I have a 1-dimensional homogeneous heat equation: $ u''(x, t) = \dot u(x, t)$
The initial value is $u(x, 0) = \exp\left(-x^2\right)$.
I plugged this into the solution formula: $ u(x, t) = \frac{1}{\sqrt{4 \pi t}} \int_{-\infty}^\infty \mathrm dy \, \exp\left(-\frac{(x-y)^2}{4t} - y^2\right)$
Since I am not sure how to calculate this integral, I typed it into Mathematica and got:
1/Sqrt[4 Pi t] Integrate[ Exp[((y - x)^2)/(4 t) - y^2], {y, -Infinity, Infinity}]
$\text{ConditionalExpression}\left[\frac{\exp\left(\frac{x^2}{4 t-1}\right)}{\sqrt{4-\frac{1}{t}} \sqrt{t}},\left(4 \Re(t)\neq 1\lor \Re\left(\frac{x}{t}\right)>0\right)\land \Re\left(\frac{1}{t}\right)<4\right]$
Plotting this looks strange at best:
http://wstaw.org/m/2012/11/10/1_1.png
For $t=0$, it is not even close to the initial value.
How do I solve this integral?
Is what Mathematica gave me correct in some sort?