Applied Partial Differential Equations: An Introduction

by Jeffrey

• ISBN13: 9780123822529
• ISBN10: 0123822521
• Format: Hardcover
• Copyright: 2002-11-04
• Publisher: Elsevier Science

Summary

This book is written to meet the needs of undergraduates in applied mathematics, physics and engineering studying partial differential equations. It is a more modern, comprehensive treatment intended for students who need more than the purely numerical solutions provided by programs like the MATLAB PDE Toolbox, and those obtained by the method of separation of variables, which is usually the only theoretical approach found in the majority of elementary textbooks. This will fill a need in the market for a more modern text for future working engineers, and one that students can read and understand much more easily than those currently on the market.

Table of Contents

Preface	xi
Introduction to Partial Differential Equations	1	(58)
What Is a Partial Differential Equation?	1	(11)
Representative Problems Leading to PDEs, Initial and Boundary Conditions	12	(24)
The traffic flow problem	13	(4)
The heat equation with a source term	17	(6)
Transverse vibrations of a string	23	(3)
Vibrations of a membrane	26	(4)
The telegraph equation	30	(1)
Longitudinal vibrations of a free elastic rod with a variable cross section	31	(1)
Electromagnetic wave propagation in free space	32	(1)
Acoustic waves in a gas	33	(3)
What Is a Solution of a PDE?	36	(4)
The Cauchy Problem	40	(4)
Well-Posed and Improperly Posed Problems	44	(2)
Coordinate Systems, Vector Operators, and Integral Theorems	46	(13)
Cartesian coordinates	47	(2)
Cylindrical polar coordinates and plane polar coordinates	49	(1)
Spherical polar coordinates	50	(2)
The Gauss divergence theorem	52	(1)
Green's and Stokes' theorems	53	(1)
Useful identities involving vector operators	54	(1)
Examples of PDE applications of vector integral theorems	55	(4)
Linear and Nonlinear First-Order Equations and Shocks	59	(44)
Linear and Semilinear Equations in Two Independent Variables	59	(14)
Quasi-Linear Equations in Two Independent Variables	73	(10)
Propagation of Weak Discontinuities by First-Order Equations	83	(10)
Propagation of weak discontinuities	86	(7)
Discontinuous Solutions, Conservation Laws, and Shocks	93	(10)
Classification of Equations and Reduction to Standard Form	103	(24)
Classification of PDEs and Their Reduction to Standard Form	103	(15)
The hyperbolic case, d = B2 - AC > 0	107	(4)
The parabolic case, d = B2 - AC = 0	111	(2)
Elliptic equations, d = B2 - AC < 0	113	(1)
Elliptic case, y > 0	114	(1)
Hyperbolic case, y < 0	115	(1)
Timelike and spacelike arcs	115	(3)
Classification of Second-Order PDE in Many Independent Variables	118	(6)
Well-Posed Problems for Hyperbolic, Parabolic, and Elliptic Partial Differential Equations	124	(3)
Linear Wave Propagation in One or More Space Dimensions	127	(46)
Linear Waves and the Wave Equation	127	(6)
The D'Alembert Solution and the Telegraph Equation	133	(15)
Mixed Initial and Boundary Value Problems for the Wave Equation	148	(9)
The Poisson Formula for the Wave Equation, the Method of Descent, and the Difference between Waves in Two and Three Space Dimensions	157	(9)
Kirchhoff's Solution of the Wave Equation in Three Space Variables and Another Representation of Huygens' Principle	166	(3)
Uniqueness of Solutions of the Wave Equation	169	(4)
Fourier Series, Legendre and Bessel Functions	173	(44)
An Introduction to Fourier Series	173	(15)
Some simple properties of Fourier Series	174	(2)
Orthogonality and the Euler formulas	176	(12)
Major Results Involving Fourier Series	188	(13)
A Summary of the Properties of the Legendre and Bessel Differential Equations	201	(16)
Legendre polynomials	202	(5)
Bessel functions	207	(10)
Background to Separation of Variables with Applications	217	(60)
A General Approach to Separation of Variables	217	(19)
Case (a): A Sturm--Liouville problem obtained from the heat equation in plane polar coordinates (r,θ), with k = constant and p = 0	230	(2)
Case (b): Spherical polar coordinates (r,θ,φ), k= constant, p = 0, and w = 0	232	(4)
Properties of Eigenfunctions and Eigenvalues	236	(6)
Applications of Separation of Variables	242	(35)
General Results for Linear Elliptic and Parabolic Equations	277	(48)
General Results for Elliptic and Parabolic Equations	277	(1)
Laplace Equation	278	(16)
The Heat Equation	294	(5)
Self-Similarity Solutions	299	(6)
Fundamental Solution of the Heat Equation	305	(11)
Duhamel's Principle	316	(9)
Hyperbolic Systems, Riemann Invariants, Simple Waves, and Compound Riemann Problems	325	(32)
Properly Determined First-Order Systems of Equations	325	(3)
Hyperbolicity and Characteristic Curves	328	(8)
Riemann Invariants	336	(8)
Simple Waves	344	(7)
Shocks and the Riemann Problem	351	(6)
Answers to Odd-Numbered Exercises	357	(30)
Bibliography	387	(2)
Index	389