# Runge-Kutta Method – Numerical Differentiation with MATLAB

Runge-Kutta method is a famous numerical method for the solving of ordinary differential equations. This method was developed in 1900 by German mathematicians C.Runge and M. W. Kutta. The RK method is valid for both families of explicit and implicit functions.

The Runge-Kutta method is shortly written as RK method. It is based on solution procedure of initial value problem with given initial conditions. There are different RK method according to distinct order such as RK method of order 1, RK method of order 2, RK method of order 3 and RK method of order 4.

At here, we write the code of Runge-Kutta Method in MATLAB step by step. MATLAB is easy way to solve complicated problems that are not solve by hand or impossible to solve at page. MATLAB is develop for mathematics, therefore MATLAB is the abbreviation of MATrix LABoratory.

``````function a = runge_kutta(df)
x0 = input('Enter initial value of x : ');
y0 = input ('Enter initial value of y : ');
x1 = input( 'Enter value of x at which y is to be calculated : ');
tol = input( 'Enter desired level of accuracy in the final result : ');

%choose the order of Runge-Kutta method
r = menu ( ' Which order of Runge Kutta u want to use', ...
' 2nd order ' , ' 3rd order ' , ' 4th order ');

switch r
case 1
% calculating the value of h
n =ceil( (x1-x0)/sqrt(tol));
h = ( x1 - x0)/n;
for i = 1 : n
X(1,1) = x0; Y (1,1) = y0;
k1 = h*feval( df , X(1,i), Y(1,i));
k2 = h*feval( df , X(1,i) + h , Y(1,i) + k1);
k = 1/2* ( k1+ k2);
X( 1, i+1) = X(1,i) + h;
Y( 1 ,i+1) = Y(1,i) + k;
end

case 2
% calculating the value of h
n =ceil( (x1-x0)/nthroot(tol,3));
h = ( x1 - x0)/n;
for i = 1 : n
X(1,1) = x0; Y (1,1) = y0;
k1 = h*feval( df , X(1,i), Y(1,i));
k2 = h*feval( df , X(1,i) + h/2, Y(1,i) + k1);
k3 = h*feval( df , X(1,i) + h, Y(1,i) + k2);
k = 1/6* ( k1+ 4*k2 + k3);
X( 1, i+1) = X(1,i) + h;
Y( 1 ,i+1) = Y(1,i) + k;
end

case 3
% calculating the value of h
n =ceil( (x1-x0)/nthroot(tol,3));
h = ( x1 - x0)/n;
for i = 1 : n
X(1,1) = x0; Y (1,1) = y0;
k1 = h*feval( df , X(1,i), Y(1,i));
k2 = h*feval( df , X(1,i) + h/2, Y(1,i) + k1);
k3 = h*feval( df , X(1,i) + h/2, Y(1,i) + k2);
k4 = h*feval( df , X(1,i) + h, Y(1,i) + k3);
k = 1/6* ( k1+ 2*k2 + 2*k3 + k4);
X( 1, i+1) = X(1,i) + h;
Y( 1 ,i+1) = Y(1,i) + k;
end

end

%displaying results
fprintf( 'for x = %g \n y = %g \n' , x1,Y(1,n+1))

%displaying graph
x = 1:n+1;
y = Y(1,n+1)*ones(1,n+1) - Y(1,:);
plot(x,y,'r')
xlabel = (' no of interval ');
ylabel = ( ' Error ');``````