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, 201114 yr by 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).

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,