The general solution for heat equation
$$u_t = - \frac{1}{\alpha} u_{xx}$$
with Duhamel's theorem yields to
$$u(x,t) = \frac{x}{\sqrt{4\alpha t}} \int_{\tau=0}^t \frac{f(\tau)}{(t-\tau)^{3/2}}\exp\Big( \frac{x^2}{4 \alpha (t-\tau)}\Big) d\tau$$
How can I go further by applying the actual boundary condition of
$$f(t) = k e^{-t}$$
I assumed that the steps of solution to a famous PDE is known and I avoided, but if it is needed, I can post the whole solution.