Suppose $\Omega = [0, 1]^3$ is a cube in $R^3$. Consider the problem $$u_t = \Delta u$$ $$u(0,x) = f(x)$$ $$u \mid_{\partial \Omega} = 0$$
Then by taking the convolution of $f$ with the fundamental solution we see that $$u(t,x) = \int_\Omega \Phi(x-y)f(y) \, dy$$
where $\Phi(x) = \frac{1}{(4\pi t)^{3/2}}e^{-\|x\|^2/4t}. $
Is there a similar formula for the situation with the non-homogenous boundary condition $u \mid_{\partial \Omega} = g(t,x)$?
Thanks.