explicit solution to heat equation without an integral sign Consider the 1-dimensional heat equation:
$$\left\{ \begin{align}
  & {{u}_{t}}\left( x,t \right)={{u}_{xx}}\left( x,t \right),\text{      }x\in R,\text{   }t>0 \\ 
 & u\left( x,0 \right)={{e}^{a{{x}^{2}}}},\text{      }x\in R \\ 
\end{align} \right.$$  
Find an explicit solution without integral signs.  
I have tried separation of variables, Green's function, and Fourier transform but just couldn't resolve the integral because of the term ${{e}^{a{{x}^{2}}}}$.  Please help.
 A: Inspired by Heat equation with initial value answered by Mercy King.  
To simplify, let $a=1$.  
Setting
$$\xi=\sqrt{\frac{1-4t}{4t}}\left(y-\frac{x}{1-4t}\right),$$
we have:
\begin{eqnarray}
\frac{(x-y)^2}{4t}-y^2&=&\frac{(1-4t)y^2-2xy+x^2}{4t}\\
&=&\frac{1-4t}{4t}\left[y^2-\frac{2x}{1-4t}y+\frac{x^2}{1-4t}\right]\\
&=&\frac{1-4t}{4t}\left[\left(y-\frac{x}{1-4t}\right)^2+\frac{x^2}{1-4t}-\frac{x^2}{(1-4t)^2}\right]\\
&=&\frac{1-4t}{4t}\left[\left(y-\frac{x}{1-4t}\right)^2-\frac{4x^2t}{(1-4t)^2}\right]\\
&=&\xi^2-\frac{x^2}{(1-4t)}.
\end{eqnarray}
It follows that
\begin{eqnarray}
u(x,t)&=&\frac{1}{\sqrt{4\pi t}}\int_{-\infty}^\infty\exp\left(-\frac{(x-y)^2}{4t}+y^2\right)\, dy\\
&=&\frac{1}{\sqrt{4\pi t}}\sqrt{\frac{4t}{1-4t}}\exp[\frac{x^2}{(1-4t)}]\int_{-\infty}^\infty\exp(-\xi^2)\, d\xi\\
&=&\frac{1}{\sqrt{\pi(1-4t)}}\exp[\frac{x^2}{(1-4t)}]\int_{-\infty}^\infty\exp(-\xi^2)\, d\xi.
\end{eqnarray}
Using the fact that
$$\int_{-\infty}^\infty\exp(-\xi^2)\, d\xi=\sqrt{\pi},$$
we get
$$u(x,t)=\frac{1}{\sqrt{1-4t}}\exp\left(\frac{x^2}{(1-4t)}\right).$$
