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After studying both ODEs and PDEs, it did strike me that teaching this subject is incredibly counter-productive and outdated. The most important skill in this

• j7ake 6y ago
  Agreed! The hyper focus on solving by hand puts the wrong emphasis on ODEs. They’re supposed to be used as a tool to do quick modelling of a process to get qualitative insights to the behaviour of the system. I recommend James Murray mathematical biology book for nice examples of how ODEs can quickly give insights into complex phenomena.
• enriquto 6y ago
  > The most important skill in this area is ability to compose differential equation from the description of the problem. This is called "modeling", and together with analysis and simulation is one of the three main parts of the theory. Where did you study differential equations? From my experience, it is typically taught that way, at least in the european universities that I know. Even in the most hard-core mathematics departments they may focus strongly on the analysis part (existence, uniqueness, regularity) to the detriment of a bit of modeling or simulation; but I have never seen those hidden. The "algebraic" solution of the equations is a fringe subject that is most often omitted, (except when it leads to a simple analysis, e.g. linear equations).
  • d_silin 6y ago
    In UK. We definitely spent a lot of time solving DEs of various classes, and very little on "modelling" part. But I remember that numerical methods were covered pretty well too.
• djaque 6y ago
  I think this gets at the discussion of whether education should provide you with a broad base of knowledge to appreciate a subject as well as use it or whether it should just, in a sense, add to your toolbox the minimum required methods to go into the workforce or move onto more advanced classes. IMO, I really enjoyed learning about differential equations (DEs) and found that solving them analytically, besides being a nice intellectual challenge, also helped me with advanced courses later on. All of AC circuit analysis and the signals theory classes you take as an electrical engineer, for example, are firmly rooted in the language of DEs and just being able to plug some equations into a computer misses out on all of the real understanding you need to have in those fields to use them in practice. One of the big red flags a junior electrical engineer can show, in fact, is too much reliance on simulation tools without having a good background understanding of what to expect from them.
  • dan-robertson 6y ago
    Maybe solving some DEs analytically can help one build up intuition as to how the solution may behave, and some algebraic transformations may be useful, but maybe if the goal is for people to know if a solution makes sense then we should put more of an emphasis on the techniques to learn qualitatively about the solution (eg phase diagrams with isoclines and equilibrium points, perturbation analysis, various linearisations). I wonder how important it is to focus on linearity. On the one hand, lots of equations aren’t linear but on the other hand, most equations are locally linear and knowing about linearity helps with Sturm-Liouville theory.
    • neutronicus 6y ago
      Perturbation theory and linearization all end with you needing to crank a solution to a memorized DE
• gaze 6y ago
  When I was a grad student, another student was trying to simulate a particular stochastic differential equation. He spent a few weeks writing a parallelized solver, learned to use MPI, and distributed the computation over our school’s cluster, got the results he needed and moved on with his life. When I had encountered the same problem later after studying SDEs in a bit more detail, I found that I could reduce the problem into something I could solve on my laptop in about 10 minutes. We can argue about how much time he and I put into the problem from ground zero. The thing I’m arguing is that one can spend a lot of time misusing the building blocks if one doesn’t understand them. Existence and uniqueness seem hoity toity until you try to numerically solve an equation over a domain for which a solution doesn’t exist, or you try to solve it and you get one of many solutions and think you’re done. It’s hard to measure the value of knowing the fundamentals... especially 10 years out when it seems like you’re reusing the same tiny subset of your toolset over and over again. The fundamentals though exist as a little voice in the back of your head reminding you not to do something that won’t work.
  • pirocks 6y ago
    Mathematica has a function called NDsolve which attempts to reduce/simplify/solve fully symbolically and then falls back on numeric methods. Never actually used it for anything real, but could have potentially avoided this situation.
    • lowkeynthorough 6y ago
      Grad student here. I recently did the obvious NDSolve thing but Mathematica kept beachballing, and after heating up my computer tremendously returned an empty set. Turns out, my problem was really simple and I could solve it by hand.
    • gaze 6y ago
      These are SDEs. Mathematica doesn’t solve them. In the end I needed a way for a computer to solve them faster, as an analytical solution wasn’t really possible. Mathematica wi never provide such insights. Maybe one day it will?
