Partial Differential Equations for Scientists and Engineers (Dover Books on Mathematics) Reprint Edition

If you'd like to teach yourself the subject of partial differential equations, and you have a decent background in calculus and ordinary differential equations, this book is perfect. It is composed of 47 chapters each of which is only a few pages long and covers an important topic, with exercises. The author is very good at explaining potentially complicated ideas in simple terms. It's all very practical, with no theorems or proofs. At the end of each chapter is suggested reading for exploring the topic in more detail. An auto-didact couldn't ask for more. I had so much fun going through this book!One of the reviewers mentioned that the answers to the exercises had a lot of errors, and I agree. I've listed the ones I found below, with the caveat that maybe a "typo" reflects my faulty understanding. You can decide for yourself. Other than this, I can't find anything to criticize in this marvelous book.Some specific comments:Table 13-2: although the separation of variables method is listed as being inapplicable to nonhomogeneous boundary conditions, in fact it can be used to solve Dirichlet problems on a rectangle with one non-homogeneous boundary.Lesson 32 p. 251: Laplacian in spherical coordinates fourth term should be cot(phi), not cot(theta).Lesson 39 p. 320: step 2 of implicit algorithm for heat problem: u11 and u16 should be zero, not 1, so first and fourth equations equal zero, not 1, and final result is u22 and u25 are 0.2, not 0.6, and u23 and u24 are 0.6, not 0.8. These results are closer to the results given by the analytic solution u=pi/4 times sum n odd sin(n pi x)/n times exp(-n^2 pi^2 t).Lesson 41 p. 338: step 3, the coefficients of the new canonical form are computed from equations (41.3), not (41.5).Lesson 44 p. 359: J(y)=1.28, not 0.46.Lesson 45: p. 369 problem 2: I believe new function z(t)=(1-t)y(t), not (1-x)y(t).Problem 5: A=.004, not .06, and B=.097, not .04. The values given in the book do not satisfy the boundary condition u(x,1)=0. The correct values can be calculated from the analytic solution u(x,y)=((cosh(pi y)-1)/pi^2 - (cosh(pi)-1)/(pi^2 sinh(pi))sinh(pi y))sin(pi x).Lesson 47 p. 385: I think gamma=t/((x-t)^2 + y^2), not 2t/(...). This gives results for u^2+v^2 close to those listed in (47.6), whereas using the result for gamma given in the book gives u^2+v^2=3.95 and 23.9.Page 386: phi(u,v) and phi(x,y)=0.53 ln(u^2+v^2)+1, not 0.57 ln etc.Answers to Problems:8.1: u(x,t)=4/pi exp(1/2(x-t/2)) etc, not 4/pi exp(-1/2(x-t/2)) etc. Also in the sum there should be a term exp(-n^2 pi^2 t).9.3: sum should be from n=1 to infinity, not n=0 to infinity.9.5: T subscript n (t) = (-1)^(n+1) etc, not (-1)^n.12.3: denominator should be sqrt(4 alpha^2 t + 1), not sqrt(4 alpha^2 + 1).13.3: alpha should be 1.20.5: both terms should include 8h, not 4h.24.2: given solution doesn't satisfy initial conditions. I believe u(x,t) should be 1/2((x+ct)+(x-ct)).25.2: the exponents of e should be minus and plus (n^2 pi^2 alpha^2 - b)t, respectively, not minus and plus (n^2 pi^2 alpha^2)t.25.6: second equation should equal 6 pi + 1 for n=3, not 8 pi + 1.28.4: log term for u(x,t) = ln(abs(1-t/x)), not -ln(t+1).35.5: calculation for a subscript n can be taken further to get (-1)^((n-1)/2) times(2n+1)/2^n for n odd, zero for n even.37.3: u i,j = 1/4 (etc etc) not 1/2 (etc etc).37.4: denominator is 2(h^2-2), not 2(h-2).39.2: u i,1 = 1, not zero.41.3: I got u epsilon epsilon + u nu nu +(nu^2/(2 sqrt(2)) u nu = 1/2 exp(-nu^2/4), but this is so different from the book that it may be my bad.45.2: should be (z'/(1-x) + z/(1-x)^2)^2, not z'/(1-x) + z/(1-x)^2.Appendix 3: 3-d spherical Laplacian all thetas should be phi's and vice versa.

I have studied advanced math and practice it in my professional life, but this book was a "fun read" for me (I don't have a need for PDEs in everyday work), and hence my reading (and review) was focused on the presentation of ideas rather than rigor. This book was a fascinating tour of both solving PDEs, understanding what PDEs conceptually represent (in natural physical systems and geometric terms) and advanced analytical and numerical problem solving techniques. Its format is setup as 47 lessons which are succinct and incrementally build on each other. The book also contains fantastic refreshers on key ideas in mathematical problem solving, such as matrices in linear algebra, coordinate system transformations (cylindrical, spherical, conformal mappings), functionals, monte carlo simulations, the Laplacian, Fourier transformation concepts, and probably a few more I'm forgetting. My point here is this book is a gem beyond just its primary purposes (PDE solutions), and its focus on ensuring readers comprehend the concepts behind the math.

A note to the autodidact: the title of this book is not "Introduction to Partial Differential Equations." You may be frustrated if you buy this hoping to use it to introduce yourself to this type of equation. On the other hand, I bought this because it is the textbook for a course I'm about to take in partial differential equations. That course has a prerequisite course, and the textbook for the prerequisite course (which I feel prepared me well to understand this text) is Zill's Differential Equations. That's an introduction to differential equations, and really only touches a little bit on PDEs. The reason it's a good idea to work through an introductory text like that before moving on to this one is that a lot of the techniques presented here involve changing a PDE problem into an ODE problem - the techniques for solving the ODE problem are then assumed known. So, the reader is expected to know how to solve ordinary differential equations right out the gate. This book also assumes the reader is familiar with the Laplace transform and has at least heard of the Fourier transform.I still haven't digested this text fully - I just read through it once at a surface level (not doing any of the work, just reading the text and taking notes on what it covers and what I need to brush up on). What I can tell you is that this text really only covers second degree PDEs. It's split up into 47 short lessons, and there is one lesson on first degree PDEs, and one lesson with a fourth degree PDE, but all other lessons are about second degree PDEs. The author takes a strictly physical approach, both in examining the equations and stylistically (it is clearly written more like a physics text than a math text). Equations are never introduced without first introducing a physical situation they can model. Instead of trying for a general method of solving PDEs (which does seem impossible), the author shows a few methods of solving a particular family of PDE, then moves on to show a few techniques of solving another family of PDE, and continues this until all simple families of second degree are covered. Essentially, this adds up to a textbook which shows you how to solve three second degree PDEs, in many different ways. There is also a lot of attention paid to various methods of transforming a problem so that it fits into one of the families covered, generalizing those results pretty well. I'm pretty confident that if I learn the material covered in this text very well, I'll be able to solve pretty much any solvable second degree problem.There is only one lesson that covers matrix methods of solving a system. Of course, another limitation is that the author thinks so physically, you will not see any work in this text dealing with more than four-dimensional independent variables. Any time a special case of a solution doesn't make physical sense, the author discards that special case even if there is mathematical meaning to it. Not a problem for you, if you're a physicist!