A First Course in Partial Differential Equations: with Complex Variables and Transform Methods (Dover Books on Mathematics)

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 = 05.4: answer can be reduced further to 157/4096, which is how my calculator displayed it7.5: last line: dA/dt greater than or equal to 010.2: 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 two32.3: 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/3Happy self-instructing!

Me agrada. Uso de números complejos para comprender propiedades electromagnéticas.

Excellent textbook

This textbook presents a number of issues in attempting an all-encompassing review. Let us read from the preface: "It also contains most of the concepts of rigorous analysis that are usually found in a course in Advanced Calculus." That line speaks volumes regarding the prospective reader to whom this text is aimed. If such is kept uppermost in mind, one finds that Weinberger has attained his goal. Might I suggest that a student attempting this text first complete Chapter 14 of Boas (1966, first edition) or, Chapter 7 of Sokolnikoff & Redheffer (1966), then this introduction will be much more palatable. In any event, what is here accomplished is a tricky exposition of weaving threads pertaining to partial differential equations, real analysis and complex variables. Placing that trilogy together is not easy. Be that as it may, there are reasons to retain this text as (at minimum) a useful resource for more advanced offerings:(1) The first Chapter spans the gamut: Taylor Expansions, Dimensional Analysis, Chain Rule, Even/ Odd Functions, Change of Variable. (keep in mind, that is merely the initial twenty-five pages ! ). Pay attention to the solved examples, they are indicative of the exercises. In fact, I make a stronger statement: attention paid to the solved examples will provide ample preparation for the exercises to be solved. Another hint, or at least something that I do: create your own simplified notation as a substitute for Weinberger's involved notation.(2) First chapter is the toughest ! If you complete it successfully, all else will be easier to swallow. The second chapter will describe linearity and superposition in the two-variable case (page 36, deriving a law of energy conservation, will prove useful.) This chapter will pose few issues to the reader, it is both lucid and detailed. The student exercises are (again) patterned after the examples. Pay attention.(3) Elliptic and Parabolic Equations, next (that is: no waves as of yet). Green's Theorem introduced (page 53). Maximum Principle (section 12) is one of my favorites. You will meet inequalities. That is another big hint: Do not be frightened by inequalities, learn all you can of them. Notice the discussion of heat equation (page 58-61), dimensional analysis will serve one well in untangling these formulas.(4) Next, the oft-discussed separation of variables technique. Meet: eigenfunctions, convergence in the mean, Parseval and Bessel Equations. Again, inequalities ! Highlight: A most useful discussion of Schwarz inequality and uniform convergence (pages 81-87). Again, some dimensional analysis (page 91), easier if the reader discovers a simplifying notation of their own. Keep pencil and paper handy when deriving the Poisson integral formula (page 102), the author provides most intermediate steps. Few details omitted. If the reader would take a glance at page 109, much here might appear frightening to a beginner. Facility with inequalities, their meaning and manipulation, will serve one well.(5) Much algebraic manipulation in the fifth chapter: nonhomogeneous problems. You meet Green's functions. You will revisit Parseval and Schwarz (met earlier in the text), you will revisit more Green's Functions. Read: "It follows that the integrals obtained by differentiating under the integral sign converge uniformly." (page 134) and "apply the divergence theorem." (page 139). So it goes: the student will need to check those assertions. The "advanced calculus" concepts introduced earlier are continually utilized throughout the remainder. All that was done for two dimensions is re-done for three dimensions in chapter (six: cubes and cylinders, cartesian and cylindrical coordinates, respectively).(6) Chapter seven is more theoretical. Page 170: while rather detailed, does provide a step-by-step construction of the solution. I highlight the section on properties of eigenvalues and eigenfunctions (Pages 171-175). Also, those special functions of mathematical physics: Bessel, Legendre, Associated Legendre, all preliminary to Green's functions for the sphere. A more difficult chapter than heretofore encountered.(7) The introduction to complex variables begins: Sixty-five pages, inclusive exercises for the student. A mini-course, of sorts. Of course, this chapter is preparation for the following, evaluating integrals by complex variable methods. Thirty pages of basic skill to be mastered (the example, page 295, which concludes, is an interesting application). And, that behind us, continue with:(8) Transform methods. All we learned in the previous two chapters now placed into service. Read: "...for a particular problem must frequently be evaluated by a special means, such as contour integration or even numerical methods." (page 319). Three-Dimensional wave equation is presented most effectively and segues to more complex variable techniques (pages 333-337). Fourier and Laplace transforms in much detail, here. In fact, you will utilize the Laplace transform for ordinary differential equations as preliminary to those for partial differential equations. Happily, the author provides many intermediate steps and details. The Diffraction problem is most useful (pages 362-369).(9) Approximation methods. Read: "The solutions which we obtained have the appearance of being exact. However, they require limiting procedures which cannot usually be carried out, so that an error occurs in actual computation." (page 374). What follows is an elaboration of this insightful statement. This is a nice conclusion to an interesting textbook.(10) Happily, solutions or hints to every exercise are presented at the end of the text (forty pages). Very useful, indeed. So concludes a challenging textbook. With adequate preliminary foundation, this is a useful textbook. One must be diligent in following textual derivations and solved-examples. If that be done, the text is approachable. That is not to say it is easy. Recall, the text is trying to teach three things simultaneously: Partial Differential Equations, Advanced Calculus, Complex Variables. It is structured differently than others at the same level. My copy is an eighth printing of the Wiley 1965 edition, so there was a market for the text. There remains such ! For a challenging introduction, I do recommend perusal of this textbook. It is definitely linear: that is, each section must be assimilated prior to the next.Also, it serves as a fine reference for advanced studies.

This book was a gift for the boyfriend. He uses it a lot, and says that it is very useful.

NIce book

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.

Doesn't motivate the subject well ! Better if one has a good command of Complex Analysis .