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

Tyn Myint-U and Lokenath Debnath: Linear Partial Differential Equations for Scientists and Engineers (4th Edition)

Published $\text {2007}$, Birkhauser

ISBN 978-0817643935

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