Customer Reviews

Partial Differential Equations: An Introduction

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I've spent the past seven years or so working on analytical and numerical solutions to the various partial differential equations 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 (viz., one and two graduate semesters, respectively, at Pennsylvania and Harvard).

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) or Segal and Kunze in real variables and integration theory; Lax or Reed and Simon 1 (Functional Analysis) in functional analysis; Lang in algebra; and Kelley or Milnor (Topology from the Differentiable Viewpoint) in topology. (Full citations are in Listmania; see my Amazon profile. All of these books are great texts for very different reasons, as my Listmania remarks suggest.)

I've yet to find a single reference for PDEs that addresses all of my questions, but a number of 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 or review: Farlow, Myint-U and Debnath, Brown and Churchill;

:: Need more theory: Stakgold (Green's Functions), Evans, Folland, Jost;

:: Need help on modeling: Strang, Stakgold (BVPs), Haberman;

:: Don't understand how concepts relate: John, Levine, Garabedian, Strauss, Carrier and Pearson;

:: Can't find tough enough exercises: Carrier and Pearson, 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 their relevance, completeness, clarity, and ease-of-use. And I should stress that I'm no doubt ignoring many fine favorites either through unfamiliarity or because I haven't actually used them.

SO WHERE DOES STRAUSS FIT? I repeat, all of these books address each of the needs in some measure, but no one is adequate for them 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 boundary-value problems 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 1, which is the original synthesis of work that began with Weyl. (The notes to Sec. XIII.15 of Reed and Simon 4 [Analysis of Operators] have the history of Dirichlet-Neumann bracketing, the main technical 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 perhaps be discounted to an extent. If I chose Strauss as a text, however, 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 long trip, 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, despite their having been written independently, insofar as I know. Both approach the subject 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 that period that publication date may not be as much a factor in selecting a single reference as it might 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 or Crank. 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 but lacks a thorough treatment of diffusion, Fourier's work having long been available. Because of its age it's possible to see in the books' discussion much of the early intuition of Hilbert and Sobolev spaces, and of weak solutions, 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. For example, the first and second chapter, respectively, in his two books cited above lay out in the cleanest way the variational approach to PDEs by means of Dirichlet's Principle and its relationship to orthogonality and Hilbert space. (It's worth noting in passing that he returns to this problem yet again in one of his more recent books [not cited], where he uses Dirichlet's Principle in electrostatics as an archetype for rigorous modeling in quantum field theory. Once more the exposition is simply superb. Details are on this review's Listmania page.)

TRANSITIONING TO NUMERICS. If I also took Gil Strang's new book to ease the transition to 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, less so perhaps for existence, but certainly for construction.

What makes this book a tour de force, however, is the way in which it consistently applies a four-part approach to mathematical simplification across a diverse set of interesting problems via constructs like "circulant" and "stiffness" matrices. In this approach nonlinear becomes linear; continuous becomes discrete; multidimensional becomes one-dimensional; and variable coefficients become constant. Thinking in this way is great for structuring numerical problems and for coping with the confusion that so frequently accompanies the initial burst of incomplete and uneven numerical results.

Actually producing those results is another matter entirely, of course, and I've given some idea of the books I've found useful for numerical problems in my brief review of Chung's book on computational fluid dynamics and its accompanying Listmania page.

HOW ABOUT PERTURBATION SOLUTIONS? Finally, if I thought I'd usually be able reduce my PDEs to ordinary differential equations (say by separating variables or using a transform), 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 thought I'd be stuck in the general case, and so couldn't reduce my PDEs, I'd take Kevorkian and Cole, which deals with perturbation solutions of PDEs directly, and Verhulst, which is a bit longer on intuition.

The brave might also consider Van Dyke, which deals specifically with singular perturbations of the Navier-Stokes equations and other equations of fluid dynamics. 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 (or possibly Myint-U and Debnath) 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 Garabedian cleans things up nicely without doing violence to the concepts, which is critical, once one moves to treatment of the general Cauchy and (especially) Dirichlet Problems. Indeed, I would be suspicious of any discussion of these problems that's much simpler than Garabedian's. 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. 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 having 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. All that's needed is a result that answers the real question, a notion that itself is sometimes quite elastic. 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.

This 1992 title by Walter A. Strauss (professor at Brown) has become a standard for teaching PDE theory to junior and senior applied math and engineering students in many American universities. Having been the actual class grader for two terms in 2004-2005, (and another year an informal teaching assistant), I found many of the students struggling with the concepts and exercises in the book. Admittedly the style of writing here is dense and if the reader does not have a strong background in the requisite topics (including physics), 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.

