Considering the 1-D semi-infinite domain, $$ \begin{aligned} & \frac{\partial^2 T(x, t)}{\partial x^2}=\frac{1}{\alpha} \frac{\partial T(x, t)}{\partial t} \quad \text { in } \quad 0<x<\infty, \quad t>0 \\ & \mathrm{BC} 1: T(x=0, t)=f(t) \\ & \mathrm{BC} 2: T(x \rightarrow \infty, t)=0 \\ & \mathrm{IC}: T(x, t=0)=0 \end{aligned} $$
The solution is given by
$$ T(x, t)=\frac{x}{\sqrt{4 \pi \alpha}} \int_{\tau=0}^t \frac{f(\tau)}{(t-\tau)^{3 / 2}} \exp \left[-\frac{x^2}{4 \alpha(t-\tau)}\right] d \tau $$
Then I notice that There's discontinuity at $x=0$, the integration of $ \int_{\tau=0}^t\left(f(\tau) /(t-\tau)^{3 / 2}\right) d \tau $ does not exist since
$$ \int_{\tau=0}^t\left(f(\tau) /(t-\tau)^{3 / 2}\right) d \tau=\sqrt{4 \pi \alpha} \frac{f(t)}{x} $$
Therefore we cannot calcute heat flux at surface, i.e.
$$ q_x^{\prime \prime}(x=0)=-\left.k \frac{\partial T}{\partial x}\right|_{x=0} $$ is unstable.
I read some references saying that the solution is not a closed form and you can use chnage of variable to get a stable form of the heat flux. Can anyone please tell me what does 'closed form' mean and how to transform a solution to a closed form?
Thank in advance.