// declare libraries

#include <stdio.h>

#include <math.h>

#include <stdlib.h>

#include <cpgplot.h>

#define N 200

//declared functions outside main

double gamma(double p), M(double r, double m, double p), rho(double r, double m, double p);

double M2(double r, double m, double p), M3(double r, double m, double p), M4(double r, double m, double p);

double rho2(double r, double m, double p), rho3(double r, double m, double p), rho4(double r, double m, double p);

double h = 0.01; //the initail parameter h, the step size

void main() //main code

{

printf("\nRUNGE-KUTTA METHOD\n\nR M Rho\n"); //printing titles for values displayed

double c = 10.0; //the parameter rho(c)

int n; //the integer steps, n

//declare arrays

float x[N];

float y[N];

float z[N];

//r,m,p as the radius, mass and density

double r,m,p;

//declare initial conditons for arrays

x[0] = h; //first array is for r=h

y[0] = (h*h*h/3)*c; //initial conditon for scaled mass (m)

z[0] = c*(1-((h*h*c)/(6*gamma(c)))); //initial conditon for rho

//for loop for n=0,1,...,200

for(n=0;n<N;n++)

{

//declared how x(n+1) relates to x(n), y(n+1) relates to y(n), z(n+1) relates to z(n)

x[n+1] = x[n]+h;

y[n+1] = y[n]+(h/6)*(M(x[n],y[n],z[n])+2*M2(x[n],y[n],z[n])+2*M3(x[n],y[n],z[n])+M4(x[n],y[n],z[n]));

z[n+1] = z[n]+(h/6)*(rho(x[n],y[n],z[n])+2*rho2(x[n],y[n],z[n])+2*rho3(x[n],y[n],z[n])+rho4(x[n],y[n],z[n]));

if(isnan(z[n+1]))

{

break;

}

//r,m,p will be declared in pg-plot

r = x[n+1];

m = y[n+1];

p = z[n+1];

printf("%.2e %.2e %.2e\n",x[n+1],y[n+1],z[n+1]); //printed values for x and y respectively

}

//Use pg-plot to plot mass and density

// cpgbeg starts a plotting page, in this case with 2x1 panels

cpgbeg(0,"?",2,1);

// sets colour: 1-black, 2-red, 3-green, 4-blue

cpgsci(1);

// sets line style: 1-solid, 2-dashed, 3-dot-dashed, 4-dotted

cpgsls(1);

// sets charachter height, larger number = bigger

cpgsch(1.);

// cpgpage() moves to the next panel

cpgpage();

// sets the axes limits in the panel

cpgswin(0,r,0,m);

// draw the axes

cpgbox("BCNST", 0.0, 0, "BCNST", 0.0, 0);

// label the bottom axis

cpgmtxt("B",2.,.5,.5,"radius");

// label the left axis

cpgmtxt("L",2.5,.5,.5,"saclaed mass");

// connect N points in ax and ay with a line

cpgline(n,x,y);

// cpgpage() moves to the next panel

cpgpage();

// sets the axes limits in the panel

cpgswin(0,r,0,c);

// draw the axes

cpgbox("BCNST", 0.0, 0, "BCNST", 0.0, 0);

// label the bottom axis

cpgmtxt("B",2.,.5,.5,"radius");

// label the left axis

cpgmtxt("L",2.5,.5,.5,"density");

// connect N points in ax and ay with a line

cpgline(n,x,z);

// close all pgplot applications

cpgend();

// end program

return;

}

//declared the function for gamma(x) = x^2/(3*sqrt(1+x^2))

double gamma(double p)

{

double e = cbrt(p);

double x = ((e*e)/(3*(sqrt(1+e*e))));

return x;

}

//declared the function for dm/dr

double M(double r, double m, double p)

{

double b = r*r*p; //coupled equation for dm/dr

return b; //return dm/dr

}

//declared the function for d(rho)/dr

double rho(double r, double m, double p)

{

double t = -((m*p)/(gamma(p)*r*r)); //coupled equation for d(rho)/dr

return t; //return d(rho)/dr

}

//declared the function for M2 from M

double M2(double r, double m, double p)

{

double Fn = M(r+h/2,m+(h/2)*M(r,m,p),p+(h/2)*rho(r,m,p));

return Fn;

}

//declared the function for rho2 from rho

double rho2(double r, double m, double p)

{

double Fn = rho(r+h/2,m+(h/2)*M(r,m,p),p+(h/2)*rho(r,m,p));

return Fn;

}

//declared the function for M3 from M2

double M3(double r, double m, double p)

{

double Fn = M(r+h/2,m+(h/2)*M2(r,m,p),p+(h/2)*rho2(r,m,p));

return Fn;

}

//declared the function for rho3 from rho2

double rho3(double r, double m, double p)

{

double Fn = rho(r+h/2,m+(h/2)*M2(r,m,p),p+(h/2)*rho2(r,m,p));

return Fn;

}

//declared the function for M4 from M3

double M4(double r, double m, double p)

{

double Fn = M(r+h,m+h*M3(r,m,p),p+h*rho3(r,m,p));

return Fn;

}

//declared the function for rho4 from rho3

double rho4(double r, double m, double p)

{

double Fn = rho(r+h,m+h*M3(r,m,p),p+h*rho3(r,m,p));

return Fn;

}