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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.Read more ›
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.
In the introduction of this book the author says the text was meant for an undergraduate level course... we are currently using the text for my graduate level class. The text is vague and there are virtually no examples. Many of the proofs normally spelled out in a text book are actually exercises. There is a solutions manual, but the manual does not contain all the solutions--just the work for the ones which already have the answers in the back of the book. If you are looking for a challenge or perhaps a review of PDEs this is the book you want. However, if this is the first time you've ever seen PDEs or you are unsure of your math capabilities you might want to have another back up text for clarification like "Applied Partial Differential Equations" by Haberman or "Partial Differential Equations for Scientists and Engineers" by Farlow.
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If you have any say in the matter read Bleecker and Csordas. Strauss is not a bad book, but it glosses over some relevant details in it's concision. You will find Bleecker and Csordas more comprehensive. Bleecker and Csordas is comparable to John M Lee's text in smooth manifolds in terms of readability.
I took a course on partial differential equations last semester using this book. At first, I was put off by how small the book was - I did not think it would be enough for me to learn from - but after working with it for a couple of weeks, I realized that it was just the right length. The chapters are very concise, but they have everything they need and a few examples per idea.
I found that most of the practice problems in the book are just slightly beyond the examples given, so you do have to work to get the right answers. It wasn't easy, but it definitely added more to my understanding than plug-and-chug questions would have. All in all, I am definitely keeping this book as a reference and would recommend it to others.
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