I Solving BVP with Analytical Solution in (0,1)^2

  • I
  • Thread starter Thread starter ty1998
  • Start date Start date
ty1998
Messages
2
Reaction score
0
TL;DR Summary
How to determine the boundary conditions to this elliptic BVP given an analytical solution
For this BVP in ##(0,1)^2##,
$$
-u_{xx} - u_{yy} = 0
$$
subject to some boundary data it is said the analytical solution is ##u(x,y) = \theta##. I've thought about this for awhile I can't seem to figure out how to determine the boundary conditions for this BVP. Moreover, ##\theta## is illustrated in figure attached. Some comments would be greatly appreciated.
 

Attachments

  • 156752681_844655712751349_5368146245785657842_n.jpg
    156752681_844655712751349_5368146245785657842_n.jpg
    23.2 KB · Views: 188
Physics news on Phys.org
In polar coordinates, x = r \cos \theta \qquad y = r \sin \theta. Since you're in the first quadrant both x and y are positive, so you don't lose any information by dividing: <br /> \frac yx = \tan \theta.
 
Thank you for the reply. However, given that I am also trying to also solve this numerically, I was curious what BCs I would need to satisfy the equations? This is my main concern.
 
The question is given that ##u=\theta## is the solution to the differential equation given unknown boundary conditions, what must those unknown boundary conditions be?

It's stupidly obvious actually, the boundary condition must be ##u=\theta## on the boundary. I feel like I must not understand the question right.
 
Substitute into the given equation: u = 0 on y = 0, u = \pi/2 on x = 0, u = \arctan(y) on x = 1, and u = \arctan(x^{-1}) on y = 1.
 
Transform the differential equation (Laplace's equation) to cylindrical polar coordinates and see what it says.
 
Back
Top