• ssivark 6y ago
  What you say is important, but far from sufficient. Recipes which serves as exercises at one level serve as building blocks for the next level. I would argue that the reason it is important to understand the solutions of a few typical forms of differential equations is that they correspond to problems that show up in many different contexts. So not only can the solution methods be applied in all those contexts, but also the intuition for that class of problems/solutions ports over nicely, and allows you to use these as building blocks in bigger systems. > Because in this way you can obtain solution to any DE with any degree of precision (within reasonable assumptions about complexity of the problem). IMHO one of the key goals of education is exactly to understand why/when this is not the case, so you don't shoot yourself in the foot. Numerical methods will often generate junk results (too many examples to count, of people running useless simulations to problems whose solutions don't exist, or algorithms don't converge, so that the results of numerics are fundamentally misleading). Further, probably one of the most valuable topics is an understanding of linear differential equations (and linear shift-invariant systems more generally), including typical solutions and how they compose. A huge chunk of our science & engineering knowledge is built on this edifice. This is hardcore theoretical stuff (ideas & concepts), that can't be derived from being able to numerically solve differential equations. Further, a solid understanding of these concepts allows one to decompose a problems solutions into eigen-modes, and build much better solvers specialized to a situation.
  • d_silin 6y ago
    Thanks for your reply! I definitely can see the gaps in my understanding of the subject now.
• barumi 6y ago
  > And then instead of attempting (and failing) to solve the DE analytically through the algebraic methods, apply modern numerical methods (Runge-Kutta, etc). Your assertion doesn't make sense. Runge-Kutta does not solve differential equations. That's not what they are used for. A solution to a differential equation is an equation that satisfies both the differential equations and the boundary conditions. Runge-Kutta, or any other iterative numerical method, does nothing if the sort. The best they do is provide estimates on what values the solution might exhibit at a given point within their domain. These estimates may be of use for some application, but they are not solutions. If you care about solutions to differential equations but for some reason don't want to learn methods that allow you to determine their solutions then the next best thing is getting up to speed with approximate methods such as the Galerkin method, which for some classes of problems and choices of trial functions can actually output exact analytical solutions.
  • d_silin 6y ago
    Semantics. I want some numbers out of the equations that will match the real-life engineering problem. Approximations/estimates are fine as long as I know the limits of their applicability.
    • barumi 6y ago
      > Semantics. Nope, don't confuse semantics with being just plain wrong. The definition of a solution to a differential equation is quite straight-forward, and a value within the problem domain is not it, even if you use the solution to determine that value. > I want some numbers out of the equations that will match the real-life engineering problem. If that's what you want then what you want is not the solution to the differential equation. You seek something else, and in a scenario where you tolerate a significant margin of error. Meanwhile, you can be taught to determine the exact solution. > Approximations/estimates are fine as long as I know the limits of their applicability. That's not what Runge-Kutta gives you. You might get it with Galerkin methods, but not with numerical approximations.
      • deleted 6y ago
• ianhorn 6y ago
  One of the last classes I ever took was nonlinear dynamical systems. It should have been one of the first after basic calc/vector calc! The typical progression is to start with systems you can solve by hand and then move to ones you can’t usually solve, but can still usefully describe, teaching how to analyze them. It should be the other way around! Start with generally useful tools that apply broadly and then teach the special cases you happen to be able to solve analytically. General tools like basins of attraction and nondimensionalization, should be considered more rudimentary than perfect closed form solutions.
  • AareyBaba 6y ago
    I agree. The biggest eye-opener was learning that the output of some dynamical systems described by deterministic diff equations cannot be predicted because infinitesimal changes in initial conditions lead to exponential changes in the output (popularly known as chaos). This was profound just like Godel's Theorem - we cannot know everything.
• dreamcompiler 6y ago
  If your DEs are linear, you can use Laplace transforms to find an exact solution. This is much simpler than inexact numerical methods.
  • btrettel 6y ago
    Not quite. The Laplace transform only works for constant coefficient linear equations as far as I'm aware. And it is inappropriate for many boundary value problems, because it solves a problem over a domain from 0 to infinity when that might not be what you want. With this being said, I'm a big fan of integral transforms in general (Fourier and Laplace particularly) and have used them many times to solve both ODEs and PDEs.
