Consider the problem $u_t=u_{xx}, 0<x<\infty, t>0$ with the initial and boundary data given by $u(x,0)=f(x), x>0$ and $u(0,t)=g(t),\ t>0$. The solution for the above problem is given by $u(x,t)=\int_0^{\infty} G(x,\xi,t)f(\xi)d\xi -2\int_0^{\infty} \frac{\partial K}{\partial x}(x,t-\tau)g(\tau)d\tau$, where $K(x,t)=\frac{1}{\sqrt{4\pi t}} exp\{-\frac{x^2}{4t}\}$ and $G(x,\xi,t) = K(x-\xi,t)-K(x+\xi,t)$.
What is the corresponding solution if the boundary data is given by $u_x(0,t)+\alpha(t)u(0,t)=g(t),\ t>0$.