Being inspired by some other question of the Heat equation in 2 spatial dimensions: $$\Delta T + \frac{\partial T}{\partial t} = 0\\\left(\frac{\partial^2 }{\partial x^2}+\frac{\partial^2 }{\partial y^2}\right)T + \frac{\partial T}{\partial t}=0$$
I made a small experiment with the following setup:
A rectangular area,
heat source as a point in the middle (constant high temperature)
the edges have a constant low temperature,
the starting temperature for everything in between is the same as the rectangular border.
How can I analytically derive the solution that this will have? I have been thinking about Fourier analysis where I think is common to solve problems like this, but it was such a long time I don't remember it all.
I am mostly curious to see if I can compare to / verify my home built numerical scheme which does find a solution on a discrete grid.