Constant Coefficients
 * 4.4 General Solutions
 * 4.5 Summary and Further Simplification
 * 4.6 Exercises


 * 5 The Cauchy Problem and Wave Equations
 * 5.1 The Cauchy Problem
 * 5.2 The Cauchy–Kowalewskaya Theorem
 * 5.3 Homogeneous Wave Equations
 * 5.4 Initial Boundary-Value Problems
 * 5.5 Equations with Nonhomogeneous Boundary Conditions
 * 5.6 Vibration of Finite String with Fixed Ends
 * 5.7 Nonhomogeneous Wave Equations
 * 5.8 The Riemann Method
 * 5.9 Solution of the Goursat Problem
 * 5.10 Spherical Wave Equation
 * 5.11 Cylindrical Wave Equation
 * 5.12 Exercises


 * 6 Fourier Series and Integrals with Applications
 * 6.1 Introduction
 * 6.2 Piecewise Continuous Functions and Periodic Functions
 * 6.3 Systems of Orthogonal Functions
 * 6.4 Fourier Series
 * 6.5 Convergence of Fourier Series
 * 6.6 Examples and Applications of Fourier Series
 * 6.7 Examples and Applications of Cosine and Sine Fourier Series
 * 6.8 Complex Fourier Series
 * 6.9 Fourier Series on an Arbitrary Interval
 * 6.10 The Riemann–Lebesgue Lemma and Pointwise Convergence Theorem
 * 6.11 Uniform Convergence, Differentiation, and Integration
 * 6.12 Double Fourier Series
 * 6.13 Fourier Integrals
 * 6.14 Exercises


 * 7 Method of Separation of Variables
 * 7.1 Introduction
 * 7.2 Separation of Variables
 * 7.3 The Vibrating String Problem
 * 7.4 Existence and Uniqueness of Solution of the Vibrating String Problem
 * 7.5 The Heat Conduction Problem
 * 7.6 Existence and Uniqueness of Solution of the Heat Conduction Problem
 * 7.7 The Laplace and Beam Equations
 * 7.8 Nonhomogeneous Problems
 * 7.9 Exercises


 * 8 Eigenvalue Problems and Special Functions
 * 8.1 Sturm–Liouville Systems
 * 8.2 Eigenvalues and Eigenfunctions
 * 8.3 Eigenfunction Expansions
 * 8.4 Convergence in the Mean
 * 8.5 Completeness and Parseval's Equality
 * 8.6 Bessel's Equation and Bessel's Function
 * 8.7 Adjoint Forms and Lagrange Identity
 * 8.8 Singular Sturm–Liouville Systems
 * 8.9 Legendre's Equation and Legendre's Function
 * 8.10 Boundary–Value Problems Involving Ordinary Differential Equations
 * 8.11 Green's Functions for Ordinary Differential Equations
 * 8.12 Construction of Green's Functions
 * 8.13 The Schrödinger Equation and Linear Harmonic Oscillator
 * 8.14 Exercises


 * 9 Boundary-Value Problems and Applications
 * 9.1 Boundary-Value Problems
 * 9.2 Maximum and Minimum Principles
 * 9.3 Uniqueness and Continuity Theorems
 * 9.4 Dirichlet Problem for a Circle
 * 9.5 Dirichlet Problem for a Circular Annulus
 * 9.6 Neumann Problem for a Circle
 * 9.7 Dirichlet Problem for a Rectangle
 * 9.8 Dirichlet Problem Involving the Poisson Equation
 * 9.9 The Neumann Problem for a Rectangle
 * 9.10 Exercises


 * 10 Higher-Dimensional Boundary-Value Problems
 * 10.1 Introduction
 * 10.2 Dirichlet Problem for a Cube
 * 10.3 Dirichlet Problem for a Cylinder
 * 10.4 Dirichlet Problem for a Sphere
 * 10.5 Three-Dimensional Wave and Heat Equations
 * 10.6 Vibrating Membrane
 * 10.7 Heat Flow in a Rectangular Plate
 * 10.8 Waves in Three Dimensions
 * 10.9 Heat Conduction in a Rectangular Volume
 * 10.10 The Schrödinger Equation and the Hydrogen Atom
 * 10.11 Method of Eigenfunctions and Vibration of Membrane
 * 10.12 Time-Dependent Boundary-Value Problems
 * 10.13 Exercises


