We would like to use centered finite differences and the method of lines to solve the unsteady diffusion–reaction problem

subject to no-flux on the left boundaries, ∂c/∂x = 0 at x = 0, a constant concentration on the right boundary, c(1, t) = 1, and an initial concentration c(x, 0) = 1. We will use n = 3 nodes for this problem and a reaction rate k = 2.

(a) Convert the PDE into a system of coupled ODEs governing the concentrations at each node, c₁, c₂, and c₃. Remember to state the initial conditions.

(b) Let us do one time step of size h = 1/4 using implicit Euler. Write down the system of two equations you need to solve to compute c₁ and c₂ after the first time step. You do not need to compute c₁ or c₂.

Transcribed Image Text:

ас at a²c ах2 - kc