Trouble deciding how to integrate a PDE

[Schrödinger's hat]

Couldn't quite figure out whether to put this in Physics, Calculus, Applied Mathematics, Computer Science, or Engineering :/

 

I'm trying to come up with a good scheme for integrating something similar to a thin beam equation:

Something along the lines of:

[math] A\frac{d^4y}{dx^4}+\text{Possibly other x derivatives}=q(x) + B\frac{d^2y}{dt^2}[/math]

 

But unfortunately comp-phys is but a hazy memory.

 

I can remember/find notes on enough to do similar things which are first order in time with finite difference methods.

Would the obvious thing of adding degrees of freedom to make:

[math]v = \frac{dy}{dt}[/math]

[math]\frac{d}{dt} \left[\begin{array}{c} v\\y \end{array}\right] = \left[\begin{array}{c}\frac{1}{B}(A\frac{d^4y}{dx^4} - q(x))\\v\end{array}\right][/math]

 

Then applying the same methods be an effective approach? Or would it be horribly unstable?

Is there an out-of-the-box method somewhere that I'm completely missing?

Edited September 30, 2011 by Schrödinger's hat

[Schrödinger's hat]

Then applying the same methods be an effective approach? Or would it be horribly unstable?

 

In case anyone else cares/comes here looking for an answer to the same question, this seems to work quite well.

The resulting system was horribly unstable with Euler method, but anything backwards-time or mixed seemed to work remarkably well.

 

I suppose I won't know for sure until it's compared to some data.

 

If anyone is watching/curious I can elaborate upon request, as there seems to be a bit of a jump in the learning curve (for what I can find online at least).

[khaled]

Take a look at these references:

 

Solution using separation of variables

 

Wilmott

 

PDE Integration with Physics Model

 

Scientific Notes on PDEs

 

PDE Integration using Discretization

 

Wolfram Notes on Numerical Solutions to PDE

 

PDE as a Linear System

 

PDE and Relaxation Methods

 

.. good luck,