## Heat Equation Help

Consider the heat equation below:

$u_t = \alpha^2 u_xx, 0
$u(0,t) = 1, u(\frac{1}{2}, t) = 4$
$u(x,0) = 0$, $0
$u(x,0) = 1$, $\frac{1}{4}

By first finding the steady state solution $u_\infty (x)$ and then considering $v(x,t) = u(x,t)- u_\infty (x)$, using separation of variables to determine the solution for $u$. You will have to determine the coefficients of a Fourier sine series.