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 **MAT**rix **LAB**oratory.

```
function a = runge_kutta(df)
% asking initial conditions
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 ');
```

## Other Numerical Methods with MATLAB Coding

- Bisection Method with MATLAB
- Newton Raphson Method with MATLAB
- Secant Method with MATLAB
- Regula Falsi Method with MATLAB
- Fixed Point Iteration with MATLAB
- Trapezoidal Rule with MATLAB
- Simpson 1/3 Rule with MATLAB
- Simpson 3/8 Rule with MATLAB
- Bool’s Rule with MATLAB
- Weddle’s Rule with MATLAB
- Euler Method with MATLAB
- Modified Euler Method with MATLAB
- Midpoint Method with MATLAB
- Runge-Kutta Method with MATLAB
- Millen’s Method with MATLAB
- Adams Bashforth Moulton Method with MATLAB
- Newton Forward Difference Interpolation with MATLAB
- Newton Backward Difference Interpolation with MATLAB
- Lagrange Interpolation with MATLAB
- Newton Divided Difference Interpolation with MATLAB
- Hermite Interpolation with MATLAB
- Natural Cubic Spline Interpolation with MATLAB
- Gauss Jacobi Method with MATLAB
- Gauss Seidal Method with MATLAB
- Power Method with MATLAB