Find the explicit formula for $u:[0,\infty)\times \mathbb{R}\to \mathbb{R}$ satisfying the heat equation with the following initial conditions:
\begin{align*} u_t&=u_{xx} \mathrm{\ \ \ \ \ \ in\ (0,\infty)\times \mathbb{R}^n}\\ u(t,0)&=0 \mathrm{\ \ \ \ \ \ \ \ \ for\ any\ }t\geq 0\\ u(0,x)&=f(x) \mathrm{\ \ \ \ for\ any\ }x\in [0,\infty) \end{align*} where $f(x):[0,\infty)\to \mathbb{R}$ is a continuous and bounded function satisfying $f(0)=0$
Note that this is an exercise right after learning the fundamental solution of the heat equation. So the very first thing I do is to write $$u(t,x)=\frac{1}{\sqrt{4\pi t}}\int_{\mathbb{R}}f(y)e^{-\frac{|y-x|^2}{4t}}dy$$ and substitute the additional condition $u(t,0)=0$ but clearly I cannot proceed after this.