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

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

Published $2007$, 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