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Nonlinear second order ODE 
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#1
Feb1013, 01:34 PM

P: 1

Hi all,
I have a nonlinear ODE in the following form: a x'' + b x'x' + c x' + d x = y where x and y are functions of time and a,b,c and d are constants. As far as I can tell the only way to solve this is numerically, something I've managed to do successfully using a RungKutta scheme. This however is a lengthy calculation, since the driving function (y) is oscillatory I am mostly interested in the phase and magnitude of the response I am looking for an alternative method. I'm wondering if I can use a Fourier series to represent the driving function y and then calculate the resulting Fourier series for x (actually it is the x' term I am most interested in) using an FFT? But I'm not sure if this is possible due to the nonlinear term in x'. Any help/suggestions would be greatly appreciated. 


#2
Feb1413, 09:11 AM

P: 29

[itex] x'(t) = 3t^2, x''(t) = 6t [/itex] Then let [itex]y(t) = t^4 + d t^3 + c t^2 + 6at + 9b [/itex], and you have an x(t) and y(t) such that [itex]a x''(t) + b x'(t)x'(t) + c x'(t) + d x(t) = y(t)[/itex], but there are in fact an infinite number of (x(t),y(t)) pairs, so this solution is somewhat trivial. Might you mean x is a function of y? Then: [itex]a x''(y) + b x'(y)x'(y) + c x'(y) + d x(y) = y[/itex] Or I'm just not understanding something. Let me know. 


#3
Feb1413, 10:31 AM

Mentor
P: 15,146

The damping is going to make this a stiff system, and the quadratic damping is going to make the stiffness rather interesting. The key problem with a fixed step size integrator such as basic Runge Kutta is that the stiffness mandates that the step size be very small. It's going to take a long time to solve and will lose accuracy in the process. Have you tried using adaptive techniques such as Runge Kutta Fehlberg, CashKarp, adaptive Adams Bashforth Moulton, Gauss Jackson? Have you tried using adaptive integration techniques such as KapsRentrop, various descendants of the Gear technique that are aimed specifically at stiff problems? 


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