Solution for higher order wave ODE

[Romik]

Hi guys,

Here is an equation that I have tried for few days to solve and still haven't succeeded, I'm interested to solve this 4th order wave equation to find u(x).

∫∫(A u(x) + B u(x)² + C u(x)³ +D u''(x)) dx dx=0

the 4th term is second derivative of displacement u(x). I assume constants are zero.

I would appreciate any hint or comment.
Thanks

 

[Simon Bridge]

What is the DE?
Do I take it that you mean to solve:
A u(x) + B u(x)2 + C u(x)3 +D u''(x) = 0

i.e. $$u^{\prime\prime} = au^3+bu^2+cu$$

 

[HallsofIvy]

What you have written, $\int\int ... dx dx$, is meaningless. Simon Bridge is assuming that you mean that the integral over an arbitrary region in the plane is 0. In that case the integrand must be 0. If you intend an integral over a specific region, that is not true.

 

[Romik]

Yes, obviously the DE is that expression.
Thanks for comment

What is the DE?
Do I take it that you mean to solve:
A u(x) + B u(x)2 + C u(x)3 +D u''(x) = 0

i.e. $$u^{\prime\prime} = au^3+bu^2+cu$$

 

[Romik]

You are right,

Thanks.

What you have written, $\int\int ... dx dx$, is meaningless. Simon Bridge is assuming that you mean that the integral over an arbitrary region in the plane is 0. In that case the integrand must be 0. If you intend an integral over a specific region, that is not true.

 

[Simon Bridge]

OK - so if I'm reading this correctly,

Problem Statement:
Given $$Au^3(x)+Bu^2(x)+Cu(x)+D\frac{d^2u}{dx^2}=0$$... find the general solution for u(x).

Attempt at a solution:
You tried to integrate the entire thing twice wrt x, ...

Would that be correct?
Sorry - I find it far from clear what you are trying to say and I think it is important for me to understand the problem when I am trying to help you.

If this is so then the DE is second order (not fourth) and of form:
$$\frac{d^2u}{dx^2}=f(u)$$ Have you tried: reduction of order and/or separation of variables?

Aside: You have indicated that this is associated with the wave equation. Which seems to suggest that $$-\frac{1}{c^2}\frac{d^2u}{dt^2}=Au^3(x)+Bu^2(x)+Cu(x)$$ ... is that correct?

 

[JJacquelin]

Hi !

This non-linear ODE is analytically solvable, leading to the inverse function of u(x).
The result on the form of x as a function of u is an elliptic integral (attachment).
Solving this elliptic integral involves the roots of a polynomial equation of the 4th degree. As a consequence, the formal result involving an elliptic function is a very complicated formula :
http://www.wolframalpha.com/input/?i=integrate+du/sqrt(g-a*u^2-b*u^3-c*u^4)&x=0&y=0
The function obtained x(u) is so big that computing the inverse function u(x) in terms of Jacobi elliptic functions is probably not realisic, nor usefull in practice. Numerical methods for solving the ODE are more convenient.

 

[Romik]

Simon Bridge

Thanks again for the comment,

These types of equations are called "Autonomous" and they are very common in classical mechanics (Hamiltonian systems)
in this case $$\frac{d^2u}{dx^2}=f(u)$$ is second order special case! (it is independent of first order derivative). terms in $$f(u)$$ come from some nonlinearities in the system. the original equation derived from Hamilton principle is order 4th, we can reduce the order to second by integration, after that we should solve the ODE with some uncommon functions and approximations like Jacobi elliptic cosine function! it is not as easy as separation of variables! A, B, C, and D are constants and they are related somehow with speed of wave.

 

[Romik]

JJacquelin

Thank you so much for your time and help.

Unfortunately I need a relation (even approximation) explicitly and I can't go with numerical solution,
I tried Mathematica before, I know, it doesn't help.
I'm thinking about Jacobi elliptic function, or generalized Riccati equation method.

Once again, thanks.

 

[Simon Bridge]

An approximate method would depend on what sort of values you expect for the constants: perhaps some of the terms dominate?