Partial Differential Equations: An Introduction | 
 enlarge | Author: Walter A. Strauss
Publisher: Wiley
Category: Book
Buy New: $49.99
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Rating:  24 reviews
Sales Rank: 460381
Media: Hardcover
Pages: 440
Number Of Items: 1
Shipping Weight (lbs): 1.6
Dimensions (in): 9.3 x 6.2 x 1
ISBN: 0471548685
Dewey Decimal Number: 515.353
EAN: 9780471548683
Publication Date: March 17, 1992
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Product Description
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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| Customer Reviews: Read 19 more reviews...
 Advanced undergraduate PDE text. April 30, 2006
Farshid Arjomandi (California, USA)
13 out of 13 found this review helpful
This 1992 title by Strauss (professor at MIT) has become a standard for teaching PDE theory to junior and senior applied maths and engineering students in many American universities. Last year, being an informal teaching assistant for the class, I found many of the students struggling with the concepts and exercises in the book. Admittedly the style of writing here is very dense and if the reader does not have a very strong background in the topic, chances are high he or she will face a grand level of frustration with the exposition and the subject as a whole. One would need perseverance and dedication working numerous hours with this text before things start to settle in. After about the second or third chapter onward, those who were still taking the class had an easier time understanding the material and doing the excercises.
Contentwise, after a brief and important introductory chapter (which should not be skipped by any reader!) the book first focuses on the properties and methods of solutions of the one-dimensional linear PDEs of hyperbolic and parabolic types. Then after two separate chapters, one on the trio of Dirichlet, Neumann, and Robin conditions and the other on the Fourier series, the author embarks upon the discussion of elliptic PDEs via the methods of harmonic analysis and Green's functions. Subsequently there is a brief introduction to the numerical techniques for finding approximate solutions to the three types of PDEs, mostly centered on the finite differences methods.
The beginning of roughly the second half of the text is devoted to the higher-dimensional wave equations and boundary conditions in plane and space, utilizing the machinery of Bessel and Legendre functions, and ending up with a section on angular momentum in quantum mechanics. In the following, Dr. Strauss brings up the discussion of the general eigenvalue problems, and then proceeds with a treatment of the advanced subject of weak solutions and distribution theory. (This topic is normally skipped in an undergraduate course.) The last two chapters are a pure delight to read, dealing with the PDEs from physics as well as a survey of the nonlinear phenomena (shocks, solitons, bifurcation theory). A few appendixes at the end, summarize the analysis background needed for the course and must be consulted before and during the first reading.
All in all this is a very splendid source for all the applied maths and engineering students, that can be used in conjuction with other references to help break through the conceptual barriers. In fact, I recommended the book by Stanley Farlow to our students and many found the presentation there very modular and accessible. For example, some of the Strauss' homework problems, such as solving the Poisson equation on an annulus, were subjects of a single chapter in Farlow. In any event, I am very much hoping to see a new and more student-friendly edition of the Strauss' text be prepared and issued in the near future.
What are you guys talking about? This book is AMAZING! October 5, 2006
Michael Harmon (NYC)
12 out of 14 found this review helpful
I have never commented on a book, up until now... and I do so only because I don't think that this book gets enough credit.
People have complained Strauss may not have explained some proofs in as much detail as he could have, people complained that he didnt give enough examples, I think this is more of a problem with the readers than the writers. If you need someone to hold your hand through every step and detail, I think you should reconsider why you are studying what you study.-
I am an undergraduate at NYU, one of the best research institutes for PDE's. I thoroughly enjoyed reading this book, it gives an amazing description of what PDE's are, how to solve them, and how they are used in science. One thing I REALLY enjoyed about this book was it did not do what many other books do: first dive into seperation of variables and focused only on that. Instead Strauss shows how to solve first and second order equations without boundary conditions, giving a very elegant prose doing so!
However, I think much of the problem that people are having with this book is that it's not a "one-size fits all." (Which I don't think any book can be!) If you are a Scienctist or Engineer and just want to learn PDE's to solve problems in science.. find another book, because this book is not the book for you.
That being said, if you are Mathematics student or interested in a more deep study of PDEs this is really a good book for you. You definitely should have taken Calc. 1-3, Linear Alegbra, ODE, and I recommend one semester of Analysis (for function spaces) before tackling this book, that is what I had, and I loved this course.
PDE is a difficult subject/course and Strauss does an amazing job at explaining it, if someone like me can get PDEs so well from this course, than I seriously believe that complaints about this book is due to fault in the readers and not the writer.
A darn good book April 7, 2003
Michael WIlls (Goleta, CA United States)
10 out of 13 found this review helpful
I honestly don't understand several of the reviews here. At the one extreme, we have people who complain that there are not enough examples, and there are too many gaps in the proofs.
Well, that is partly the point. At some point in math, you have to move beyond the spoon fed approach of a typical lower division calculus textbook and fill in the gaps and figure out the examples for yourself.