• deleted 6y ago
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• jbay808 6y ago
  > And then instead of attempting (and failing) to solve the DE analytically through the algebraic methods, apply modern numerical methods (Runge-Kutta, etc). > Because in this way you can obtain solution to any DE with any degree of precision (within reasonable assumptions about complexity of the problem). This is actually an entirely different class of thing, and it's important not to confuse them. That's kind of like saying, "why would I ever need a function that can compute square roots at run-time? Just tell me whatever number it is you want the square root of, and then using my pocket calculator, we can hard-code its square root to any number of digits of accuracy at compile-time!" A numerical solution is a specific answer only to a specific input; depending on what you're trying to do, that might be useful or it might be absolutely useless. An analytical solution is like a short "run-time" program that computes solutions for given parameters, and which can often be reversed and inverted. One example where it's especially painful is when you're trying to do the inverse: find the correct differential equation that results in a desired solution. (AKA a design problem: you need to design a filter, oscillator, linkage, control law, etc, that results in a desired behaviour). This isn't just about ivory tower mathematicians! Engineers need to do this! An analytic solution makes this easy. Imagine if you, an electrical engineer, only know how to be handed a given schematic, break it down into the DE's that govern it, and then numerically solve them. You wouldn't be able to design a new circuit, just predict the behaviour of existing ones. Imagine needing to run sweeps of SPICE simulations over wide ranges of resistor and capacitor values to design a low-pass filter just because you never learned how an RC filter works. It's not that you can't do that, and in some cases where you're dealing with nonlinear systems it's the most pragmatic approach. But that's also a very simple example; as the system gets more complex, the curse of dimensionality will make that more and more challenging, while a system of linear differential equations could continue to have exact solutions no matter how many dimensions you have, and furthermore, make it easy to pick out exactly which variables are the most critical. Or maybe you want to find an optimum point. You can do numeric optimization, but again, you'll get hit with the same curse of dimensionality if the system gets complicated. But if you have an algebraic solution, you might be able to pick out a formula for it. Then not only will you get the optimum (either its coordinates, or its value), you also get an equation that lets you see, in functional form, how that optimum point moves around as you change the other variables in the problem. Then maybe you can maximize that if you want to, and so on. In other cases, being able to remove even one dimension from the analysis can be used to speed up a numerical solution over all the other parameters. Like if you're designing a plane and you have to search over various wingspans, pitches, taper, and so on... if you can get an analytic equation that gives you an optimum wingspan for a given pitch and taper, then you can reduce your search space by an entire dimension, which has a huge impact.
• btrettel 6y ago
  While I agree that the undergraduate ordinary differential equations class has major issues, numerical methods are not quite as easy and straightforward as you indicate. I've worked in computational fluid dynamics (CFD) before and often there is not a cookie-cutter algorithm you can apply to a problem like a RK scheme for an ODE. The statement "you can obtain solution to any DE with any degree of precision" is practically not true for many systems. Despite massive increases in computational power, I don't know anyone who (e.g.) believes that turbulence will be solved any time in the next few decades solely by throwing more computing power at the problem. Through a combination of experimentation, analysis, and computation, however, we have made a lot of progress. One major advantage of analytical methods is that they more easily allow you to see larger scale trends in the problem. If a computation is expensive, calculating sensitivities can be difficult, and in my experience sensitivities are found in CFD less often than they should be. Computation in some sense is more like experimentation: it gives you a local value. You need to do multiple runs to look at trends. I think most people also underestimate how far analytical methods can get. In my PhD, I produced an accurate approximate solution to a nonlinear series of ODEs relevant to the problem I was studying. This gave a lot of insight into the problem that researchers using numerical methods for decades apparently couldn't figure out. By the way, I approximated the unsolvable nonlinear system of ODEs there as a different nonlinear system of ODEs, albeit an exactly solvable system. Most people linearize, but that would have lead to a far less accurate solution in this case. Many nonlinear differential equations can be solved exactly; you can see a catalog of these at this website and in the associated books: http://eqworld.ipmnet.ru/ http://eqworld.ipmnet.ru/ It's really quite similar to integral tables. You look up the form of the equation you have and they tell you the solution (often with a proof or citation for the proof) or at the very least give you a head start if they don't have the solution. I'm working on another theory project right now and I intend to use the same "approximate an unsolveable nonlinear system with a solveable nonlinear system" trick here.
  • na85 6y ago
    Very interesting. Is there a similarity criterion or criteria you apply when searching for systems that you can use as approximates?
• Enginerrrd 6y ago
  I totally agree. I've always longed for a class called "differential reasoning" that focuses on building differential equations, proofs with differentials, and tricky arguments and mathematical justifications for all the stuff physics professors say "is technically invalid, but works". In my engineering program, they did something like this, though it was weak on the differential reasoning part. And heavy in hand-coding numerical solvers in FORTRAN.
• LudwigNagasena 6y ago
  At least in economics 1) simply solving an equation is not enough, you have to understand the dynamics, how changes in parameters change the solution 2) some models simply don’t have solutions that are straightforward to calculate