How can I plot a second order differential equation with boundary condition using fourth order Runge-Kutta method?

%%%%%%%%%%%%%%%% Runga-Kutta%%%%%%%%%%%%%%%%

h=0.0001;

xfinal=d;

x(1)=0;

y(1)=0; % initial value of y

y(xfinal)=0; % final value of y

% Let y' = z (f1) and y" = z' (f2);

f1 = @(x, y, z) z;

f2 = @(x, y, z) ky^2*y-(ky*(-2*W*(pi/d)*tan(2*pi*x/d)+2*u0*((pi/d)^2)*cos(2*pi*x/d))*y)/(OP3-ky*u0*(sin(pi*x/d).^2-1/2)+...

B*(OP3-ky*u0*(sin(pi*x/d).^2-1/2)-A*(-2*W*(pi/d)*tan(2*pi*x/d)+2*u0*((pi/d)^2)*cos(2*pi*x/d)-ky*(OP3-ky*u0*(sin(pi*x/d).^2-1/2))))*(1-...

M*(opi^2)/(M*OP3^2-gi*Ti*ky^2)));

for i=1:ceil(xfinal/h)

x(i+1)=x(i)+h;

K1y = f1(x(i), y(i), z(i));

K1z = f2(x(i), y(i), z(i));

K2y = f1(x(i)+0.5*h, y(i)+0.5*K1y*h, z(i)+0.5*K1z*h);

K2z = f2(x(i)+0.5*h, y(i)+0.5*K1y*h, z(i)+0.5*K1z*h);

K3y = f1(x(i)+0.5*h, y(i)+0.5*K2y*h, z(i)+0.5*K2z*h);

K3z = f2(x(i)+0.5*h, y(i)+0.5*K2y*h, z(i)+0.5*K2z*h);

K4y = f1(x(i)+h, y(i)+K3y*h, z(i)+K3z*h);

K4z = f2(x(i)+h, y(i)+K3y*h, z(i)+K3z*h);

y(i+1) = y(i)+(K1y+2*K2y+2*K3y+K4y)*h/6;

z(i+1) = z(i)+(K1z+2*K2z+2*K3z+K4z)*h/6;

end

plot(x,y,'-','linewidth',1)

hold on

[John D'Errico]

It looks like you already solved the ODE, and plotted it. Where is the problem? (Even so, if this were not homework, as it surely is, you should be using an ODE solver, not writing your own code.)

Answer 1

[Cameron]

Here is a list of built-in ODE solvers within MATLAB.

Choose an ODE Solver - MATLAB & Simulink (mathworks.com)