A singular reaction–diffusion problem with two boundary layers

A singular reaction–diffusion problem with two boundary layers. Solve the following singular perturbation problem: d2cb/ dξ 2 = ξcb − 1 which arises in a model for a consecutive reaction in a liquid film (Deen, 2011). Here cb is the concentration variable, ξ is the distance variable, and is a small parameter. The boundary conditions are that there is no flux for cb at both ends, viz., ξ = 0 and ξ = 1. Show that there are boundary layers at both sides (0 and 1), and that a different scaling is needed for each side. Obtain these solutions together with that for the inner region to the first needed level of approximation.

Plot your results for = 0.01. Answers from a numerical solution generated by BVP4C are provided for comparison: cb(0) = 5.9782 and cb(1) = 1.1439. You may also wish to run the BVP4C code to verify these results for extra practice