I find in a Centrale's school document this solution for "short" time of the heat equation, I have not MAPLE or other calculus softwares, and I just want to be sure if my hand verification is correct id est its a correct solution.
I was a bit surprise by the $\sqrt{\frac{4at}{\pi}}$ that's why I want to be sure.
The solution is :
$$T(x,t)=T_0+\frac{\phi}{\lambda}\left[\sqrt{\frac{4at}{\pi}}\exp\left(-\frac{x^2}{4at}\right)-x\left(1-\mathrm{erf}\left(\frac{x}{\sqrt{4at}}\right)\right)\right]$$
Thank you in advance
EDIT : Like robert said its $\sqrt{\frac{4at}{\pi}}$ and not $\frac{\sqrt{4at}}{\pi}$
It's not quite right. I think you mean
$$ T(x,t) = T_0 + \dfrac{\phi}{\lambda} \left[ \sqrt{\dfrac{4at}{\pi}} \exp \left(-\dfrac{x^2}{4at}\right) - x \left(1 - \text{erf}\left(\dfrac{x}{\sqrt{4at}}\right)\right)\right] $$
for the heat equation in the form
$$ \dfrac{\partial T}{\partial t} = a \dfrac{\partial^2 T}{\partial x^2} $$