Register to reply

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

by meldraft
Tags: analysis, complex, laplacian, pdes, solve
Share this thread:
meldraft
#1
Feb6-12, 11:26 AM
P: 280
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..!
Phys.Org News Partner Science news on Phys.org
Scientists develop 'electronic nose' for rapid detection of C. diff infection
Why plants in the office make us more productive
Tesla Motors dealing as states play factory poker
osnarf
#2
Feb6-12, 04:42 PM
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 [itex]\zeta_{i} = a_{i}x + b_{i}y + c_{i}z,[/itex] where one parameter[itex] (a_{i}, b_{i}, c_{i})[/itex] is equal to[itex] \sqrt{-1}[/itex] 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.
meldraft
#3
Feb7-12, 03:28 AM
P: 280
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!

fluidistic
#4
Feb8-12, 04:49 PM
PF Gold
fluidistic's Avatar
P: 3,207
Can complex analysis be used to solve PDEs other than the Laplacian?

Quote Quote by meldraft View Post
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..!
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.
meldraft
#5
Feb8-12, 05:24 PM
P: 280
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?


Register to reply

Related Discussions
Finite Difference method to solve PDEs Differential Equations 1
[complex analysis] domain coloring applet? (visualizing complex functions) General Math 0
A question on orthogonality relating to fourier analysis and also solutions of PDES Calculus & Beyond Homework 1
Solving PDEs involving complex variables Differential Equations 0
Complex Laplacian General Math 3