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I'm looking for examples of Helmholtz and Biharmonic equations in Cartesian co-ordinates with exact solutions, in order to compare my numerical solutions with it.

I was able to find quite a few examples on the internet, where the problem with boundary conditions were precisely defined. Those were, unfortunately only illustrative examples and exact solutions were not shown.

I was encouraged about manufacturing the solution (like on math.stackexchange.com, and I did that successfully). I was afraid in that case some interesting examples the specialists in PDEs are aware of wouldn't be treated, like some solutions given by infinite series (which I would truncate when some level of accuracy is reached). For example, the one given on Wikipeda article on elliptic BVPs is interesting.

Any particular example, or a useful link to a web-page or a paper is appreciated.

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Johntra Volta
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2 Answers2

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Look for the book Vibration of plates by Arthur Leissa. It has explicit solutions for square and circular plates. Including tables with approximated eigenvalue for different boundary conditions.

Christian Clason
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creyesk
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Does an axis aligned rectangular box (length/width/height = a/b/c) with dirichlet boundary conditions ($\phi=0$) on the walls admit a closed form / exact solution? Maybe a tensor product of sinusoids, e.g. $\phi(x,y,z) = sin(k_x x)sin(k_y y)sin(k_z z)$. Pick $k_x/k_y/k_z$ judiciously to realize the dirichlet condition, e.g. $k_x = n\pi/a$, $k_y = m\pi/b$, $k_z = p\pi/c$, for some integers (n,m,p). Plug that solution into the $div \, grad$ operator, then the resulting equation, $div\,grad \, \phi = k^2 \phi$, should give you the separation condition for $k_x,k_y,k_z$ (probably $k_x^2+k_y^2+k_z^2=k^2$).

This is kinda standard fare for the vector wave equation / maxwell's equations (electromagnetics), I've not messed around much with the scalar helmholtz equation but I'd expect it to work very similarly. For electromagnetic resonators / the VWE, I'd recommend Balanis' "Advanced Engineering Electromagnetics". There's probably a comparable reference for the the scalar Helmholtz equation (a graduate level acoustics text, perhaps?) but I wouldn't know what it is.

I have no experience with the Biharmonic equation.

Johntra Volta
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rchilton1980
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  • This is the direction I'm striking from right now. Check out this link for solution of biarmonic eq. in Fenics (I'm not involved in it however): http://fenicsproject.org/documentation/dolfin/dev/python/demo/pde/biharmonic/python/documentation.html. – Johntra Volta Oct 22 '12 at 17:24
  • The exact solution for this is u(x,y) = sin(pix)sin(piy), but it's not mentioned though... – Johntra Volta Oct 22 '12 at 17:25