# 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

## 2 Answers

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.

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.