Numerical Integration (Simpson's 1/3 Rule)

The Runge-Kutta Method was developed by two German men, Carl Runge (1856-1927) and Martin Kutta (1867- 1944), in 1901.

Runge Kutta Method implementation in c

Carl Runge developed numerical methods for solving the differential equations that arose in his study of atomic spectra.

Today his name is associated with the Runge-Kutta methods to numerically solve differential equations. Another German applied mathematician, Kutta, is also remembered for contributing to the differential equations-based Kutta-Joukowski theory of airfoil lift in aerodynamics.

The 4th-Order Runge-Kutta method is a standard numerical method used to solve differential equations with a known initial condition. The method starts at the initial condition and proceeds stepwise to develop successive points in the function based on the previous point and the calculated Runge-Kutta parameters. The method lends itself to spreadsheet calculations.

The code for the Runge Kutta method has been written in c language.

CODE for Runge Kutta method in C:

#include<stdio.h>

#include<conio.h>

#include<math.h>

float fun(float x, float y)

{

    return(0.1*x*x+0.1*y*y);       /* here we will give our function which to be solved*/

}

int main( )

{

    int i,j,N;

    float x0,y0,x,h,y,k[20][20],Y[20],n,p,a;

   printf("\n Enter the initial value of X:");

    scanf("%f",&x0);

    printf("\n Enter the initial value of Y:");

    scanf("%f",&y0);

    printf("\n Enter the given value of X:");

    scanf("%f",&x);

    printf("\n Enter the value of step size h:");

    scanf("%f",&h);

    n=(x-x0)/h;

        N=(int)n;

    printf("\nNumber of iterations is:%d\n",N);

    for(i=1;i<=N;i++)

    {

        k[i][1]=h*fun(x0,y0);

        k[i][2]=h*fun(x0+0.5*h,y0+0.5*k[i][1]);

        k[i][3]=h*fun(x0+0.5*h,y0+0.5*k[i][2]);

        k[i][4]=h*fun(x0+h,y0+k[i][3]);

        k[i][5]=(1.0/6.0)*(k[i][1]+2*k[i][2]+2*k[i][3]+k[i][4]);

        x0=x0+h;

        y=y0+k[i][5];

        Y[i]=y;

        y0=y;

    }

    printf("\nk1\t\tk2\t\tk3\t\tk4\t\tk\t\tY\n");

    x0=x0-n*h;

    p=h;

    for(i=1;i<=N;i++)

    {

        for(j=1;j<=5;j++)

        {

                        printf("%f\t",k[i][j]);

        }

        a=x0+h;

        printf("\tY(%f)=%f",a,Y[i]);

        printf("\n");

        h=h+p;

    }

    printf("\nThe final value of Y:\t%f",y);

   getch();

}

Input and Output:

If the function is 0.1*x*x+0.1*y*y, then what will happen in the program is given below

Runge Kutta Method implementation in c

APPLICATION OF RUNGE KUTTA METHOD:

Runge-Kutta methods are widely used to integrate initial value problems for ordinary differential equations. They can also incorporate initial value problems for time-dependent partial differential equations by applying the so-called method of lines. 

Related Post:

Numerical Integration (Simpson's 1/3 Rule)