Runge Kutta Method

Algorithm Of Runge Kutta Method Order2

Runge_Of_order2(a,b,h,y0)
{
This algorithm takes the following inputs where h is the width of each division y0
is the value of y when x=0 and a&b are the limits
while(x<=b)
{
  x1=x+h
  k1=h*(f(x,y0))
  k2=h*f(x+h,y0+h)
  k=(k1+k2)/2
  y1=y0+k
  print(i,x1,y1)
  i=i+1
  x=x1
  y0=y1
}
}


Algorithm Of Runge Kutta Method Order4

Runge_Of_order4(a,b,h,y0)
{
This algorithm takes the following inputs where h is the width of each division y0
is the value of y when x=0 and a & b are the limits

while(x<=b)
{
  x1=x+h
  k1=h*(f(x,y0))
  k2=h*f(x+h/2,y0+k1/2)
  k3=h*f(x+h/2,y0+k2/2)
  k4=h*f(x+h,y0+k3)
  k=(k1+2*k2+2*k3+k4)/2
  y1=y0+k
  print(i,x1,y1)
  i=i+1
  x=x1
  y0=y1
}
}



C Program For Runge Kutta Method

Source Code:

#include<stdio.h>
#include<conio.h>
#include<math.h>

void R_K(float,float,float,int,int);
float f(float,float);
void main()
{
 int n,order;
 float a,b,y0;
 clrscr();
 printf("\nSOLVE :\n");
 printf("\n (dy/dx)=x-y     using the given limits.\n");
 printf("\nEnter the upper limit:");
 scanf("%f",&b);
 printf("\nEnter the lower:");
 scanf("%f",&a);
 printf("\nEnter the value of y for the first interval:");
 scanf("%f",&y0);
 printf("\nENTER THE NUMBE3R OF SUB-DIVISIONS:");
 scanf("%d",&n);
 printf("\nEnter order(2 or 4):");
 scanf("%d",&order);
 R_K(a,b,y0,n,order);
 getch();
}

void R_K(float a,float b,float y0,int n,int order)
{
 float h,x,y1,y2,k=0,k1,k2,k3,k4;
 int i=0;
 h=(b-a)/n;
 x=a;
 printf("\nSOLUTION :");
 printf("\n\tXr\tK\t\tYr\t");
 printf("\n------------------------------------------------");
 printf("\nr=%d\t%g\t%g\t\t%g",i,x,k,y0);
 y1=y0;
 for(i=1;i<=n;i++)
 {
  if(order==4)
  {
  k1=h*f(x,y1);
  k2=h*f((x+(h/2)),(y1+(k1/2)));
  k3=h*f((x+(h/2)),(y1+(k2/2)));
  k4=h*f((x+h),(y1+k3));
  k=(k1+2*k2+2*k3+k4)/6;
  }
  else if(order==2)
  {
   k1=h*f(x,y1);
   k2=h*f((x+h),(y1+h));
   k=(k1+k2)/2;
  }
  y2=y1+k;
   x=a+i*h;
  printf("\nr=%d\t%g\t%f\t%g",i,x,k,y2);
  y1=y2;
 }
}

float f(float x,float y)
{
 return x-y;
}



 OUTPUT(order 2):

SOLVE :

 (dy/dx)=x-y     using the given limits.

Enter the upper limit:1

Enter the lower:0

Enter the value of y for the first interval:1

ENTER THE NUMBE3R OF SUB-DIVISIONS:10

Enter order(2 or 4):2

SOLUTION :

 Xr      K               Yr
------------------------------------------------
r=0     0       0               1
r=1     0.1     -0.100000       0.9
r=2     0.2     -0.080000       0.82
r=3     0.3     -0.062000       0.758
r=4     0.4     -0.045800       0.7122
r=5     0.5     -0.031220       0.68098
r=6     0.6     -0.018098       0.662882
r=7     0.7     -0.006288       0.656594
r=8     0.8     0.004341        0.660934
r=9     0.9     0.013907        0.674841
r=10    1       0.022516        0.697357


 OUTPUT(order 4):

SOLVE :

 (dy/dx)=x-y     using the given limits.

Enter the upper limit:1

Enter the lower:0

Enter the value of y for the first interval:1

ENTER THE NUMBE3R OF SUB-DIVISIONS:10

Enter order(2 or 4):4

SOLUTION :

 Xr      K               Yr
------------------------------------------------
r=0     0       0               1
r=1     0.1     -0.090325       0.909675
r=2     0.2     -0.0722132      0.837462
r=3     0.3     -0.055825       0.781637
r=4     0.4     -0.0409963      0.740641
r=5     0.5     -0.0275787      0.713062
r=6     0.6     -0.015438       0.697624
r=7     0.7     -0.00445263     0.693171
r=8     0.8     0.00548734      0.698659
r=9     0.9     0.0144814       0.71314
r=10    1       0.0226196       0.73576