$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\Aex}{{A_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\dfc}{\alpha} % diffusion coefficient
$$
The 1D diffusion equation
The famous diffusion equation, also known as the heat equation,
reads
$$ \frac{\partial u}{\partial t} =
\dfc \frac{\partial^2 u}{\partial x^2}
$$
Here,
- \( u(x,t) \): unknown
- \( \dfc \): diffusion coefficient
Alternative, compact notation:
$$ u_t = \dfc u_{xx} $$
