Theorem Let $u(x,t)$ be the temperature at a point on a long solid metal rod lying along the $x$ axis at time $t>0$. It can be shown that $u$ satisfies the following [[Partial differential equations|PDE]]. $\frac{ \partial u }{ \partial t } = \alpha \frac{ \partial^{2} u }{ \partial x^{2} } $where $\alpha$ is a constant called the thermal conductivity of the metal.