Difference Method. Using the same method, solve the 2D diffusion problem with the prescribed multivariate Gaussian initial condition at the following prescribed boundary conditions. Show how U(x,y,t) changes with time, i.e. time slices of the U(x,y,t) function.
$\frac{\partial u(x, y, t)}{\partial t} = \alpha^2 \left(\frac{\partial^2 u(x, y, t)}{\partial x^2} + \frac{\partial^2 u(x, y, t)}{\partial y^2}\right)$
$u(x, y, 0) = e^{-\frac{(x-\mu_1)^2}{2\sigma_1^2} - \frac{(y-\mu_2)^2}{2\sigma_2^2}}
\frac{1}{\sqrt{2\pi\sigma_1^2 2\pi\sigma_2^2}}$
$\sigma_1 = \sigma_2 = 0.05$
$\mu_1 = \mu_2 = 0.5$
a) u(0,0,t) = u(1,0,t) = u(0,1,t) = u(1,1,t) = 0;
b) u(0,0,t) = u(1,0,t) = u(0,1,t) = u(1,1,t) = 1;
