Solving Blasius Equation using Newton Raphson method

Solution to Blasius Equation for flat plate, a 3rd order non-linear ODE by Newton Raphson in combination with ODE45

Blasius equation for flat plate is a Third Order Non-Linear Ordinary Differential Equation governing boundary layer flow : f'''(η)+(1/2) f(η) f''(η) = 0 where η is similarity variable. This equation can be solved numerically by converting to three simulatneous Ordinary Linear Differential Equations : { f(η) = f(η) ; g(η) = f'(η) ; h(η) = f''(η) } then f'(η) = g(η) ; g'(η) = h(η) ; h'(η) = -(1/2) f(η) h(η) with f(0) = 0 , g(0) = f'(0) = 0 , and h(0) = ? (to be found) such that g(∞)=1.

We handle this problem as Initial Value Problem approached by numerical methods by Choosing h(0) such that it shoots to g(∞)=1. Initial guesses may give an error: 1- g(∞) ≠ 0 . with subsequent iterations of numerical methods resolves the error. This method is called shooting technique.

Here, Newton Raphson approximation is used to refine values of h0 then using ODE45 or Rk4 method to find solution.

Reference : https://nptel.ac.in/content/storage2/courses/112104118/lecture-28/28-7_blasius_flow_contd.htm