Solving Laplace's equation using conformal mapping

[lonewolf5999]

I'm trying to use conformal mapping to solve for a function u(x,y) satisfying Laplace's equation ∇²u = 0 on the outside of the unit circle (i.e. the complement of the unit disk), with boundary conditions:

u = 1 on the unit circle in the first quadrant,
u = 0 on the rest of the unit circle.

To start with, I'm trying to simplify the boundary conditions by conformal mapping, so I first tried mapping the outside of the unit circle to the upper half plane with a linear fractional transformation w = -i(z-1)/(z+1), and then sending the upper half plane to the rectangle using an elliptic integral transformation W(w) = ∫ dt/sqrt[(1-t²)(1-k²t²)], integrated from 0 to w.

I did this because I was trying to send the unit circle in the first quadrant to one full edge of the rectangle, and the same for the part of the unit circle in the other quadrants, but that doesn't seem to work, because after doing the elliptic integral transformation, it turns out the unit circle in the first quadrant gets sent to only half of the bottom edge of the rectangle, so that doesn't simplify my boundary conditions much. Rotating the unit circle before transforming it doesn't seem to help either, since none of the quarter-circles are sent to a full edge of the rectangle.

Any suggestions on how I can proceed, or what other sorts of transformations would help me simplify the boundary conditions, would be helpful!

 

[mighty2000]

I would recommend first checking your calculations and transformations to ensure they are correct. It is possible that there may be an error in your process that is causing the boundary conditions to not simplify as expected.

If your calculations are correct, you may want to consider using a different conformal mapping that can better simplify the boundary conditions. You could also try breaking up the unit circle into multiple sections and mapping each section separately to see if that helps.

Another option could be to use a different method to solve Laplace's equation on the outside of the unit circle, such as the method of images or separation of variables.

It may also be helpful to consult with other scientists or mathematicians who have experience with conformal mapping and solving Laplace's equation for guidance and suggestions. Collaboration and feedback from others can often lead to new insights and solutions.

Overall, it is important to carefully analyze and troubleshoot your approach, and to be open to trying different methods and seeking assistance when needed. Good luck with your research!