I am a student helping to develop a remedial course for other students who have recently failed the undergraduate PDE course at our university. The topics are provided from the syllabus in the pictures. What specific mathematical topics (not programming) do you recommend that students be proficient in, in order to succeed in this course?
They must understand the quadratic equation and how to factor it, solve it and manipulate it formally in both the real and complex case. Also, how to solve simple trigonometric equation and knowledge of the graphs of basic trig. functions such as sine, cosine and tangent. In addition, they must understand the solution to 2nd order constant coefficient ODEs in depth. In particular, the use of hyperbolic cosine and sine for the formulation of solutions in parallel to the usual introduction of sine and cosine and exponentials is helpful. Of course they must have a strong grounding in partial differentiation and power series calculation. Of course, I speculate, since I have no idea as to why the students you target have failed. But, in my experience, students struggle a lot with things which are actually from before calculus. Algebra and trigonometry are important.
I would say that a good knowledge of linear algebra is paramount (for many reasons). About equally important is the full mastery of the differential and integral calculus of one variable. With these two, you can introduce the differential and partial derivatives pretty fast even if the students haven't seen them before and just wave your hands about multidimensional integration (though, of course, if they've seen the multivariate calculus too, it will be helpful as well).
Beyond that they should just have common sense. That is not taught in any cookbook courses, so some course developing it should be taken (discrete mathematics, elementary number theory, etc.; just something where you need to think and put a few things together for solving problems, not just to follow a prescribed algorithm).
With these three, you should be just fine filling some gaps in on the fly, if necessary. Without one of those, you'll struggle no matter what else you put as formal prerequisites. Warning: students who have successfully completed all three courses I mentioned are generally rare.
In my experience, the difficulty with this course (the course description is very standard) is that students need to have actually mastered all of the prerequisite skills to a fairly high standard.
To solve a BVP by separation of variables you will need to solve several ODE problems, each of which requires solving multiple calculus problems, each of which requires solving multiple algebra and trig problems. If you aren’t 95% accurate in each of the sub problems then there is little chance you’ll get a correct solution to PDE boundary value problem.
I think polling the (very probably small) group of students would be a better way to determine the answer than looking for some general trend, here. The reason I say this is because PDEs are really quite advanced compared to the average juco/4-year remediation situation (students trying to get through calculus who have basic problems in algebra and even arithmetic).
Even in a situation with significant amounts of social promotion and students getting by predecessor classes with low grades, you are talking about students who have passed calculus and ODEs. Even if not all-stars, they have to have some reasonable skills to have not washed out previously. With that in mind, I'd even say that remediation of previous courses (yes, even if not flawlessly understood!) is NOT the key metric for helping the students. Much more likely is to just give them a second go through and have them concentrating more on the topic.
P.s. I disagree with the " more linear algebra" answer. This is a very math-y answer, not a pedagogically sound answer. While there may be some interesting foundations/connections of LA to PDEs, this is not helpful/needed for a struggling learner in an engineering style PDE course. Pulling down my engine math book (similar to the text you have here) and looking at the PDE chapters, there are asymptotically approaching zero matrices. It's really more like ODEs, with some annoyances of series futzing (which is a lot of x-pushing algebra) and the engineering based derivations (which I disagree with, think it's harder to learn a math technique in the context of an applied word problem...prefer the ODE approach, where you learn the math first and then practice it, as math...and then later do applied problems with the added cognitive load of translating to/from physical to equation land.)