Conceptual trouble--please help me understand what my matrix solution is telling me
 

Hello everyone:

I'm having trouble with a simple conceptual issue.

I am helping a friend to write a code that solves the implicit 1-D advection-diffusion equation. I'm a programmer without much physics experience and there's one fundamental issue I don't understand... What does my solution mean physically?

I have constructed the appropriate tridiagonal matrix A and vector b to solve Ax = b. As far as I know, each row represents a new time and each column represents a different point in space. I have a solution for the x vector using Gauss elimination, but I don't know what it means! How do I get from the solution (x vector) to a plot that shows how T varies in time and space? Do I go back to the coefficient matrix (A) and multiply each coefficient by the appropriate element in x?