Regular perturbation for diffusion with second-order reaction

Regular perturbation for diffusion with second-order reaction. Consider for illustration the diffusion–reaction problem given by d2c /dx2 = Mc2 (14.54) with the same boundary conditions as before. Use the expansion for c in terms of M as in Eq. (14.24) for small values of M, which is a regular perturbation problem. Show that by substituting into the differential equation we obtain Mf 1 + M2f 2 +···= M + 2M2f1 + O(M3) (14.55) By equating the powers of M, write out the governing equations for f1 and f2. Integrate the equations to obtain the following results: f1 = x2 − x and f2 = (x4/4 − x3 + 2x)/3 Derive an expression for the effectiveness factor, and show that it can be approximated as 1 − 2M/3 to an approximation of O(M2). Derive the next approximation. Also solve the problem numerically using BVP4C and compare the results.