I'd like to solve the following non-autonomous, non-linear first order differential equation, which is a result of (quite straightforward) chemical kinetics:
dy/dt = a*exp(-k*t) - b*y^2 - c*y with a,b,c,k > 0 and y(0) = 0.
Is there an algebraic solution? I tried all general methods including Laplace transforms, to no avail. WolframAlpha gives a solution using Bessel and Gamma functions, which is completely unstable in my region of interest.
Dear Dr. Becker,
The analytical solution of this equation exists, but the resulting algebraic expression is cumbersome. I doubt the usefulness of this expression for practical calculations
See screenshot (Maple 18).
Dear Mr. Velázquez - León,
Thank you so much for your thorough explanation! I do not necessarily need the analytical solution; I just need to calculate the concentration y(t) given a,b,c,k, which I then use in a least-squares optimization routine. Since I am implementing this in Matlab at the moment, I might try the numerical approach using ode15s. If this is too slow, then I will try to understand your last suggestions and implement them. Thanks again!
---------------
Dear Dr. Korobov,
Thank you for your answer. This is indeed one of the solutions Maple/Mathematica/WolframAlpha gives, but, as you already state, can be troublesome to implement in a numerical algorithm. Theoretically the Bessel functions give the correct solution, but the implementation of the Bessel functions is again a similar problem as solving the differential equation I started with (correct me if I'm wrong).
Dear Mr. Velázquez - León, dear Luis,
Using ode15s works perfectly! And it is very fast and accurate as well! I love it :). I can now easily run ode15s inside an optimization routine, calling it more than 10000 times and have the optimization finished in less than a minute. Gotta love Matlab sometimes...
Thank you so much!
All the best, René
Dear Luis,
That's exactly what I already did last week, and indeed, it runs perfectly. I'm using ode45 now since it is supposed to be a bit more accurate (and it's just as fast).
Thanks again,
René
- Badges
- Science topic
- Similar topics
- Mathematics