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.