$$\frac{\partial u}{\partial t} = \frac{\partial ^2 u}{\partial^2x}$$ where $u=u(x,t)$, $0 \leq x \leq L$

BC1: $u(x=0,t>0)=T_f$; BC2: $u(x=L,t>0)=T_i$; IC: $u(x,t=0)=f(x)$