Nonlinear differential equation (Laplace transform?)

In summary, the conversation is about the use of Laplace transform to solve a non-linear differential equation. The person asking the question initially thought it was not possible, but after doing some research, they found that there are methods for solving non-linear equations using Laplace transform, although they involve linearizing the equation. They also discuss the challenges of finding analytic solutions for non-linear equations and the use of numerical methods in MATLAB to solve them. The conversation ends with the mention of MATLAB's functions for solving stiff/non-stiff systems.
  • #1
fnsaceleanu
9
0
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!
 
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  • #2
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
 
  • #3
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..
 
  • #4
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.
 
  • #5
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.
 
  • #6
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 [Broken]

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?
 
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  • #7
looks like the paper only considers nolinear DE including the first derivative only..
 
  • #8
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
 

1. What is a nonlinear differential equation?

A nonlinear differential equation is a mathematical equation that involves an unknown function and its derivatives, where the derivatives are not necessarily proportional to the function itself. In other words, the equation is not a straight line and cannot be solved using basic algebraic methods.

2. What is the Laplace transform?

The Laplace transform is a mathematical tool used to solve linear differential equations. It converts a function of time into a function of complex frequency, making it easier to solve differential equations by transforming them into algebraic equations.

3. How is the Laplace transform used to solve nonlinear differential equations?

The Laplace transform can be used to solve nonlinear differential equations by first transforming the equation into a linear form, solving it using the Laplace transform, and then transforming the solution back into the original form using the inverse Laplace transform.

4. What are the advantages of using the Laplace transform to solve nonlinear differential equations?

The Laplace transform allows for the use of algebraic methods to solve differential equations, which can be more efficient and accurate than traditional methods. It also allows for the solution of initial value problems and systems of differential equations.

5. Are there any limitations to using the Laplace transform for solving nonlinear differential equations?

Yes, the Laplace transform can only be used to solve linear differential equations, so it cannot be directly applied to nonlinear equations. Additionally, the inverse Laplace transform may produce multiple solutions, so it is important to verify the solution using other methods.

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