# Suggestions of a non-linear example for a small research project on numerical solution of ODEs?

I'm a first year undergrad and I'm doing a small research extension on numerically solving ODEs. I have done the main ODE course at my university, as well as physics. The second part of the project constitutes using a method (linear multistep in my case) to solve a system of our choice and writing a report about it. I spoke to my supervisor and she said that a nonlinear example would be instructive since I am interested in plasma physics in the future. Also I'd like to do something involving fluids.

Problem is I have no idea where/how to find a specific example of a system modelled by a nonlinear ODE (I have found plenty of nonlinear PDE examples). So I was wondering if someone could tell me or guide me to an example, or how to find an example.

I am aware that this style of question may be frowned upon, but I've been looking for a while without success (and posted on other forums) so I thought I'd give it a shot.

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The motion of an ordinary pendulum, i.e. a weight on a string, is an example of a non-linear ODE. You probably learned in school that it's linear, but this is only an approximation for small amplitudes.

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I'd say this one is mandatory. –  David H Jun 30 '13 at 11:08
In case the OP is unclear why it is only an approximation: en.wikipedia.org/wiki/…. –  Dimensio1n0 Jun 30 '13 at 11:20
Yeah we already looked at this example in class when we learnt about nonlinear equations. This was the first idea that popped into my head but I want to do something that's a bit more unique and also relevant to fluid dynamics and whatnot. Should I fail to find another example though I'll certainly consider a pendulum or something similar (inverted? double?). –  kurtgirdle Jun 30 '13 at 11:25
@Rife168 Try a realistic model of a damped, driven physical pendulum for a challenge. –  David H Jun 30 '13 at 12:06

Interested in the plasma physics, you can try to understand the Debye-Hückel equation. I'm not sure if the Wikipedia page is the best resource to start with. Most of the time people linearises this equation to find the Debye-length (this length has plenty fathers, but I do not remember neither who nor how many :-), but it may have some interesting things to do with the non-linear problem (or at least the higher order perturbed one). It's the problem of the density of screened free charges in a plasma.

This is an ODE only in 1D of course (note nevertheless the possibility to discuss the radial component when circular symmetry is supposed). I'm not sure if it's the more relevant one for plasma, but that's the only non-linear plasma or fluid example I know which is not a system of partial differential equations.

Have fun.

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Motion in a Newtonian gravitational field. Seems quite linear if you make some approximations (gravitational field being constant) but if the distances involved are large enough, then, the non-linearity would be obvious. The ODE is :

$$x^2\frac{\mbox{d}^2x}{\mbox{d}t^2}=-GM$$.

Yes, its Newton's law of universal gravitation itself.

Also, check out:

http://en.wikipedia.org/wiki/Differential_equation#Linear_and_non-linear

There are some examples, over there.

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This simplest non-linear ODE problem and one that is rarely handled properly is one with step functions such as modeling friction. A simple spring mass system on a friction plane will provide the opportunity to try either

• Adjust numeric steps to discontinuity
• Plow over the discontinuity
• Adjust state directly or with iterations before next step is allowed
• Create a smooth function form of the discontinuity as an approximation

$$\ddot{x} = -\omega^2 x -\mu m g \frac{ \dot{x} }{|\dot{x}|}$$

See example 6 here for a visual and a solution by inspection.

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