Nonlinear differential equation (Laplace transform?)

[fnsaceleanu]

Hi,

Part of my research, I nondimensionalized an ODE to eventually arrive at this form:

sin(τλ) = q^((n+2)/(n+1)) + κq' + q''

where q' = dq/dτ

The problem is of course the nonlinear q^n. n is an integer greater than 0.
Is there a Laplace transform for this?
Or what solutions are there for this kind of equation?

Thanks!

 

[HallsofIvy]

I was thinking that the Laplace transform could only be used to solve linear d.e.s but wasn't sure so I "google" "Laplace Transform" and "non-linear differential equation". Somewhat to my surprise, I got a number of hits. Unfortunately, when I opened pages on "solving non-linear differential equations by the Laplace Transform method", I found that the first instruction was to linearize the equation. I didn't read further- I sure they gave further instructions for getting better solutions than just to the linearized version- but it seems that the Laplace Transform cannot be directly used on non-linear differential equations.

If you are still interested, here is the simplest of the pages:
http://wikis.lib.ncsu.edu/index.php/Laplace_Transforms

 

[fnsaceleanu]

Thanks for the reply.
Once linearized it would be easy to solve with the Laplace transform.
I've also read that there are no analytic solution to nonlinear differential equations unless they have specific forms..

 

[Mute]

Nonlinear ODEs typically don't have nice, analytic solutions. You'll either have to solve it numerically or develop some approximation scheme for the parameter regime of interest. One thing you can note is that for n very large you approximately have a linear ODE, so you might be able to develop some sort of approximation scheme where you solve the linear ODE and then calculate corrections of order 1/n or higher.

 

[haruspex]

I doubt that it helps, but if you substitute q = p^-n-1 you get rid of the fractional power. It looks a whole lot better but for the appearance of a (p')².

 

[fnsaceleanu]

Thanks to those who replied.. looks like linearization via taylor series approximation could work but then i have to linearize about a point, an usually it's the equilibrium.

I have found this as well http://www.new.dli.ernet.in/rawdataupload/upload/insa/INSA_2/20005a65_421.pdf

Looks interesting, as it can solve nonlinear differential equations via the laplace transform, BUT it needs a harmonic input of the form x(t) = exp (s_1 + s_2 + s_3 +...+ s_n)t... whereas my input is of sin form..

Any thoughts there?

 

[fnsaceleanu]

looks like the paper only considers nolinear DE including the first derivative only..

 

[fnsaceleanu]

I've done a correction to the exponent that makes the DE nonlinear...

It's actually q^(n+2)/((n+1)^2)... this makes it interesting since if n=0.618 or n=-1.618, then the DE is linear... but that n depends on a curve of best fit.

Has anyone used numerical methods in Matlab to get the system response if it's nonlinear?
That seems to work ,and MATLAB has a bunch of functions for stiff/nonstiff systems