What are specific set of tools Partial Differential Equations provide in studying a system? [closed]

I know what are PDEs, but I am looking to identify the major strengths of PDEs. If I have to convince a pool of engineers to use PDEs for solving a problem, let's say stress distribution in a body. What are the top three strengths I should pitch to convince them? Can I mathematically explain those strengths?

• "[...] to use PDEs for solving a problem" Instead of what?
– JRN
Sep 11, 2022 at 12:57
• PDEs are just mathematical formulas which express the relation between variables which characterize the physical system. The change in one thing prompts a change in another and how those changes are implemented is set forth in the governing PDE. How else would you understand the relation between state variables ? Machine learning ? (I jest, but I'm sure you could get funding for that) Sep 13, 2022 at 22:07
• I've voted to close this as "needs details or clarity." Why do you "have to convince a pool of engineers to use PDEs for solving [...] stress distribution in a body"? What else could they use instead of PDEs?
– JRN
Jan 8, 2023 at 12:44
• @JRN: I've voted to close this ... --- I upvoted guest troll's answer just now (not sure why I didn't when it appeared, unless I overlooked doing so or missed seeing it) because I thought the points were useful for what the questioner was probably seeking and it has what seemed to be useful ideas for others who might land here from a google search. However, when I did this I had forgotten that the question was closed (in fact, that aspect might have simply not registered with me prior to scrolling down to the answer), but apparently one can upvote an answer to a closed question . . . Jan 8, 2023 at 16:55
• @DaveLRenfro, I cannot see how one can answer "How do I show that X is better than Y or Z" when it isn't clear what Y or Z are.
– JRN
Jan 9, 2023 at 10:17

1. Dimensionality. Instead of just y as a function of x, little baby problems like first year physics, you are looking at x, y, z (sometimes). The world of engineers (well at least the whole mech-E branch, mechE and all the associated ones like aero/chem/narch/civil/etc.) is very physical and three dimensional. Things like bridges and frac jobs and nuclear reactors are all systems with multiple dimensions. You can also look at multiple output dimensions (heat, pressure, etc.)

2. Use in their curriculum. This is a COMPLETELY LEGITIMATE and TRENCHANT argument. You don't need to wax on about the wonders of math and Andrew Wiles and proofs. But be honest. There is a reason why they have an engine math course with a sampler of PDEs (or get an intro in their ODE class). You need it ESPECIALLY for topics with flow (fluids and heat transfer), which are required basic courses for most engineers. [Not the EEs, but they need a butt-ton of math anyhow since they are modeling signals and such.] Learning the basic tools in a math course makes it a lot easier to apply in a majors course. If they have to learn all that stuff while they are learning the content, it makes it much harder...learning two things at once. Also, of course straight math problems are much easier than "word problems". And their majors courses are full of word problems. So this is a chance to learn the techniques first. So, they are not scared of the freaking "Joe" (J(0)) Bessel function the first time they see it. Let them have it straight. Remember this is WHY there are jobs for math profs. To teach techniques to STEM students (math majors are a TINY minority of the population). And engineers are practical and money oriented. "This will help you pass fluids class" is a GOOD argument.

3. Engineers can't count.

P.s. Even when the PDE is ugly and hard to solve analytically, you can get good insights from the PDE itself. For example setting rate of change to zero (at equilibrium) and assessing final conditions. (This does apply to ODEs as well.) The reason I didn't list this as "3" is because it's a bit of a fine point and makes more sense with kids who are late in their course. Think it will not sell as well with a neophyte. But keep it in reserve for later on.

• "Engineers can't count." I don't understand. Are you saying that engineers aren't good at math?
– JRN
Jan 9, 2023 at 10:11
• Engineers can't count. -- I missed this earlier, and I too don't understand the point. I'm used to people saying "mathematicians can't count" as a silly way of implying that their pursuits are so ratifiedly beyond basic arithmetic that they can't count (or can't add, etc.). That interpretation doesn't seem to fit here. Maybe the answer could start off with something like "As someone who has an engineering background, let me give you two examples." :) Jan 9, 2023 at 11:35
• TYPO: "so ratifiedly beyond" was intended to be "so rarefiedly beyond". And yes, "rarefiedly" is not a legitimate word (as far as I can tell), but the usage here should be clear -- beyond in a way that is primarily of a rarefied nature. Jan 9, 2023 at 16:44