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

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## Tyn Myint-U and Lokenath Debnath:

## 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

### 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