Question about approximation of the solution of heat equation on the half-line

Question

We have the heat equation: $\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial^2 x}$ on the positive half-line $x>0$.

with the boundary condition $u(t,0)=0$.

The initial condition is compactly supported.

I want to show that the solution has the following approximation: $u(t,x) \approx \frac{x}{t^\frac{3}{2}}e^{-\frac{x^2}{4t}} $

I am not sure how to go about proving this, any help would be very much appreciated!