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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 results of 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.

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

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What problem are you considering exactly? If you can treat inhomogeneous equations and arbitrary boundary conditions, you can just start with whatever exact solution you want, and then use the equation to generate the inhomogeneous term and the boundary condition. –  timur Oct 18 '12 at 16:55
    
Yes I was thinking about manufacturing the solution, but I was afraid in that case I wouldn't treat some interesting examples the specialists in PDEs are aware of. –  Johntra Volta Oct 18 '12 at 17:12
    
Like some solutions given by infinite series, for example... –  Johntra Volta Oct 18 '12 at 17:14
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For the purpose of checking if your code is working correctly, there is nothing wrong with manufacturing a solution. That's what people do. –  timur Oct 19 '12 at 13:41
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