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

[ty1998]

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.

 

[pasmith]

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: $$\frac yx = \tan \theta.$$

 

[ty1998]

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.

 

[Office_Shredder]

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.

 

[pasmith]

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 $y = 1$.

 

[Chestermiller]

Transform the differential equation (Laplace's equation) to cylindrical polar coordinates and see what it says.