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Partial Differential Equations : Second Edition - Emmanuele DiBenedetto

Partial Differential Equations

Second Edition

Hardcover

Published: 1st September 2009
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This self-contained text offers an elementary introduction to partial differential equations (pdes), primarily focusing on linear equations, but also providing some perspective on nonlinear equations. The classical treatment is mathematically rigorous with a generally theoretical layout, though indications to some of the physical origins of pdes are made throughout in references to potential theory, similarity solutions for the porous medium equation, generalized Riemann problems, and others.

The material begins with a focus on the Cauchy-Kowalewski theorem, discussing the notion of characteristic surfaces to classify pdes. Next, the Laplace equation and connected elliptic theory are treated, as well as integral equations and solutions to eigenvalue problems. The heat equation and related parabolic theory are then presented, followed by the wave equation in its basic aspects. An introduction to conservation laws, the uniqueness theorem, viscosity solutions, ill-posed problems, and nonlinear equations of first order round out the key subject matter.

Large parts of this revised second edition have been streamlined and rewritten to incorporate years of classroom feedback, correct errors, and improve clarity. Most of the necessary background material has been incorporated into the complements and certain nonessential topics have been given reduced attention (noticeably, numerical methods) to improve the flow of presentation.

The exposition is replete with examples, problems and solutions that complement the material to enhance understanding and solidify comprehension. The only prerequisites are advanced differential calculus and some basic Lp theory. The work can serve as a text for advanced undergraduates and graduate students in mathematics, physics, engineering, and the natural sciences, as well as an excellent reference for applied mathematicians and mathematical physicists.

From Reviews of the First Edition:

"The author's 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: Laplace's 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

"This is a well-written, self-contained, elementary introduction to linear, partial differential equations."

-ZentrallblattMATH

"This book certainly can be recommended as an introduction to PDEs in mathematical faculties and technical universities."

-Applications of Mathematics

"The book under review, the second edition of Emmanuele DiBenedetto's 1995 Partial Differential Equations, now appearing in Birkhauser's '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 DiBenedetto's fine taste in the subject and his experience in teaching this difficult material.

Make no mistake: the book is neither chatty nor discursive, but there's 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, DiBenedetto's 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 author's 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: Laplace's 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)

Preface to the Second Edition.- Preface to the First Edition.- Preliminaries.- Quasi-Linear Equations.- The LaPlace Equation.- Boundary Value Problems by Double Layer Potentials.- Integral Equations and Eigenvalue Problems.- The Heat Equation.- The Wave Equation.- Quasi-Linear Equations of First Order.- Non-Linear Equations of First Order.- References.- Index.

ISBN: 9780817645519
ISBN-10: 0817645519
Series: Cornerstones
Audience: Tertiary; University or College
Format: Hardcover
Language: English
Number Of Pages: 389
Published: 1st September 2009
Publisher: BIRKHAUSER BOSTON INC
Country of Publication: US
Dimensions (cm): 23.5 x 15.5  x 2.39
Weight (kg): 1.66
Edition Number: 2
Edition Type: Revised