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1998 Paperback (ISBN 9781571460653) Published: 1998 Publisher: International Press of Boston, Inc. Language: English Paperback 72 pages List Price: $30.00 Table of Contents (PDF) Preview (PDF) Pub. Date ISBN-13 ISBN-10 Medium Binding Size, Etc. Status List Price 1998 9781571460653 Print paperback In Print US$30.00 (The starred sections form the basic part of the book.) Chapter 1Where PDEs Come From 1.1 What is a Partial Differential Equation? 1 1.2 First-Order Linear Equations 6 1.3 Flows, Vibrations, and Diffusions 10 1.4 Initial and Boundary Conditions 20 1.5 Well-Posed Problems 25 1.6 Types of Second-Order Equations 28 Chapter 2Waves and Diffusions 2.1 The Wave Equation 33 2.2 Causality and Energy 39 2.3 The Diffusion Equation 42 2.4 Diffusion on the Whole Line 46 2.5 Comparison of Waves and Diffusions 54 Chapter 3Reflections and Sources 3.1 Diffusion on the Half-Line 57 3.2 Reflections of Waves 61 3.3 Diffusion with a Source 67 3.4 Waves with a Source 71 3.5 Diffusion Revisited 80 Chapter 4Boundary Problems 4.1 Separation of Variables, The Dirichlet Condition 84 4.2 The Neumann Condition 89 4.3 The Robin Condition 92 Chapter 5Fourier Series 5.1 The Coefficients 104 5.2 Even, Odd, Periodic, and Complex Functions 113 5.3 Orthogonality and General Fourier Series 118 5.4 Completeness 124 5.5 Completeness and the Gibbs Phenomenon 136 5.6 Inhomogeneous Boundary Conditions 147 Chapter 6Harmonic Functions 6.1 Laplaces Equation 152 6.2 Rectangles and Cubes 161 6.3 Poissons Formula 165 6.4 Circles, Wedges, and Annuli 172 (The next four chapters may be studied in any order.) Chapter 7Greens Identities and Greens Functions 7.1 Greens First Identity 178 7.2 Greens Second Identity 185 7.3 Greens Functions 188 7.4 Half-Space and Sphere 191 Chapter 8Computation of Solutions 8.1 Opportunities and Dangers 199 8.2 Approximations of Diffusions 203 8.3 Approximations of Waves 211 8.4 Approximations of Laplaces Equation 218 8.5 Finite Element Method 222 Chapter 9Waves in Space 9.1 Energy and Causality 228 9.2 The Wave Equation in Space-Time 234 9.3 Rays, Singularities, and Sources 242 9.4 The Diffusion and Schro¨ dinger Equations 248 9.5 The Hydrogen Atom 254 Chapter 10Boundaries in the Plane and in Space 10.1 Fouriers Method, Revisited 258 10.2 Vibrations of a Drumhead 264 10.3 Solid Vibrations in a Ball 270 10.4 Nodes 278 10.5 Bessel Functions 282 10.6 Legendre Functions 289 10.7 Angular Momentum in Quantum Mechanics 294 Chapter 11General Eigenvalue Problems 11.1 The Eigenvalues Are Minima of the Potential Energy 299 11.2 Computation of Eigenvalues 304 11.3 Completeness 310 11.4 Symmetric Differential Operators 314 11.5 Completeness and Separation of Variables 318 11.6 Asymptotics of the Eigenvalues 322 Chapter 12Distributions and Transforms 12.1 Distributions 331 12.2 Greens Functions, Revisited 338 12.3 Fourier Transforms 343 12.4 Source Functions 349 12.5 Laplace Transform Techniques 353 Chapter 13PDE Problems from Physics 13.1 Electromagnetism 358 13.2 Fluids and Acoustics 361 13.3 Scattering 366 13.4 Continuous Spectrum 370 13.5 Equations of Elementary Particles 373 Chapter 14Nonlinear PDEs 14.1 Shock Waves 380 14.2 Solitons 390 14.3 Calculus of Variations 397 14.4 Bifurcation Theory 401 14.5 Water Waves 406 Appendix A.1 Continuous and Differentiable Functions 414 A.2 Infinite Series of Functions 418 A.3 Differentiation and Integration 420 A.4 Differential Equations 423 A.5 The Gamma Function 425 References 427 Answers and Hints to Selected Exercises 431 Index 446 © , Amazon.com, Inc. or its affiliates The book under review, the second edition of Emmanuele DiBenedettos 1995 Partial Differential Equations, now appearing in Birkhäusers Cornerstones series, is an example of excellent timing. This is a well-written, self-contained, elementary introduction to linear, partial differential equations.So it is that DiBenedetto, whose philosophical position regarding PDE is unabashedly that although a branch of mathematics, it is closely related to physical phenomena, presents us with marvelous coverage of (in order), quasi-linearity and Cauchy-Kowalevski, Laplace, BVPs by double-layer potentials, and my favorite three chapters: integral equations and the eigenvalue problem, the heat equation, and the wave equation. Then he returns to quasi-linearity (for first order equations), goes on to non-linearity, linear elliptic equations with measurable coefficients..., and, finally...DeGiorgi classes.PDE is beautifully written, in clear and concise prose, the mathematics is cogent and complete, and the presentation testifies both to DiBenedettos fine taste in the subject and his experience in teaching this difficult material.Make no mistake: the book is neither chatty nor discursive, but theres something more or less ineffable about it, making it appear somehow less austere than other texts on PDE. Check it out.DiBenedetto has also included a decent number of what he calls Problems and Complements, and, to be sure, these should capture the attention of the conscientious student or reader.Thus, DiBenedettos PDE is indeed a cornerstone text in the subject. It looks like a rare gem to me.—MAA Reviews (Review of the Second Edition)The authors intent is to present an elementary introduction to pdes...In contrast to other elementary textbooks on pdes...much more material is presented on the three basic equations: Laplaces equation, the heat and wave equations...The presentation is clear and well organized...The text is complemented by numerous exercises and hints to proofs.—Mathematical Reviews (Review of the First Edition)This is a well-written, self-contained, elementary introduction to linear, partial differential equations.—Zentrallblatt MATH (Review of the First Edition)This book certainly can be recommended as an introduction to PDEs in mathematical faculties and technical universities. —Applications of Mathematics (Review of the First Edition) Page 2 Physical AssumptionSmall VibrationAnisotropic Porous MediumInstantaneous EquilibriumExterior UnitThese keywords were added by machine and not by the authors. 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