# Reference: solved problems and exercises on PDEs

I'm looking for books/lecture notes that contain solved problems and exercises on PDE.

More specifically, I'm interested in

• basic 'computational' exercises;
• more theoretical/advanced problems (also with some functional analytical flavor);
• problems that are more "numerical" in nature (maybe based on software like Matlab or Mathematica).

Some references can be found at Supplemental reference request-Graduate level PDE problems and solutions book, but I'm looking for something more both basic and advanced.

• Do you know finite difference method for example ? Commented Sep 8, 2017 at 11:57
• Please, tell me how my answer did not fulfill your expectations and maybe I can improve it. Commented Sep 12, 2017 at 13:59

Have you tried Folland's book? It is very theoretical and he leaves a lot of details as exercises. I believe it is a good way to learn to use the main text theorems and examples as exercises, since they are solved there, and then proceed to the book's problem themselves.

Brezis' Functional Analysis, Sobolev Spaces and Partial Differential Equations may be an excelent option for you, since you look for theory connected to funtional analysis. The last three chapters are about PDE's, using functional analysis tools developed in the beggining of the book. It has a huge amount of beutiful exercises, with most of them solved in the end of the book. May be just what you are looking for.

At last, Zachmanoglu's Introduction to Partial Differential Equations with Applications may be useful for practicing computations. It has plenty examples and sections regarding computations.

For computational methods, I konw about Jeffrey Cooper's Intro to PDEs with MATLAB (I have never read it, but all Bikhauser's books I have ever read were very good, so maybe it is worth a try).

EDIT: I would also add Salsa's book, which has plenty of examples and a solutions manual.

The notes of Igor Yanovsky contain a lot of problems. All the problems are solved, albeit they contain almost no theory. You can download the notes from his homepage.
I like also the books of Salsa 'Partial differential equations in action: from modeling to theory' and 'Partial differential equations: complements and exercises'. The latter has a lot of solved problem, the first is for the theory. In these books you can find problems of computational and theoretical nature. For the numerical part... Well, I leave it to someone else.

The Problem Book in Mathematics by Komech & Komech should have what you're after.

The Schaum's Outline of Partial Differential Equations is also very good too.

This is more of a lengthy comment than an answer, because I fear you won't get a satisfactory one. Solved problems in PDE-books are a kind of rare beast. Apart from some simple calculations in the beginning of a course, in general the problems are theoretical in nature and at this point in your studies, you should be able to judge the correctness of your proof alone. (At least this would be my attempt at an explanation. The real answer is probably that books on PDE normally are quite lengthy anyway, as are the solutions to many exercises, and neither author nor publisher want to write or respectively pay for even more pages...)

Brezis has already been mentioned, apart from this, of all the PDE-books I know, there are none with solutions. The only thing which also comes to mind is "Linear functional analysis" by Alt, which is a recent translation of a German classic. I have only read the original, but it has solved exercises and it touches some topics also related to PDE.

Apart from this, if you want the really basic "solve this equation explicitly" kind of problem, you should look more in the direction of the applications, as those problems are a kind of staple for the more advanced "mathematics for physicists/engineers"-books.

However I would say that those kind of calculations are a bit outdated, since they usually only solve simple cases and do neither help with understanding the general concepts nor really help in practical applications. Modern study of PDE instead mostly consists of proving existence with an abstract argument and then maybe showing some further properties of the solution, for example regularity. There is also the numerical analysis side of things, however while some of the methods are the same (After all, if you want your numerical algorithm to converge to a solution, there needs to exist one in the first place), I have always experienced them as a different kind of crowd. Usually they are interested in solving the same few classical problems over and over again, but with more and more improved algorithms. And I am not sure, if you would count the source code of some numerical solver as solution to a given problem.

For basic problems solved out, plus a good overview of introductory PDEs, see http://www.math.ucla.edu/~yanovsky/handbooks/PDEs.pdf which is a study guide for a qualifying exam on PDEs.

For more advanced problems around the level of 1st or 2nd year graduate studies, looking up solutions to end of the chapter problems in Evans "PDEs" text is quite easy. There are many many resources online for finding answers, more so than any book I am familiar with.

I am not sure what is expected for the 'numerical' problems... are you talking about solved problems in writing finite difference schemes for PDEs and the alike? This is a whole separate field of mathematics, so some guidance on whether you're interested in certain methods (finite differences, finite element, finite volume, spectral methods, etc.) would help locate solved exercises in that topic.