I am trying to solve the heat equation for a semi infinite rod with lateral surfaces insulated and $u(x,0)$ = $u_0$ for $x>0$, $u(0,t)=u_1$ for $t>0$, and the $\lim_{t\to\infty} u(x,t)=u_0$.
I think I need to start with:
$$\frac{d^2u}{dx^2}=\frac{1}{\alpha^2}\frac{du}{dt}$$
Then I take Laplace Transform:
$$\frac{d^2U}{ds^2}-\frac{s}{\alpha^2}U=0$$
Can someone help me with the following steps?