# What “boundary conditions” can make a rectangle “look” like a circle?

I posted the question below in Stackoverflow but then realized that it perhaps would find a better audience here.

I am solving a fourth order non-linear partial differential equation in time and space (t, x) on a square domain with periodic or free boundary conditions with MATHEMATICA.

Without using conformal mapping, what boundary conditions at the edge or corner could I use to make the square domain "seem" like a circular domain for my non-linear partial differential equation which is cartesian?

The options I would not like to use are:

• Conformal mapping
• changing my equation to polar/cylindrical coordinates?

This is something I am pursuing purely out of interest just in case someone screams bloody murder if misconstrued as a homework problem! :P

Edit:

If I have a result from solving a PDE in cartesian coordinates, how do I transfer these results or view them in polar coordinates?

• Your question is quite imprecise; I don't understand what you mean for a square to "look" or "seem" like a circle. It might help if you explained what you find unsatisfactory about conformal mapping and polar coordinates. – Rahul Feb 20 '12 at 4:32
• @RahulNarain If I have results of a PDE in cartesian coordinates, how do I transfer them to polar coordinates? – dearN Feb 20 '12 at 16:53