Help understanding ODE code

I found this code online, and I'm trying to understand exactly how it works. All I've really been able to understand is that f2zero finds f"(0) with a guess of '1'. the function odesol and the input f2zeroiter is the part I can't really understand.

Here's the code:

function f2zero = boundarylayerblasius dydx = @(t,f) [f(2); f(3); -0.5 * f(1) * f(3)]; f2zeroguess = 1; function f1infm1 = odesol(f2zeroiter) [T,F] = ode45(dydx,[0,5.2],[0,0,f2zeroiter]); f1infm1 = F(end,2)-1; end f2zero = fzero(@odesol, f2zeroguess); plot(T,F, 'linewidth',2); legend('f','f''','f"', 'location', 'northwest') end

Answer 1

[Hari]

Hi Josh,

I understand that you are trying to comprehend a MATLAB code that solves a boundary layer problem using the Blasius equation. You are specifically confused about the "odesol" function and its interaction with "f2zero".

Defining the ODE System:

The function "dydx" defines the system of ordinary differential equations (ODEs) for the Blasius boundary layer problem. It is a system of three first-order ODEs.

dydx = @(t,f) [f(2); f(3); -0.5 * f(1) * f(3)];

Initial Guess for f''(0):

"f2zeroguess" is the initial guess for the unknown boundary condition f''(0). This is a common approach in shooting methods for boundary value problems.

f2zeroguess = 1;

ODE Solution Function (odesol):

The function "odesol" solves the ODEs using ode45 with the guessed value of f''(0). It calculates the solution over the interval [0, 5.2] with initial conditions [0, 0, f2zeroiter].

The purpose of "odesol" is to return the difference between the computed and desired boundary condition at infinity (f'(infinity) = 1).

function f1infm1 = odesol(f2zeroiter)
[T,F] = ode45(dydx,[0,5.2],[0,0,f2zeroiter]);
f1infm1 = F(end,2) - 1;
end

Finding the Correct f''(0) with fzero:

fzero is used to find the root of odesol, i.e., the value of f''(0) that makes f'(infinity) = 1. It iteratively adjusts f2zeroiter until the condition is satisfied.

f2zero = fzero(@odesol, f2zeroguess);

Plotting the Solution:

The solution is plotted to visualize the functions f, f', and f''.

plot(T,F, 'linewidth',2);
legend('f','f''','f"', 'location', 'northwest');

Refer to the documentation of "ode45" for more details: https://www.mathworks.com/help/matlab/ref/ode45.html

Refer to the documentation of "fzero" for more details: https://www.mathworks.com/help/matlab/ref/fzero.html

Hope this helps!