$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\Aex}{{A_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\dfc}{\alpha} % diffusion coefficient
$$
The initial-boundary value problem for 1D diffusion
$$
\begin{align}
\frac{\partial u}{\partial t} &=
\dfc \frac{\partial^2 u}{\partial x^2}, \quad x\in (0,L),\ t\in (0,T]
\tag{1}\\
u(x,0) &= I(x), \quad x\in [0,L]
\tag{2}\\
u(0,t) & = 0, \quad t>0,
\tag{3}\\
u(L,t) & = 0, \quad t>0\tp
\tag{4}
\end{align}
$$
Note:
- First-order derivative in time: one initial condition
- Second-order derivative in space: a boundary condition at each
point of the boundary (2 points in 1D)
- Numerous applications throughout physics and biology
