## Aprroximate Solution to Non-Linear System Dynamics

Here is an  example for the field of Fluid Mechanics.  This is the solution to the laminar boundary layer over the flat plate.  it is a non-linear two point boundar value problem

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.

Approximate Solutions to Optimal Control Problems