Solving for two variables using 4th order Runga Kutta Method

I need to use a 4th order Runga Kutta method to find variables u and t.

du/dr = f(r,t)

dt/dr = f(r,t,u)

u(j+1) = +

where

and

where

[Walter Roberson]

http://www.mathworks.com/matlabcentral/answers/6200-tutorial-how-to-ask-a-question-on-answers-and-get-a-fast-answer

Answer 1

[Torsten]

Define your functions for du/dr and dt/dr in the line f = @(r,y) ... and adapt the settings in the calling program according to your needs.

rstart = 0.0;
rend = 1.0;
h = (rend - rstart)/20;
R = (rstart:h:rend).';
Y0 = [1 -1]; 
B = 4;
f = @(r,y) [y(2) -exp(-B*r)-y(1)+5*exp(-2*r)-2*exp(-(B+2)*r)+exp(-B*r)+r];
Y = runge_kutta_RK4(f,R,Y0);
plot(R,Y)                           

function Y = runge_kutta_RK4(f,R,Y0)
  N = numel(R);
  n = numel(Y0);
  Y = zeros(N,n);
  Y(1,:) = Y0;
  for i = 2:N
    r = R(i-1);
    y = Y(i-1,:);
    h  = R(i) - R(i-1);
    k0 = f(r,y);
    k1 = f(r+0.5*h,y+k0*0.5*h);
    k2 = f(r+0.5*h,y+k1*0.5*h);
    k3 = f(r+h,y+k2*h);
    Y(i,:) = y + h/6*(k0+2*k1+2*k2+k3);
  end
end