 * 11 Green's Functions and Boundary-Value Problems
 * 11.1 Introduction
 * 11.2 The Dirac Delta Function
 * 11.3 Properties of Green's Functions
 * 11.4 Method of Green's Functions
 * 11.5 Dirichlet's Problem for the Laplace Operator
 * 11.6 Dirichlet's Problem for the Helmholtz Operator
 * 11.7 Method of Images
 * 11.8 Method of Eigenfunctions
 * 11.9 Higher-Dimensional Problems
 * 11.10 Neumann Problem
 * 11.11 Exercises


 * 12 Integral Transform Methods with Applications
 * 12.1 Introduction
 * 12.2 Fourier Transforms
 * 12.3 Properties of Fourier Transforms
 * 12.4 Convolution Theorem of the Fourier Transform
 * 12.5 The Fourier Transforms of Step and Impulse Functions
 * 12.6 Fourier Sine and Cosine Transforms
 * 12.7 Asymptotic Approximation of Integrals by Stationary Phase Method
 * 12.8 Laplace Transforms
 * 12.9 Properties of Laplace Transforms
 * 12.10 Convolution Theorem of the Laplace Transform
 * 12.11 Laplace Transforms of the Heaviside and Dirac Delta Functions
 * 12.12 Hankel Transforms
 * 12.13 Properties of Hankel Transforms and Applications
 * 12.14 Mellin Transforms and their Operational Properties
 * 12.15 Finite Fourier Transforms and Applications
 * 12.16 Finite Hankel Transforms and Applications
 * 12.17 Solution of Fractional Partial Differential Equations
 * 12.18 Exercises


 * 13 Nonlinear Partial Differential Equations with Applications
 * 13.1 Introduction
 * 13.2 One-Dimensional Wave Equation and Method of Characteristics
 * 13.3 Linear Dispersive Waves
 * 13.4 Nonlinear Dispersive Waves and Whitham's Equations
 * 13.5 Nonlinear Instability
 * 13.6 The Traffic Flow Model
 * 13.7 Flood Waves in Rivers
 * 13.8 Riemann's Simple Waves of Finite Amplitude
 * 13.9 Discontinuous Solutions and Shock Waves
 * 13.10 Structure of Shock Waves and Burgers' Equation
 * 13.11 The Korteweg–de Vries Equation and Solitons
 * 13.12 The Nonlinear Schrödinger Equation and Solitary Waves
 * 13.13 The Lax Pair and the Zakharov and Shabat Scheme
 * 13.14 Exercises


 * 14 Numerical and Approximation Methods
 * 14.1 Introduction
 * 14.2 Finite Difference Approximations, Convergence, and Stability
 * 14.3 Lax–Wendroff Explicit Method
 * 14.4 Explicit Finite Difference Methods
 * 14.5 Implicit Finite Difference Methods
 * 14.6 Variational Methods and the Euler–Lagrange Equations
 * 14.7 The Rayleigh–Ritz Approximation Method
 * 14.8 The Galerkin Approximation Method
 * 14.9 The Kantorovich Method
 * 14.10 The Finite Element Method
 * 14.11 Exercises


 * 15 Tables of Integral Transforms
 * 15.1 Fourier Transforms
 * 15.2 Fourier Sine Transforms
 * 15.3 Fourier Cosine Transforms
 * 15.4 Laplace Transforms
 * 15.5 Hankel Transforms
 * 15.6 Finite Hankel Transforms


 * Answers and Hints to Selected Exercises
 * 1.6 Exercises
 * 2.8 Exercises
 * 3.9 Exercises
 * 4.6 Exercises
 * 5.12 Exercises
 * 6.14 Exercises
 * 7.9 Exercises
 * 8.14 Exercises
 * 9.10 Exercises
 * 10.13 Exercises
 * 11.11 Exercises
 * 12.18 Exercises
 * 14.11 Exercises


 * Appendix: Some Special Functions and Their Properties
 * A-1 Gamma, Beta, Error, and Airy Functions
 * A-2 Hermite Polynomials and Weber–Hermite Functions


 * Bibliography
 * Index