The diffusion equation $u_t=Du_{xx}$ can be nondimensionalized via the scaling parameters $L$ and $T$,
$$
T^{-1}\dfrac{\partial u^*}{\partial t^*}=DL^{-2}\dfrac{\partial^2u^*}{\partial x^{*2}}
$$
Hence we can naturally introduce quantity $C:=\dfrac{L}{\sqrt{DT}}$. The PDE can be rewritten as
$$
C^2\dfrac{\partial u^*}{\partial t^*}=\dfrac{\partial^2u^*}{\partial x^{*2}}
$$
For simplicity, we write $u$, $t$ and $x$ instead of $u^*$, $t^*$ and $x^*$. Let $c$ denotes $\dfrac{x}{\sqrt{Dt}}$,
$$
\left\{\begin{align*}
\dfrac{\partial u}{\partial x}\,\,&=\dfrac{\partial u}{\partial c}\cdot\dfrac{\partial c}{\partial x}\\
\dfrac{\partial^2 u}{\partial x^2}&=\dfrac{1}{Dt}\cdot\dfrac{\partial^2 u}{\partial c^2}\\
\dfrac{\partial u}{\partial t}\,\,&=-\dfrac{x}{2\sqrt D\sqrt t^3}\cdot\dfrac{\partial u}{\partial c}\\
\end{align*}\right.
$$
The equation $C^2\dfrac{\partial u^*}{\partial t^*}=\dfrac{\partial^2u^*}{\partial x^{*2}}$ becomes:
$$
u_c=-\dfrac{2}{c}u_{cc}
$$
Hence
$$
u(x,t)=u_0+(u_\infty-u_0)\cdot\dfrac{2}{\sqrt\pi}\int_0^{x/2\sqrt{Dt}}e^{-s^2}\mathrm ds
$$
where $u_0:=u|_{c=0}$, $u_\infty:=u|_{c=\infty}$.