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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?

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    $\begingroup$ What are the formal prerequisites for the course? In the US curriculum, I would think these prerequisites would be Calc 3 (multivariable calculus), Linear Algebra, and perhaps a course in ordinary differential equations. The notion of a "remedial" course is jarring, but this might be cultural or regional. "Remediation" for an upper division math course is a strange framework. $\endgroup$
    – user52817
    Jul 30, 2023 at 23:35
  • $\begingroup$ The formal pre-reqs include Calc 1, 2, and 3, as well as Linear Algebra and Differential Equations 1 (ODEs). The remedial course is something specific at our university. $\endgroup$ Jul 31, 2023 at 13:11
  • $\begingroup$ @brodybjones What is so "strange" about it is that remediation is typically a more common framework when the students have changed institutions or tracks educationally and what is an earned credential on paper is not deemed sufficient by the new party involved (e.g., nominal completion of algebra in high school but entrant could not pass a university proficiency test, getting a low score on an AP exam and having to re-take a course already taken). Is this point in the curriculum a common point of entry for students at your school? .... $\endgroup$
    – Steve
    Jul 31, 2023 at 17:38
  • $\begingroup$ If not, it seems like the problem could be with the prior courses' curriculum or grading... could be that students are taking and passing multiple service courses calc, ODE, linear algebra, etc. but necessary material is not covered or not tested thoroughly enough in those courses to ensure future success. $\endgroup$
    – Steve
    Jul 31, 2023 at 17:40
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    $\begingroup$ In the US, university/college 3rd-yrs ("juniors") in 2022-2023 would have been 1st-yrs in 2020-2021, a bad vintage in math learning here, with knock-on effects. Even if they completed calculus in spring 2020, many AP courses just stopped in March and did not complete the curriculum. Just as the grown-ups didn't know what to do back then, now they're unsure how to deal with the effects. Remediation is a reasonable thing to try, even though as a vogue word, it is way out of fashion. $\endgroup$
    – user12357
    Aug 1, 2023 at 14:45

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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.

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    $\begingroup$ For Fourier series and orthogonality, I would add integration techniques, especially integration by parts. And a review of even and odd functions might help. $\endgroup$
    – J W
    Jul 29, 2023 at 5:28
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    $\begingroup$ @JW excellent points. Integration by parts with symbolic variables are a must for the calculation of Fourier coefficients and the even/odd trick is very important when we're working on a domain which is symmetric about the origin. $\endgroup$ Jul 29, 2023 at 15:04
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    $\begingroup$ Seconding (thirding?) integration by parts. I didn't really grok why anyone should care about IBP until I took PDEs in grad school, and saw what a powerful theorem it is. I have often said that I would like to write a musical about PDE. It would be called Ad Hoc, and the first act would include an "I want" song called "Integration by Parts". $\endgroup$
    – Xander Henderson
    Jul 29, 2023 at 21:43
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    $\begingroup$ For me (and it's been ~20 years now), I felt like my biggest struggles in PDE were how intensely we used the later ODE topics we had quickly skimmed over. I'd always felt strong with the earlier tools. Though I do wonder if PDE courses tend to be a little more quirky in what they cover/lectures (mine certainly was; I know any course can be that way, but maybe lower undergraduate courses, taught more, to wider audiences, offer hopes of being more consistently defined and taught by profs perhaps more driven/experienced to teach to a "wider" audience) So wonder if it may be hard to narrow down $\endgroup$ Jul 29, 2023 at 22:45
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    $\begingroup$ I can see IBP being a fair bit too... I did feel we didn't use it much in latter courses after learning it, vs things like the quadratic equation, trig functions/derivs, etc. And some other tools less often used also felt tough to dust off and use intensely (things like maybe partial fractions, Taylor functions, Green's function... don't remember which were used more/less). But I specifically remember piles of LaPlacians that we hadn't been so intensive at in ODE. And that it seemed, while other algebraic tools seemed more pattern recognizable, ODE tools maybe just randomly popped in wherever $\endgroup$ Jul 29, 2023 at 22:57
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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.

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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.

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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.)

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  • $\begingroup$ "This is a very math-y answer, not a pedagogically sound answer." That depends on the definition of "pedagogically sound". If you want the students just to follow algorithms for solving various standard types of PDE's, you are absolutely right, but then we do not need those students: Wolfram Alpha does it better. But even if you need them, for example, to change the coordinates in a non-linear (or even linear) way to reduce an equation to some manageable form (say, wave in 2D to $u_{xy}=0$), which is a level zero skill in PDE, matrices do jump in immediately. :-) $\endgroup$
    – fedja
    Aug 4, 2023 at 2:49
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    $\begingroup$ Why do you call yourself "guest troll"? $\endgroup$
    – Dominique
    Aug 4, 2023 at 7:41

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