I want a whole code for solving the Blasius equation(similarity variable 'eta') using shooting method with Runge Kutta 4th order numerical technique

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naygarp on 26 Oct 2017

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Commented: Walter Roberson on 19 Jun 2018

I tried to write a brief code for the Blasius equation but I am unable to proceed further, it will be helpful if improvements are done in the code that I have written

% f'''+1/2*f*f''=0
% converting to 3 1st order odes
% f'=G
% G'=H= f''
% H'= -1/2*f*H = f'''
clc;
close all;
h=0.01; % step size
F1=@(eta,f,G,H) G;
F2=@(eta,f,G,H) H;
F3=@(eta,f,G,H) -1/2*f*H;
%initial conditions
eta(1)=0;
f(1)=0;
G(1)=0;
%initial guess
H(1)= 0.332;
N=10/h; % eta ranges from 0 to 10
for i=1:N
    eta(i+1)= eta(i)+h;
    k1= F1(f(i),  G(i),  H(i),   eta(i));
    l1= F2(f(i),  G(i),  H(i),   eta(i));
    m1= F3(f(i),  G(i),  H(i),   eta(i));
    k2= F1(f(i)+1/2*h*k1, G(i)+1/2*h*l1, H(i)+1/2*h*m1, eta(i)+h/2);
    l2= F2(f(i)+1/2*h*k1, G(i)+1/2*h*l1, H(i)+1/2*h*m1, eta(i)+h/2);
    m2= F3(f(i)+1/2*h*k1, G(i)+1/2*h*l1, H(i)+1/2*h*m1, eta(i)+h/2);
    k3= F1(f(i)+1/2*h*k2, G(i)+1/2*h*l2, H(i)+1/2*h*m2, eta(i)+h/2);
    l3= F2(f(i)+1/2*h*k2, G(i)+1/2*h*l2, H(i)+1/2*h*m2, eta(i)+h/2);
    m3= F3(f(i)+1/2*h*k2, G(i)+1/2*h*l2, H(i)+1/2*h*m2, eta(i)+h/2);
    k4= F1(f(i)+h*k3, G(i)+h*l3, H(i)+h*m3, eta(i)+h);
    l4= F2(f(i)+h*k3, G(i)+h*l3, H(i)+h*m3, eta(i)+h);
    m4= F3(f(i)+h*k3, G(i)+h*l3, H(i)+h*m3, eta(i)+h);
    f(i+1)= f(i)+ h/6*(k1+2*(k2+k3)+k4);
    G(i+1)= G(i)+ h/6*(l1+2*(l2+l3)+l4);
    H(i+1)= H(i)+ h/6*(m1+2*(m2+m3)+m4);
end
plot(eta,f,'r')
hold on
plot(eta,G,'b')
hold on
plot(eta,H,'g')