heat equation with exponentially time dependent boundary condition

Question

Consider the one dimensional heat equation on a semi-infinite bar such that

$u_t=u_{xx}$

with initial condition of

$u(x,0)=0$.

What is the solution for all x and t if the boundaries are

$u(0,t)=e^{-t}$ and $u(x,t)=0$ as x approaches infinity?