The second edition (2007) adds new exercises, subject material, comments, and corrections throughout. 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, water waves). 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 math 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, please make sure to check out this book's official accompanying student solutions manaual for extra help on doing the homework problems, and hopefully learning your course's target material in a more effective manner.

May I begin by stating that my critique is based on having read and used the first 7 chapters, with some familiarity with chapters 9, 11, and 13. With that said, just about every NEGATIVE comment and review posted prior to this review, I believe, is for the most part quite accurate and nightmarishly true. In particular, Strauss states the obvious, while omitting key and crucial steps (this isn't limited just to his proofs). One might notice that the last comment is similar to the Rudin style. Let me assure you that unlike Rudin, Strauss' presentation is not elegant, it does not inspire, and simply cannot be compared to Rudin. Some other major flaws include: hasty organization, lack of depth and breath in theory, and the problem sets consist mostly of trivial proofs and unimaginative applications. I would not recommend this book under any circumstances. If you want to learn PDEs, take the graduate course.

(Continue reading only if you have to use this book for a class)

If you are unfortunate enough to be forced to read this book, here is some advice:

::Prerequisites::
It is explicitly stated in the preface that this book is intended
for undergraduates at the junior/senior level. I believe that in order to learn anything meaningful from Strauss, it requires that you have already had the following courses: calculus, multivariate calc (vector calc), linear algebra, analysis, and ordinary differential eqns. (Complex analysis, is not necessary, but does illuminate specific areas. Fourier analysis, is not necessary. Since half the books tries to establish main theorems of Fourier analysis--may I add, not at a rigorous level.)

Part of the reason why this book is abhorred so much, is that it assumes the reader is somewhat 'mathematically mature'. To many this process begins after multivariable calc (unless you took the rigorous honors) with an *upper* division linear algebra or analysis course (lower division lin. alg. doesn't count). This is where the student is first asked to write his/her own proofs. After such courses, expressions such as "it is easy to show..." and "the reader can verify for him/herself..." imply that the student is encouraged, if not expected, to actually do it for themselves.

The student who has taken the aforementioned courses, should be adequately prepared to read Strauss. However, students with this much (actually its not much at all) preparation will probably find Stauss' book a joke and lacking in rigor. Ch 5 on Fourier Series attempted to develop L_2 theory and tried to set Fourier series on a rigorous base, but I feel it failed miserably.

::Conclusion::
Only mathematics majors will probably get something meaningful out of this book, but only if they are sufficiently prepared. Other like engineers or physicists, could learn PDEs from this book, but it is highly unlikely.

::Recommendations::
If you really want to learn PDEs, you should skip taking an undergraduate PDE course that uses this book and take a Graduate PDE course. You will heavily on analysis, thus try to take an Analysis Honors course at your school.

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.

This text is probably quite useful if you already understand partial differential equations and just need to review topics that you have already covered and grasped in the past. It outlines all aspects of introductory PDE's well, and in the appropriate order. There are well-thought out problems at the end of each chapter with answers to selected exercises that will reinforce your recall of the material. However, if you have never studied partial differential equations before, you will never learn the subject merely by reading this book. Yes, the style is comprehensible and conversational, but the derivations leave out many important steps, and you will never be able to work the excellent problems at the end of each section merely by reading and understanding each section of this book. Instead, if you are a newby to this subject and you are forced to use this book as the result of taking a course in which it is the assigned textbook, I recommend that you use "Partial Differential Equations : An Introduction" by David Colton (ISBN 0486438341) in conjunction with this text. Colton's book does a pretty good job of mopping up after Strauss, in that Colton's text takes the time to show adequate proofs and examples of sufficient complexity that you can understand the material. In addition, it pretty much covers the same subjects as Strauss in the same order only with much more detail. In addition, Colton also has good exercises for each chapter and also has answers to selected exercises. In summary, read Strauss as an outline for review and problem sets, read Colton's book for a good explanation.

I have only read bits of this book, but every time I read it I come to the same conclusion. I think the previous reviewers have highlighted the key problem with this book. A previous reviewer wrote: 'If you are a Scientist 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.' There is some truth in this. However, in the preface Strauss wrote: 'This is an undergraduate textbook. It is designed for juniors and seniors who are science, engineering or mathematics majors.' It would appear that the book is simply not very suitable for the wider audience it was intended for - in particular it is too advanced for less experienced undergraduates. Also, despite containing numerous applied examples, these examples are dealt with so briefly that they would make little sense to a scientist or engineer who has not already studied the applied material in depth on other courses. This is somewhat inevitable as the book attempts to include a wide multitude of examples, and so its strength is also its weakness. This book is not a 'one size fits all' but was clearly advertised as one and ends up simply belittling itself in trying to be one thing whilst actually being another. It is a very good reference, however, and is very valuable to scientists and engineers who have already studied much of the material in an applied context, but the text is somewhat disjointed as it reads more like a catalogue of PDEs rather than as a 'how-to-do-it from first principles' manual. The book looks attractive, but every time in the past that I started reading it I soon put it down again! However, I have begun to read the book seriously from cover to cover. Maybe a taught course that is constructed around the book would work, indeed it is a valuable source of examples, but this is not a good book for self-taught study, except perhaps to those with more maths than a typical junior undergraduate. I would however recommend it as a must in the library of anyone who deals with PDEs on a frequent basis, or who wishes to teach the subject. In short - great for the right audience but the right audience is not exactly the one advertised in the preface! (This is a common failing of many textbooks). Since I began to seriously read this book from cover to cover and I am finding it fairly straight-forward, but then I have studied PDEs before. However, those who do not like the book should be comforted, because at first I never liked it either - but give it a chance and read it carefully and remember that many of the examples will not make complete sense until you have studied the science behind them in other courses. In this sense, the text is a valuable reference to senior science and engineering graduates. I think for someone new to the topic the text probably skips the odd crucial sentence of explanation. However, for an advanced mathematics, science or engineering undergraduate the strength of this book is that it puts everything together and so is a valuable reference and valuable to consolidate what you have studied on other courses. However, because it is so all-inclusive, it would need to be several hundred pages longer if it were to describe everything clearly from first principles (even assuming competence with ODEs). So, if you still don't like it, then come back to it in a year or so. I shall update this review if, as I continue with the text, my view changes.

Despite its title, this book is not really written at an introductory level -- at least not at an introductory *undergraduate* level. For example, more than a passing familiarity with rigorous "epsilon-delta" type proofs is useful when reading Strauss' book, and being comfortable with topics like the convergence of sequences of functions couldn't hurt, either. In short, a background in real analysis, at least in one variable, is probably good to have before attempting to work through this text -- not so much because the material is heavily dependent on real analysis, but because an analysis course primes you for the kind of thinking that you need to do a lot of to sift through this often-dense material at a reasonable pace.

I don't believe this book deserves to be criticised so harshly. It does tend to jump from around a bit from one topic to the next, but it covers an impressive amount of ground concisely and with adequate clarity. If you're a total newbie to PDEs, try Stanley Farlow's Dover book instead, then come back to Strauss to get to the next level. The Schaum's Outline on Fourier Analysis is also good if you want lots of practice solving actual engineering-type problems.

If you have the appropriate math background and motivation to study PDEs at an intermediate level, then Strauss' book is probably one of the better ones.

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

This text is full of very nice and very practical topics relating to solving PDEs. This information unfortunately is largely inaccessible to most readers as the writing style is just downright BAD. OK - so nobody expects a course in PDEs to be anything but a formidable challenge. As such my complaint against this book is not that it is a difficult read - rather, my complaint is that it is unnecessarily so. There are many parts where arguments take large needless "logical jumps," concepts are explained poorly and examples are often not helpfull at all. For every section of every chapter, I did not see much difference between the writings of this text and what an instructor's lecture notes may look like.
The uniqueness - and as much as I hate to admit, what makes the text "good" - is in its treatment of methods for solving PDEs other than separation of variables. Using alternative methods to solve well known equations (i.e. wave, heat etc.) has the advantage of illustrating the differences between the solutions(For example does information "travel" towards infinity or dissipate?). These differences are impossible to teach through separation alone which treats all such equations as if they were the same. There is also the obvious problem of students ending up in a bind when they eventually come across a non-separable problem.
All in all, the book illustrates a very nice outline of what a good first course in PDEs should be - but it does just that and nothing else. Its unfriendly presentation of the material makes this work pedagogically unsound.

Developing an understanding of PDE's is a tough task, don't make it tougher by using this book (unless you are already somewhat familiar with the subject)! This text is extremely difficult to follow if one is new to partial differential equations. I used it several years ago when I took an undergrad course in the subject and found that the presentation of the material was too concise and the exercises unhelpful. If you're looking for a decent intro text for self study try "Basic Partial Differential Equations" by Bleecker and Csordas.