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Partial Differential Equations: An Introduction, 2nd Edition,9780470054567

Partial Differential Equations: An Introduction, 2nd Edition

by
Edition: 2nd
Format: Hardcover
Pub. Date: 12/1/2007
Publisher(s): WILEY
Availability: This title is currently not available.

Summary

Our understanding of the fundamental processes of the natural world is based to a large extent on partial differential equations (PDEs). The second edition of Partial Differential Equations provides an introduction to the basic properties of PDEs and the ideas and techniques that have proven useful in analyzing them. It provides the student a broad perspective on the subject, illustrates the incredibly rich variety of phenomena encompassed by it, and imparts a working knowledge of the most important techniques of analysis of the solutions of the equations. In this book mathematical jargon is minimized. Our focus is on the three most classical PDEs, the wave, heat and Lapace equations. Advanced concepts are introduced frequently but with the least possible technicalities. The book is flexibly designed for juniors, seniors or beginning graduate students in science, engineering or mathematics.

Table of Contents

Where PDEs Come From
What is a Partial Differential Equation?p. 1
First-Order Linear Equationsp. 6
Flows, Vibrations, and Diffusionsp. 10
Initial and Boundary Conditionsp. 20
Well-Posed Problemsp. 25
Types of Second-Order Equationsp. 28
Waves and Diffusions
The Wave Equationp. 33
Causality and Energyp. 39
The Diffusion Equationp. 42
Diffusion on the Whole Linep. 46
Comparison of Waves and Diffusionsp. 54
Reflections and Sources
Diffusion on the Half-Linep. 57
Reflections of Wavesp. 61
Diffusion with a Sourcep. 67
Waves with a Sourcep. 71
Diffusion Revisitedp. 80
Boundary Problems
Separation of Variables, The Dirichlet Conditionp. 84
The Neumann Conditionp. 89
The Robin Conditionp. 92
Fourier Series
The Coefficientsp. 104
Even, Odd, Periodic, and Complex Functionsp. 113
Orthogonality and General Fourier Seriesp. 118
Completenessp. 124
Completeness and the Gibbs Phenomenonp. 136
Inhomogeneous Boundary Conditionsp. 147
Harmonic Functions
Laplace's Equationp. 152
Rectangles and Cubesp. 161
Poisson's Formulap. 165
Circles, Wedges, and Annulip. 172
Green's Identities and Green's Functions
Green's First Identityp. 178
Green's Second Identityp. 185
Green's Functionsp. 188
Half-Space and Spherep. 191
Computation of Solutions
Opportunities and Dangersp. 199
Approximations of Diffusionsp. 203
Approximations of Wavesp. 211
Approximations of Laplace's Equationp. 218
Finite Element Methodp. 222
Waves in Space
Energy and Causalityp. 228
The Wave Equation in Space-Timep. 234
Rays, Singularities, and Sourcesp. 242
The Diffusion and Schrodinger Equationsp. 248
The Hydrogen Atomp. 254
Boundaries in the Plane and in Space
Fourier's Method, Revisitedp. 258
Vibrations of a Drumheadp. 264
Solid Vibrations in a Ballp. 270
Nodesp. 278
Bessel Functionsp. 282
Legendre Functionsp. 289
Angular Momentum in Quantum Mechanicsp. 294
General Eigenvalue Problems
The Eigenvalues Are Minima of the Potential Energyp. 299
Computation of Eigenvaluesp. 304
Completenessp. 310
Symmetric Differential Operatorsp. 314
Completeness and Separation of Variablesp. 318
Asymptotics of the Eigenvaluesp. 322
Distributions and Transforms
Distributionsp. 331
Green's Functions, Revisitedp. 338
Fourier Transformsp. 343
Source Functionsp. 349
Laplace Transform Techniquesp. 353
PDE Problems from Physics
Electromagnetismp. 358
Fluids and Acousticsp. 361
Scatteringp. 366
Continuous Spectrump. 370
Equations of Elementary Particlesp. 373
Nonlinear PDEs
Shock Wavesp. 380
Solitonsp. 390
Calculus of Variationsp. 397
Bifurcation Theoryp. 401
Water Wavesp. 406
Appendix
Continuous and Differentiable Functionsp. 414
Infinite Series of Functionsp. 418
Differentiation and Integrationp. 420
Differential Equationsp. 423
The Gamma Functionp. 425
Referencesp. 427
Answers and Hints to Selected Exercisesp. 431
Indexp. 446
Table of Contents provided by Ingram. All Rights Reserved.

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