# Helmholtz and Biharmonic equation examples with exact solution

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.

• Are you looking for driven problems or eigenvalue problems? – rchilton1980 Oct 22 '12 at 13:01
• I'm interested for the eigenvalue problems, I don't know anything about the other one. – Johntra Volta Oct 22 '12 at 15:48

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 $k_x^2+k_y^2+k_z^2=k^2$).