This is not an answer to your question. This is an answer to what information you need to determine in order to solve a heat problem such as this one.
In general, a heat conduction problem in 2 dimensions needs 4 constraints, 2 in each direction. I will use the example in section 2 of the PDF you linked (http://geodynamics.usc.edu/~becker/teaching/557/problem_sets/problem_set_fd_2dheat.pdf) to illustrate.
I drew a diagram of the 2D heat conduction that is described in the problem.

You have mentioned before that you wish to solve the problem using an explicit finite-difference method.
The first step would be to discretize the problem area into a matrix of temperatures.
For the example linked, elements 1.5km wide by 2km high were chosen.
The boundary conditions are:
Perfect insulation (zero heat flux) on the left and right sides. In an explicit solver, all Neumann (fixed first derivative) conditions are represented by creating an extra row of "fictitious" temperatures which are outside the problem area. Since the heat flux is zero, the fictitious temperatures are held equal to the temperatures at the edge of the problem zone.
Fixed temperature (Dirichlet BC) at the upper surface for all time. The surface of the problem area will always be 0°C.
Initial temperature at lower surface of 1300°C, with the central point begin held at 1500°C for all times >eq;0. Note I have denoted the Heaviside function as $u(t)$.
Once you get the boundary conditions set up correctly, explicit solvers become simple plug and chug to solve the temperature distribution.
If you want to find out how much heat is supplied by the constant temperature heat source, you will need to use Fourier's law with the approximate derivatives given by the explicit finite-difference method.
Assuming the heat is applied at a point $T_{i,j}$, the drawing below shows how to compute the heat flux per unit depth.
Obviously, you would need to know how far in an out of the page this problem extends to compute the heat flux.
