# Can complex analysis be used to solve PDEs other than the Laplacian?

by meldraft
Tags: analysis, complex, laplacian, pdes, solve
 P: 209 Dr. Reinhart Piltner used complex analysis to find a general solution for 3d static elasticity problems in terms of complex functions, which amounted to finding a biharmonic potential function in terms of six arbitrary complex functions of three complex variables (of the form $\zeta_{i} = a_{i}x + b_{i}y + c_{i}z,$ where one parameter$(a_{i}, b_{i}, c_{i})$ is equal to$\sqrt{-1}$ for each i) that meets several other conditions. I don't know if the elasticity part interests you, but you will probably find the derivation of biharmonic solution interesting. http://math.georgiasouthern.edu/~rpi...blications.htm For some reason only the 1987 and 1989 papers work, the others all open the same paper (copy-paste web designing?). The 1987 paper is the one with the derivation, though. That's the only other PDE application to complex numbers I know of, but I'm sure there are plenty of others.