I want to show that \begin{gather} u(x,t) = \int_0^\infty \frac{f(t+s)}{\sqrt{2\pi s^3}}\sum_{n=-\infty}^\infty \left[(4n+1-x)exp\left(-\frac{(4n+1-x)^2}{2s}\right)+(4n+1+x)exp\left(-\frac{(4n+1+x)^2}{2s}\right)\right]ds\\ \text{with}\quad u(1,t)=f(t). \end{gather} fulfills the heat equation $u_t+\frac{1}{2}u_{xx}=0$ for $t>0$ and $0<x<1$.
$f(t)$ is finite and continuous.