heat kernel

The whole-line heat equation's solution for initial data concentrated at a single point -- a spreading Gaussian.

The heat kernel $S(x,t)=\dfrac1{\sqrt{4\pi kt}}e^{-x^2/(4kt)}$ solves $S_t=kS_{xx}$ for $t>0$ , is positive everywhere, integrates to $1$ over $x$ for every $t$ , and narrows toward a spike at $x=0$ as $t\to0^+$ ; the solution from any initial data $g$ is the convolution $u(x,t)=\displaystyle\int_{-\infty}^\infty S(x-y,t)\,g(y)\,dy$ .