# How to set the temperature at the vertices points for a rectangular domain?

Suppose I have to solve the 2-D heat equation in a rectangular domain using the finite difference method, for the boundary conditions say:

$T_1$ is the temperature of the right side of the rectangle,

$T_2$ is the temperature of the top side,

$T_3$ is the temperature of the left side, and $T_4$ is the temperature at the bottom.

           T2
_______________
|               |
T3 |               |T1
|               |
|_______________|
T4


at the vertices points of the rectangle, how to set the boundary conditions? do I need to take the average value of the two temperatures?

Regards

• For your finite difference, if you are you doing something like $\nabla^2 u = \frac{u_{i+1,j}-2u_{i,j} + u_{i-1,j}}{\Delta x^2} + \frac{u_{i,j+1}-2u_{i,j} + u_{i,j-1}}{\Delta y^2}$ then the corner points wont affect the solution on the interior. – Steve May 24 '16 at 14:38

• @Bill Physcally how to impose the Top edge at $T_2$ and the right edge to $T_1$ , does that really make sense? of course not – Navaro May 24 '16 at 23:20