Working through Partial Differential Equations homework questions to prepare for a final. I have created answers but hoping to compare with others that may be more complete.

1. In many applications of mathematics in science and finance, we have to solve the Poisson equation: V2 4(x, y) = f, (1) where f is a given function. Consider a simplified problem in a square region by as- suming that f = 2y. Supplement this problem with the following boundary conditions 4(0, y) = 0, 4(1,y) = y, 4(x,0) =0, 4(2,1) =22. (2) (a) On a 2 x 2 grid (2 intervals in each direction), how many unknowns interior values of 4 must be found? (b) Write the equation(s) that must be solved to find the unknown interior value(s) from the previous task (a). (c) Apply a simple Jacobi iteration to the equation(s) from the task (b). How does your answer compare with the exact solution u(x, y) = xy