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

1. Feb 6, 2012

### meldraft

Hey all,

I was reading up on Harmonic functions and how every solution to the laplace equation can be represented in the complex plane, so a mapping in the complex domain is actually a way to solve the equation for a desired boundary.

This got me wondering: is this possible for other PDEs apart for the laplacian? For instance, diffusion, or the heat equation? Thus far, my search hasn't yielded any relevant information..!

2. Feb 6, 2012

### osnarf

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/~rpiltner/sub_piltner/piltner_publications.htm [Broken]

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.

Last edited by a moderator: May 5, 2017
3. Feb 7, 2012

### meldraft

If there is such a thing as 'cool' for papers, those are its definition

I'l read them in detail this afternoon! Thnx a million!

4. Feb 8, 2012

### fluidistic

I think so. You could take a Laplace transform of for example the heat equation (or wave equation, I don't remember well) with given boundary conditions and initial conditions. In order to get the solution to the PDE, at one point you'll need to take the inverse transform wich might involve solving an integral with making use of the residue theorem.

5. Feb 8, 2012

### meldraft

Hmmm you got me there, I'll have to read up on the residue theorem. From what few I read on wiki though, you can use it to (among other things) solve real integrals. This looks indeed quite like the case of the Laplacian, since harmonic functions end up representing a real solution.

Wouldn't a strange consequence however be the following:

Every complex function has harmonic real and imaginary parts. If other PDEs can be expressed in complex form, solutions to the aforementioned equations would also be harmonic?