First Course in Partial Differential Equations with Complex Variables and Transform Methods [Hardcover]

I used this book for a yearlong course in PDE's when I was an undergraduate. I really wish I had had that course before quantum mechanics, as many of the difficult mathematics there would have been cake after weinberger. This book is fairly terse but quite complete. The text can be hard to follow by onesself and I often would refer to easier books on PDE's to get over certain humps. However, weinberger covers most of the material that any undergraduate would want and does it at a relatively high level. The problems are simply posed but are quite solvable and all have answers in the back of the book. My recollection is that the book has very few typos. The discussion is more mathematically rigourous than many more popular books but not so rigourous that it is painful for someone who views the mathematics as a tool. The book isorganized into about 90 sections, each corresponding roughly to a 50 minute lecture there are problem sets of approximately 10 problems for each of these. More recent topics touching on nonlinear PDEs and the like are not here as the book was written in 1966. If you want to get a very solid background in PDEs that will leave you in good stead for conquering books like Jackson's Electrodynamics or Schiff's quantum mechanics, this is a good book. If you want a simple introduction, for limited use go somewhere else like churchill or powers.

This is not so much a book as it is a published version of lecture notes: it might make a useful supplement to a textbook in a course, or a useful review text for someone who already knows the subject, but it is useless as a standalone textbook for a course or for someone studying the material on their own. The book dives into specific applications and problems without giving much theoretical background. Even for a student with a strong background in ODE's and some exposure to PDE's, this book will be difficult to follow and understand.

The book is essentially a walkthrough of solutions to specific problems. There is little discussion of the theoretical foundations behind the material, little motivation for any of the solutions, little discussion of why a particular line of attack was chosen for a specific problem.

The reader is left with the ability to carry out specific techniques but without any global picture of how to know which techniques are appropriate for which situations.

I got this book after finishing Farlow's Partial Differential Equations for Scientists and Engineers. This is a much more mathematically rigorous and sophisticated book, and it took me a lot longer to go through it. (I still haven't done the last two sections on the Laplace transform and approximation methods). My original goal was to learn complex analysis, and since Farlow recommended this book for that topic, I started in the middle with Section VIII, Analytic Functions of a Complex Variable, then did IX and X, then I through VII. This caused a few difficulties solving some of the problems in the later sections due to unfamiliarity with material presented in the earlier sections. E.g. problem 71.4 presumes a knowledge of Green's function, which is covered in section V. Nonetheless, this worked out OK. I now feel as though I have a pretty solid grounding in the material (by which I mean not that I can remember it all, but I can recognize when I've seen it before, and know where to look it up again). That is my operational definition of having "learned" the material.

Frankly, I'm quite surprised by the second reviewer's comment that this book would not be good for self-instruction. I thought it was great! But I had to be very patient and resist the urge to move ahead before I really understood what I had just read. Doing the problems is an excellent reality check! That's one of the features of this book that I liked the most: lots of problems, with ALL the answers in the back, so you can't fool yourself. Most of the answers can be reached with persistence, although I was stumped by a few. But don't buy this book if you don't have a background in calculus and ordinary differential equations! It would also be helpful to review the topic of partial differential equations in an easier format, like Farlow, before progressing to this text.

The material is interesting and varied. There are sections heavy on abstract math and proofs of theorems, other sections mostly practical in orientation. The section on complex analysis lives up to its billing in Farlow. In general, I found that I could skim through the really abstract material without losing the thread of the argument. I'm not good at proofs so I skipped over most of the problems that called for proving something. But at least now I know what is meant by existence and uniqueness proofs, etc.

The book has very few errors. I've listed the ones I think are errors below. Maybe they are and maybe they aren't!

p. 182 second paragraph R(r) sin(n*theta) etc. should be R(r) sin(m*theta) etc.

p. 192 fifth line from bottom "degree n in x,y, and z" should be "degree n in x,y, and r".

p. 223 second equation from bottom should be du2/de*du1/dx - dv2/de*dv1/dx + i(dv2/de*du1/dx + du2/de*dv1/dx)

p. 265 last equation right hand side integrand should be e^(-i*theta), not e^(i*theta).

Solutions to exercises:

1.4: rho subzero du1/dx + d rho subone/dt = 0

5.4: answer can be reduced further to 157/4096, which is how my calculator displayed it

7.5: last line: dA/dt greater than or equal to 0

10.2: second line: epsilon 3 = z*sqrt(alpha/D)

14.1e: e^(-n^2*pi^2*t-t-x)*sin(n*pi*(x-1))

18.4: right side of equation should be 1/2 (e^pi + e^(-pi)) i.e. cosh(pi)

31.4: one times first sum, not two times first sum, and difference between first sum and second (double) sum, not sum of the two

32.3: second line, numerator, sinh term should be sinh (sqrt((n-1/2)^2 + (2m-1)^2 +1) * (1-z))

37.1: substituting xbar = x+1 in formula for c sub-n I get c sub-n twice what answer gives.

37.4: formula for c sub-n: integrand should include (1+x)^-1, not ^-2.

44.4: the exponential t-term is e^(-(mu sub-k supra-(n+1/2))*t), not e^(-mu sub-k supra-((n+1/2)*t))

50.7: log term is log(2 plus or minus sqrt(3))

53.4: the part of the disk to the LEFT of the line....

58.2: ? pi*i*(e-2)

58.7: ? -pi/3

Happy self-instructing!