Ordinary Differential Equations
(Runge - Kutta Method)
Objective : - To write C++ Program to solve Ordinary Differential Equations using Runge – Kutta Method
Theory :
The Taylor’s series method of solving differential equations numerically restricted by the labour involved in finding the higher order derivatives. However these are a class of methods which do not require the calculations of higher order derivatives. These methods agree with Taylor’s series solution upto the terms in hr, where r differs from method to method and it is called the order of the method.
Euler’s Method, Modified Euler’s Method & Runge’s Methods are the Runge-Kutta Methods of the first, second and third order respectively. The Fourth Order Runge-Kutta Method is most commonly used and is often referred as ‘Runge-Kutta Method’ only.
Program:
#include<iostream.h>
#include<conio.h>
#include<math.h>
#include<iomanip.h>
float fun(float x, float y)
{
return (1+pow(y,2));
}
int main()
{
clrscr();
int n,i,j;
float h;
//Entering the Number of Iterations
cout<<"Enter the number of Iterations "<<endl;
cin>>n;
//Entering the width of step size
cout<<"Enter The Value of Step Size "<<endl;
cin>>h;
long double y[25],x[25],k[25][25];
//Entering the initial values of x & y
cout<<"Enter the value of x0 "<<endl;
cin>>x[0];
cout<<"Enter The Value of y0 "<<endl;
cin>>y[0];
//Calculating the value of x
for(i=1;i<=n;i++)
{
x[i]=x[i-1]+h;
}
cout<<"Solution of the given differential equation by Fourth Order Runga Kutta Method is "<<endl;
//Calculating & Printing the values of k1, k2, k3, k4 & y
for(j=1;j<=n;j++)
{
k[1][j]=h*fun(x[j-1],y[j-1]);
cout<<"K[1] = "<<k[1][j]<<"\t";
k[2][j]=h*fun(x[j-1]+(h/2),y[j-1]+(k[1][j]/2));
cout<<"K[2] = "<<k[2][j]<<"\t";
k[3][j]=h*fun(x[j-1]+(h/2),y[j-1]+(k[2][j]/2));
cout<<"K[3] = "<<k[3][j]<<"\t";
k[4][j]=h*fun(x[j-1]+h,y[j-1]+k[3][j]);
cout<<"K[4] = "<<k[4][j]<<"\n";
y[j]=y[j-1]+((k[1][j]+2*k[2][j]+2*k[3][j]+k[4][j])/6);
cout<<"y["<<h*j<<"] = "<<setprecision(5)<<y[j]<<endl;
}
getch();
}
Output:
