Partial Differential Equations: An Introduction [Hardcover]

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.

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.