An ordinary differential equation with a Neumann boundary condition.

Let's consider a simple diffusion and simultaneous first$-$order reaction in a reactor with a length of $L$ in the $z$ axis, where $A$ is converted into $B$ with a reaction rate of $k.$ The concentration distribution of $A$ is governed by the following ODE:

$D_{AB}\frac{d^2{C}_A}{dz}=k{C}_A$

where $C_A$ and $D_{AB}$ is the concentration of A and diffusion coefficient respectively.

Predict the concentration of A using the finite difference method with the centered finite difference $($first$-$order approximation$)$ for the spatial derivative. Use the boundary conditions and parameters:

$\backslash$table$[[$Boundary condition at $z=0,$Boundary condition at $z=L,k,L,D_{AB},$Step size $(h)$