At the other extreme, one person complained that the exercises were uninspired and did not lead away from the text. The only response I have to that is "are you reading the same text that I am?"
The title is not misleading. The book is a concise introduction to PDEs. One should have had some upper divison analysis, and some lower divison ODEs but that's about it.
I have had a graduate course in PDEs, which I basically failed to understand. I was able to get through the course, but without ever getting any "big picture". This is probably because I had never taken an undergraduate PDE course. I now have got to the point where I need to know undergraduate level PDEs and this textbook has been perfect. It is hard, but readable. The questions cover a lot of material and have a wide range of difficulty. As I've worked my way through the book, I feel that I am finally getting to grips with the subject, and beginning to see a big picture.
One of the better textbooks in my collection
Well organized book December 13, 1999
Nicholas E. Athanas (Ratrace, IL)
11 out of 14 found this review helpful
This book is very concise and to the point. It is exactly what its title suggests. Good reference book too, and good examples. I've checked out many books on this subject and this is probably the best. The appendices were also very helpful.
GOOD FOR WHAT IT DOES--NO ONE BOOK DOES IT ALL March 9, 2008
Peter J. Stan (Los Angeles, CA)
4 out of 4 found this review helpful
I've spent the past seven years or so working on analytical and numerical solutions to the various PDEs that price financial derivatives. My focus has been very much on getting and extending useable answers. When it comes to PDEs specifically, I'm mostly self-taught, but my background in real variables and functional analysis is solid.
In some ares of mathematics, a single, classic text whose exposition is top-notch, or that inspires despite its exposition, manages to cover the field well. For example, I'm thinking of books like Halmos (Measure Theory), Rudin (Real and Complex Analysis), or Segal and Kunze in real variables and integration; Lax or Reed and Simon I (Functional Analysis) in functional analysis; Lang (Algebra) in algebra; and Kelley (General Topology) or Milnor (Topology from the Differentiable Viewpoint) in topology. (Full citations are in Listmania; see my Amazon profile.)
I've yet to find a single reference for PDEs that addresses all of my questions, but several books taken together manage nicely. I jumped around in these books when I was learning the subject, and I'm convinced that such cherry-picking is the best approach for PDEs, since the field is so broad in theory and applications. The downsides, of course, are expense and potential confusion from conflicting notation and approach.
Ignoring just for the moment the vast area of approximate solutions by discretization and perturbation techniques, here's who seems to be best for what, when the problem involves linear PDEs:
:: Need quick intuition: Farlow, Myint-U and Debnath, Brown and Churchill;
:: Need more theory: Stakgold (Green's Functions), Evans, Folland;
:: Need help on modeling: Strang (CSE), Stakgold (BVPs), Haberman (Applied PDEs), Farlow;
:: Don't understand how concepts relate: John, Levine, Garabedian, Strauss, Carrier and Pearson (PDEs);
:: Can't find tough enough exercises: Carrier and Pearson (PDEs), Kevorkian;
:: Need inspiration or deep intuition: Courant and Hilbert (both volumes), Zeidler (Nonlinear Functional Analysis 2A [Linear Monotone Operators], Applied Functional Analysis [especially AMS 108]).
I've ranked books very subjectively within each category on a composite of things like relevance, completeness, clarity, and ease-of-use. And I should stress that I'm no doubt ignoring many fine favorites purely through unfamiliarity.
WHERE DOES STRAUSS FIT? I repeat, all of these books address each of the needs in some measure, but no one is adequate for all. The terse treatment and broad coverage in Strauss are great for tying concepts together and revealing their logical relationships. This is especially evident in the superb Chaps. 1 and 2-3, as well as in Chaps. 9 and 10, which treat the Cauchy Problem and BVPs in space, respectively.
Chapter 11's discussion of eigenvalue problems, and particularly their asymptotics, is remarkable at the book's level but nowhere near that in Garabedian or especially that in Courant and Hilbert, which is the original synthesis of work beginning with Weyl to which Courant and Hilbert each contributed in important ways. (The notes to Sec. XIII.15 of Reed and Simon IV [Analysis of Operators] have the history of Dirichlet-Neumann bracketing, the main methodological advance.) Both of Stakgold's works also discuss this problem but not as well as Strauss.
I've done very little teaching (and I wasn't very good at it!), so my views should be taken with a grain of salt perhaps larger than usual. If I chose Strauss as a text, I'd have to believe either that my lectures would fill in the gaps that Strauss so clearly has or that other books on my syllabus could take up the slack. If you're having trouble learning "from Strauss," the problem may lie not with the book, but with an incomplete course, since Strauss is, in many ways, only a good set of summary notes. Again, it's good as far as it goes, but it doesn't go the whole way; that's why you need the other books.
