Partial Differential Equations: An Introduction / Edition 2

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Overview

Covers the fundamental properties of partial differential equations (PDEs) and proven techniques useful in analyzing them. Uses a broad approach to illustrate the rich diversity of phenomena such as vibrations of solids, fluid flow, molecular structure, photon and electron interactions, radiation of electromagnetic waves encompassed by this subject as well as the role PDEs play in modern mathematics, especially geometry and analysis.
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Editorial Reviews

Booknews
An advanced-undergraduate textbook for science, engineering, or mathematics majors, providing an introduction to the basic properties of partial differential equations and to the techniques that have proved useful in analyzing them. The prerequisites are a solid knowledge of calculus, especially multivariate, and small amounts of ordinary differential equations and of linear algebra. Annotation c. Book News, Inc., Portland, OR (booknews.com)
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Product Details

  • ISBN-13: 9780470054567
  • Publisher: Wiley, John & Sons, Incorporated
  • Publication date: 12/21/2007
  • Edition description: New Edition
  • Edition number: 2
  • Pages: 432
  • Sales rank: 1,306,800
  • Product dimensions: 6.10 (w) x 9.30 (h) x 0.80 (d)

Table of Contents

Chapter 1: Where PDEs Come From
1.1 What is a Partial Differential Equation?
1.2 First-Order Linear Equations
1.3 Flows, Vibrations, and Diffusions
1.4 Initial and Boundary Conditions
1.5 Well-Posed Problems
1.6 Types of Second-Order Equations

Chapter 2: Waves and Diffusions
2.1 The Wave Equation
2.2 Causality and Energy
2.3 The Diffusion Equation
2.4 Diffusion on the Whole Line
2.5 Comparison of Waves and Diffusions

Chapter 3: Reflections and Sources
3.1 Diffusion on the Half-Line
3.2 Reflections of Waves
3.3 Diffusion with a Source
3.4 Waves with a Source
3.5 Diffusion Revisited

Chapter 4: Boundary Problems
4.1 Separation of Variables, The Dirichlet Condition
4.2 The Neumann Condition
4.3 The Robin Condition

Chapter 5: Fourier Series
5.1 The Coefficients
5.2 Even, Odd, Periodic, and Complex Functions
5.3 Orthogonality and the General Fourier Series
5.4 Completeness
5.5 Completeness and the Gibbs Phenomenon
5.6 Inhomogeneous Boundary Conditions

Chapter 6: Harmonic Functions
6.1 Laplace’s Equation
6.2 Rectangles and Cubes
6.3 Poisson’s Formula
6.4 Circles, Wedges, and Annuli

Chapter 7: Green’s Identities and Green’s Functions
7.1 Green’s First Identity
7.2 Green’s Second Identity
7.3 Green’s Functions
7.4 Half-Space and Sphere

Chapter 8: Computation of Solutions
8.1 Opportunities and Dangers
8.2 Approximations of Diffusions
8.3 Approximations of Waves
8.4 Approximations of Laplace’s Equation
8.5 Finite Element Method

Chapter 9: Waves in Space
9.1 Energy and Causality
9.2 The Wave Equation in Space-Time
9.3 Rays, Singularities, and Sources
9.4 The Diffusion and Schrodinger Equations
9.5 The Hydrogen Atom

Chapter 10: Boundaries in the Plane and in Space
10.1 Fourier’s Method, Revisited
10.2 Vibrations of a Drumhead
10.3 Solid Vibrations in a Ball
10.4 Nodes
10.5 Bessel Functions
10.6 Legendre Functions
10.7 Angular Momentum in Quantum Mechanics

Chapter 11: General Eigenvalue Problems
11.1 The Eigenvalues Are Minima of the Potential Energy
11.2 Computation of Eigenvalues
11.3 Completeness
11.4 Symmetric Differential Operators
11.5 Completeness and Separation of Variables
11.6 Asymptotics of the Eigenvalues

Chapter 12: Distributions and Transforms
12.1 Distributions
12.2 Green’s Functions, Revisited
12.3 Fourier Transforms
12.4 Source Functions
12.5 Laplace Transform Techniques

Chapter 13: PDE Problems for Physics
13.1 Electromagnetism
13.2 Fluids and Acoustics
13.3 Scattering
13.4 Continuous Spectrum
13.5 Equations of Elementary Particles

Chapter 14: Nonlinear PDEs
14.1 Shock Waves
14.2 Solitions
14.3 Calculus of Variations
14.4 Bifurcation Theory
14.5 Water Waves

Appendix
A.1 Continuous and Differentiable Functions
A.2 Infinite Sets of Functions
A.3 Differentiation and Integration
A.4 Differential Equations
A.5 The Gamma Function

References

Answers and Hints to Selected Exercises

Index

 

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