Looking for resources to help me understand the basics of PDEs for physics

TL;DR Summary: I am taking a math methods course this semester of which a large part are PDEs. I don't understand the context behind the order in which we are solving PDEs. I am interested in learning how other people were taught PDEs and any book recommendations you might have.

I am taking a math methods course this semester of which a large part are PDEs. I don't understand the context behind the order in which we are solving PDEs.

What we have done thus far is solve the wave equation for the homogenous case and some non-homogenous cases. The methods we have used are separation of variables and eigenfunction expansions. I am not sure I understand the intuition behind certain steps we are taking to solve the equations e.g. letting the solution u(x,t) = w(x,t) + v(x).

In looking for other books that are available like Farlow and Strauss, I noticed that they use different methods like the method of characteristics and that they don't consider some of the problems we have solved to be important. I would be interested in hearing the order of how other people were taught PDEs, especially in terms of the physical intuition behind certain substitutions and if there are any introductory PDE book recommendations that focus on the physical intuition.

Also, I am also interested in recommendations for books on numerical methods for PDEs.

It is not quite clear what you have trouble with. Order is a bad adjective when it comes to differential equations since it is a technical term. I would definitely search for lecture notes on university servers, e.g. by the search key "Partial Differential Equations + pdf". The "+pdf" part is necessary to end up with actual lecture notes and not on someone's homepage.

My search found:

• Toronto, ~400 pages
  https://www.math.toronto.edu/ivrii/PDE-textbook/PDE-textbook.pdf
• India, ~400 pages; a book, I'm not sure that it does not violate copyright, therefore no link
• Iran, ~650 pages; a book, I know it violates copyright, therefore no link
• Dallas, TX; a book (P.Olver), it should not violate copyrights (TU@Dallas server) but I think it does, therefore no link
• UCLA (maybe the same as Toronto), ~400 pages
  https://www.math.ucla.edu/~yanovsky/handbooks/PDEs.pdf
• Finnland, ~570 pages
  shockingly again a university server that violates copyrights, therefore no link
• UCSB (Santa Barbara, CA), ~100 pages, mainly physical applications like wave and heat
  https://web.math.ucsb.edu/~grigoryan/124A.pdf
• Austria, Vienna, ~400 pages
  https://www.mat.univie.ac.at/~gerald/ftp/book-pde/pde.pdf
• Netherlands, Nijmegem, ~100 pages
  https://www.math.ru.nl/~burtscher/lecturenotes/2021PDEnotes.pdf
• Germany, Hannover, ~400 pages, numerical approach
  https://www.repo.uni-hannover.de/bi...1/lecture_notes_Numerics_PDEs_Jan_28_2020.pdf
• Germany, Ulm, ~150 pages
  https://www.uni-ulm.de/fileadmin/website_uni_ulm/mawi.inst.010/PDE/pde-skript-1011.pdf

All these links are in English. You see, that you can find many of them even without violating copyrights.

I found Olver's book a lot, however, I remember that someone here once said that he tends to be a bit sloppy when it comes to details. On the other hand, he is good to read.

In case you have specific questions especially about the "context" of something, then ask in our technical forum
https://www.physicsforums.com/forums/differential-equations.74/

By order, I mean the content that is usually covered i.e. what methods are used, mathematical background topics like the dirac delta function or the Fourier transform etc.

I do not know much about pde's, or physics, but I can still suggest some books you might look at, judging them by their titles, their introductions, and the pedigree of their authors. My impression (from reading the intro to book 1) below) is that there is no general theory of pde's but there are important examples, heat equation, wave equation, and Laplace equation prominent among them. I myself have used the heat equation in my own research in complex algebraic and analytic geometry, and of course the Laplace equation is fundamental to complex and harmonic analysis.

I have on my shelf these two books and suggest them to you for a look;
1) Lectures on Partial differential equations by the great mathematician V. Arnol'd, and
2) Differential equations of mathematical physics, by L. Hopf, an assistant of Einstein.

1) contains far more than 2), and at a far higher level.

Different methods for solving PDEs are useful in different cases. There may be cases where the method of characteristic works well and some cases where it doesn’t. There may be cases where separation of variables works well and other cases where it doesn’t.

As for why we generally have a penchant for teaching separation of variables extensively: it is intimately tied to many concrete physical concepts such as eigenmodes of oscillation or the energy eigenstates of quantum mechanics.

Coming to why we do things like this:

Philip551 said:

I am not sure I understand the intuition behind certain steps we are taking to solve the equations e.g. letting the solution u(x,t) = w(x,t) + v(x).

This may seem like a silly approach that does not net you anything. However, the point is typically to separate a more involved problem that you may have into easier subproblems. In this case, u may be the solution to something like an inhomogeneous wave equation. Assuming some inhomogeneoties to be stationary, the ansatz would be to introduce v to be the time independent function solving the stationary problem, thus removing the stationary inhomogeneities from the problem for w. You have likely encountered similar methods in an ODE course, where you have written the solution as a homogeneous part plus some particular solution. The main difference in the example you give is that the particular solution is assumed not to depend on time.

Partial Differential Equations by Sommerfeld