= Gaussian initial data for the heat equation
{c}
{title2=$u(x,t)=A(1+4\gamma Dt)^{-1/2}\exp[-\gamma(x-x_0)^2/(1+4\gamma Dt)]$}
For $D,\gamma>0$, the displayed <function> solves the <heat equation> $u_t=Du_{xx}$ and has <initial condition> $u(x,0)=Ae^{-\gamma(x-x_0)^2}$. Set $s=1+4\gamma Dt$. Direct <differentiation> gives
$$
u_t=\left(-\frac{2\gamma D}{s}+\frac{4\gamma^2D(x-x_0)^2}{s^2}\right)u,
\qquad
u_{xx}=\left(-\frac{2\gamma}{s}+\frac{4\gamma^2(x-x_0)^2}{s^2}\right)u.
$$
These identities prove the equation. The total <integral> is $A\sqrt{\pi/\gamma}$, independent of time. The real smooth formula is valid for $t>-1/(4\gamma D)$, including a finite interval of negative time for this particular initial profile. For nonzero $A$, its value at $x_0$ diverges as the lower time endpoint is approached. The <superposition principle> combines finitely many such profiles with different centers or widths.
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