Partial Differential Equations: An Introduction, 2nd Edition

by Strauss, Walter A.

• ISBN13: 9780470054567
• ISBN10: 0470054565
• Edition: 2nd
• Format: Hardcover
• Copyright: 2007-12-21
• Publisher: WILEY

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.

Author Biography

Dr. Walter A. Strauss is a professor of mathematics at Brown University. He has published numerous journal articles and papers. Not only is he is a member of the Division of Applied Mathematics and the Lefschetz Center for Dynamical Systems, but he is currently serving as the Editor in Chief of the SIAM Journal on Mathematical Analysis. Dr. Strauss' research interests include Partial Differential Equations, Mathematical Physics, Stability Theory, Solitary Waves, Kinetic Theory of Plasmas, Scattering Theory, Water Waves, Dispersive Waves.

Table of Contents

Where PDEs Come From
What is a Partial Differential Equation?	p. 1
First-Order Linear Equations	p. 6
Flows, Vibrations, and Diffusions	p. 10
Initial and Boundary Conditions	p. 20
Well-Posed Problems	p. 25
Types of Second-Order Equations	p. 28
Waves and Diffusions
The Wave Equation	p. 33
Causality and Energy	p. 39
The Diffusion Equation	p. 42
Diffusion on the Whole Line	p. 46
Comparison of Waves and Diffusions	p. 54
Reflections and Sources
Diffusion on the Half-Line	p. 57
Reflections of Waves	p. 61
Diffusion with a Source	p. 67
Waves with a Source	p. 71
Diffusion Revisited	p. 80
Boundary Problems
Separation of Variables, The Dirichlet Condition	p. 84
The Neumann Condition	p. 89
The Robin Condition	p. 92
Fourier Series
The Coefficients	p. 104
Even, Odd, Periodic, and Complex Functions	p. 113
Orthogonality and General Fourier Series	p. 118
Completeness	p. 124
Completeness and the Gibbs Phenomenon	p. 136
Inhomogeneous Boundary Conditions	p. 147
Harmonic Functions
Laplace's Equation	p. 152
Rectangles and Cubes	p. 161
Poisson's Formula	p. 165
Circles, Wedges, and Annuli	p. 172
Green's Identities and Green's Functions
Green's First Identity	p. 178
Green's Second Identity	p. 185
Green's Functions	p. 188
Half-Space and Sphere	p. 191
Computation of Solutions
Opportunities and Dangers	p. 199
Approximations of Diffusions	p. 203
Approximations of Waves	p. 211
Approximations of Laplace's Equation	p. 218
Finite Element Method	p. 222
Waves in Space
Energy and Causality	p. 228
The Wave Equation in Space-Time	p. 234
Rays, Singularities, and Sources	p. 242
The Diffusion and Schrodinger Equations	p. 248
The Hydrogen Atom	p. 254
Boundaries in the Plane and in Space
Fourier's Method, Revisited	p. 258
Vibrations of a Drumhead	p. 264
Solid Vibrations in a Ball	p. 270
Nodes	p. 278
Bessel Functions	p. 282
Legendre Functions	p. 289
Angular Momentum in Quantum Mechanics	p. 294
General Eigenvalue Problems
The Eigenvalues Are Minima of the Potential Energy	p. 299
Computation of Eigenvalues	p. 304
Completeness	p. 310
Symmetric Differential Operators	p. 314
Completeness and Separation of Variables	p. 318
Asymptotics of the Eigenvalues	p. 322
Distributions and Transforms
Distributions	p. 331
Green's Functions, Revisited	p. 338
Fourier Transforms	p. 343
Source Functions	p. 349
Laplace Transform Techniques	p. 353
PDE Problems from Physics
Electromagnetism	p. 358
Fluids and Acoustics	p. 361
Scattering	p. 366
Continuous Spectrum	p. 370
Equations of Elementary Particles	p. 373
Nonlinear PDEs
Shock Waves	p. 380
Solitons	p. 390
Calculus of Variations	p. 397
Bifurcation Theory	p. 401
Water Waves	p. 406
Appendix
Continuous and Differentiable Functions	p. 414
Infinite Series of Functions	p. 418
Differentiation and Integration	p. 420
Differential Equations	p. 423
The Gamma Function	p. 425
References	p. 427
Answers and Hints to Selected Exercises	p. 431
Index	p. 446
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