# How to solve this BVP

• I
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.

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## Answers and Replies

Homework Helper
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.

Staff Emeritus
Gold Member
2021 Award
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.

Delta2
Homework Helper
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 $y = 1$.

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