Study guide: Finite difference schemes for diffusion processes

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$\newcommand{\uex}{{u_{\small\mbox{e}}}} \newcommand{\Aex}{{A_{\small\mbox{e}}}} \newcommand{\half}{\frac{1}{2}} \newcommand{\tp}{\thinspace .} \newcommand{\Oof}[1]{\mathcal{O}(#1)} \newcommand{\dfc}{\alpha} % diffusion coefficient$

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