The independant variables are the non-dimensional
distance from the wall. The dependant variable f
represents the stream function. The solution represents similarity
solutions. The equation is due to Blasius and generally numerical
solutions abound after assuming the missing initial value for f''(0)
as 0.33206.
Fig.1. The non-linear two-point boundary
value problem
The function that is useful as a
solution is f(n)'. This the non-dimension x-velocity profile.
Where this ratio reaches 0.99 is also used to define the edge of the boundary
layer. The standard numerical solution and the Bezier ( 4 th order
solution) is shown. On this scale it is difficult to distinguish
them.
Fig 2. Laminar Boundary Layer over
Flat Plate
Several other examples have been studied. The Bezier curves provide a reasonable approximate solution ( analytical solution) to non-linear initial and two-point boundary value problems.