# Fourth Order Hyperbolic PDE's

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David Koenig on 17 Aug 2012
Commented: David Koenig on 15 Oct 2014
I have been struggling with trying to solve the vibrating plate equation in polar coordinates with angular dependence (non symmetrical initial condition). That is,
c d2y/dt2 = d4y/dr4
except that d4y/dr4 is just short cut symbolism for the double Laplacian in polar coordinates. I have been successful using a relatively simple explicit finite difference approach for symmetrical initial conditions but the asymmetrical IC's cause a problem. I have been holding off trying to reformulate an implicit finite difference method because there is so much algebraic bookkeeping.
I see that the partial differential solver toolbox does not deal with this problem. Does anyone know about Matlab scripts that deal with this problem?
Thanks.

Deepak Ramaswamy on 17 Oct 2012
Hi
Don't know if you're still looking to solve this problem. The link below shows a static plate example of converting a 4th order PDE to a 2nd order PDE in the PDE Toolbox
The trick shown in the example can be extended to the hyperbolic problem as well. You might also want to consider using the eigensolver in the PDE Toolbox if you're interested in looking at modes.
Deepak

#### 1 Comment

David Koenig on 15 Oct 2014
Deepak, I am taking the brute force approach of replacing the partials with finite differences and then developing an explicit numerical method from the result. It seems to work OK but, of course, there is some serious bookkeeping required.
On the other hand, your suggestion of converting to two 2nd order pdes is interesting and I will pursue it. Thanks.
Dave