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I am a master student in Mathematics, and I have to prepare a seminar for a course in mathematical methods for applied sciences. I have a good background in numerical analysis for ODEs, PDEs and hence I also would like to run some simulations in Python. I also have some experience with standard numerical methods for SDEs.

The topic of such a talk could be

  • a chapter from classical Murray's books, so maybe some biological models, but that is of course not mandatory.
  • I also thought to search for some articles like this one from SIAM, but honestly I'm pretty open to other possibilities.
  • something that can be modeled by some ODEs that are also "hard" to solve (stiffness) or something that has some properties and one wants the numerical solution to satisfy such a property.

Of course, I do not want the level to be too low, but nor to high, since I'm still a student.

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  • $\begingroup$ If you want biology just do the Lotka-Volterra equations, there is lots of fun stuff there - fixed points, stability etc. Every computational scientist should do this problem at least once in a lifetime. $\endgroup$ – Maxim Umansky May 8 '20 at 3:59
  • $\begingroup$ @MaximUmansky thanks for the advice. I already studied LV at the bachelor, and it is a bit too easy. Do you have by chanche also some other topic (also biology related) ? :) $\endgroup$ – andereBen May 8 '20 at 7:34
  • $\begingroup$ How about the theory behind "embedded Runge-Kutta" methods that are the backbone of all widely used ODE solvers today? It's fascinating what you can do with cleverly chosen Butcher tableaux! $\endgroup$ – Wolfgang Bangerth May 8 '20 at 22:05
  • $\begingroup$ @WolfgangBangerth I agree, it sounds cool! But how can it be linked explicitely with some biological topic? Otherwise, is there some biological topic that you know requires specific ODE solvers in order to obtain good solutions? $\endgroup$ – andereBen May 9 '20 at 8:39
  • $\begingroup$ @andereBen -- ah, I missed from your post that it needs to be related to biology. Then maybe that's not the topic for you. How about the SIR model and its many generalizations? $\endgroup$ – Wolfgang Bangerth May 9 '20 at 15:54

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