We set following first-order system: f & = p, p& = q, q& = 1 2 f q (15) & = z,z & = 1 2 Sc f z + Sc(np + ) (16) with the boundary conditions: f(0) = S ; p(0) = 0; (0) = 1 (17) To solve (15) and (16) with (17) as an IVP we must need values for q(0) i.e. f &&(0) and z(0) i.e. & (0) but no such values are given in the boundary conditions. The initial guess values for f & &(0) and & (0) are chosen and applying fourth order Runge-Kutta method a solution is obtained. Then we compare the calculated values of f & () and () at (= 20) with the given boundary conditions f & () = 1 and () = 0 and adjust the values of f &&(0) and & (0) using Secant method to give better approximation for the solution. The step-size is taken as = 0.01. The process is repeated until we get the results correct up to the desired accuracy of 106 level.

This is taken from BOUNDARY LAYER FLOW WITH DIFFUSION AND FIRST-ORDER CHEMICAL REACTION OVER A POROUS FLAT PLATE SUBJECT TO SUCTION/INJECTION AND WITH VARIABLE WALL CONCENTRATION.

My question is, how can this be used in order to get the velocity profile? (f vs f')