CHOOSING A SINGLE REFERENCE. If I were packing for a desert island, I'd take Levine and Garabedian, since everything I need can be backed out of their presentations with some effort. In many ways these books can be thought of usefully as a set. Both are written at an intermediate level, meaning that techniques less sophisticated than those involving function spaces are fair game. Levine spends 700 pages on separation-of-variables, Fourier analysis, and transform methods, applied to parabolic and elliptic equations in general and the diffusion (heat) equation in particular. Garabedian picks up just where Levine leaves off to treat the Cauchy Problem for hyperbolic equations and the Dirichlet and Neumann Problems for elliptic equations. His book is also roughly 700 pages in length and like Levine's is a model of clarity.
Although both books have been available for some time, basic approaches to the three classic, second-order, linear equations and their variants--the gist of a first course--have changed so little in the past 50 years that publication date may not be as much a factor in selecting a single reference as it would be in some other areas. Indeed, it's well worth reading Fourier's original memoir on heat conduction, possibly modulated by a modern treatment like Carslaw and Jaeger (Conduction of Heat in Solids) or Crank (Mathematics of Diffusion). Levine (Chap. 13) also contains a technical precis of Fourier's original approach.
If I found that I needed greater depth, meaning function spaces, I'd turn first to Courant and Hilbert or to either of Zeidler's state-of-the-art books. The treatment in C&H is profound and downright majestic. Many have spent a productive professional lifetime in these books, and C&H-2 comes close to being the sort of reference I describe in the second paragraph. Because of its age, it's possible to see in the books' discussion much of the intuition of Hilbert and Sobolev spaces that was later covered with layers of rigorous abstraction. Zeidler's discussion of that abstraction is simply the clearest that I've found anywhere. It's extraordinary that any author works as hard as Zeidler to convey mathematical ideas, and for this reason his books are among my favorites across all topics.
TRANSITIONING TO DISCRETE APPROXIMATIONS. If I also took Gil Strang's new book to ease the transition to discrete approximations and eventually building and evaluating numerical code, I'd forget the rest of the list without worries. Indeed, as Courant mentions in the preface of C&H-2, there was to have been a brief third volume dealing with discrete approximations for existence and construction of solutions. Strang would stand nicely in its stead, if not for existence, then certainly for construction.
Actually doing numerical work is another matter entirely, of course, and I've given some idea of the books I've found useful for these problems in my brief review of Chung's book on computational fluid dynamics.
HOW ABOUT PERTURBATION SOLUTIONS? Finally, if I closed my eyes and pretended that I'd always separate variables, so I only needed to worry about perturbation solutions of hairy ODEs, I'd toss in Bender and Orszag and feel pretty good about analytical approaches. (If you're lost in B&O, and it does have its moments, try Holmes, which is a more accessible survey at less depth. And if you need to begin at the beginning, go to Lin and Segel [Chaps. 6-7. 9, and 11], which treats the ideas you need, before you get buried in algebra.)
If I just couldn't bring myself to make that assumption, I'd take Kevorkian and Cole (Multiple Scale and Singular Perturbation Methods), which deals in part with perturbation solutions of PDEs directly, and Verhulst, which is a bit longer on intuition.
The brave might also consider Van Dyke (Perturbation Methods in Fluid Mechanics), which deals specifically with singular perturbations of the Navier-Stokes equations, their many variants, and the other equations of fluid mechanics. To go this route you'd have to believe that you could adapt Van Dyke's results to whatever problems you ran into, which can be real work. Hinch's short book is useful as a complement.
ONCE MORE...WITH FEELING! If I were learning things from scratch again, I'd sleep with Farlow under the pillow and Garabedian under the bed, regardless of what textbook my instructor had chosen. For the careful Farlow raises at least as many questions as it answers--purists have my guarantee that they'll hate it. You need to supplement Farlow with greater depth in your areas of interest, and in most instances Garabedian cleans things up nicely without doing violence to the concepts.
An added plus is that since the books are reprints, published by Dover and AMS Chelsea, respectively, their cost is quite reasonable, even though Garabedian is beautifully printed and library-bound (would you believe sewn-in signatures and useable inside margins?).
This review is a lot longer than I'd first intended, and its recommendations are in many ways idiosyncratic but certainly worth their cost. Partly, I think that's a function of the field itself. There seem to be as many approaches to learning PDEs as there are backgrounds and interests. The diversity of sources is likewise broad, and their quality is quite high. The beauty and power of the subject have lured many first-class mathematicians, like a striking number of the authors mentioned above, into writing basic texts. In the end someone's treatment will answer your questions, pretty much no matter what they are, if you just have the patience to look around. After enough looking, of course, you'll find you can answer many of your own questions.
One of the things that makes real-world PDEs in whatever field such fun is that getting an answer is all that's important. It doesn't matter what books you use, what willing help you receive from whom, or how you reformulate a problem to make it more tractable, as long as a result that answers the real question (sometimes a rather elastic notion) eventually emerges from your efforts. There's much to be said for learning the field in the same no-holds-barred way, and I hope my remarks can get you started in that direction.
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