Study guide: Finite difference schemes for diffusion processes

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

Step 2: Fulfilling the equation at the mesh points

Require the PDE (1) to be fulfilled at an arbitrary interior mesh point $ (x_i,t_n) $ leads to $$ \begin{equation} \frac{\partial}{\partial t} u(x_i, t_n) = \dfc\frac{\partial^2}{\partial x^2} u(x_i, t_n) \tag{5} \end{equation} $$

Applies to all interior mesh points: $ i=1,\ldots,N_x-1 $ and $ n=1,\ldots,N_t-1 $

For $ n=0 $ we have the initial conditions $ u=I(x) $ and $ u_t=0 $

At the boundaries $ i=0,N_x $ we have the boundary condition $ u=0 $.