Consider the heat equation below:

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

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.