Two-Tank Mixing Problem C++ Implementation Using Runge-Kutta 4 for a System of Two Ordinary Differential Equations Implementation by James Pate Williams, Jr.

// Two-tank Mixing Problem
// x'(t) = f(t, x, y)
// y'(t) = g(t, x, y)
// x(0) = 5, y(0) = 1
// See “Ordinary Differential Equations
// from Calculus to Dynamical Systems”
// by Virginia W. Noonburg for the exact
// solution, view pages 165 – 167. Also
// see “Elementary Numerical Analysis:
// An Algorithmic Approach Third
// Edition” by S. D. Conte and Carl de
// Boor pages 398 – 491 for the Runge-
// Kutta-4 equations.

#include "RK4System.h"

void RK4System::Solve(
	real t0, real t1,
	real x0, real  z0,
	real(*f)(real, real, real),
	real(*g)(real, real, real),
	int nSteps, vector<real>& tv,
	vector<real>& xv, vector<real>& yv)
{
	real k1, k2, k3, k4, l1, l2, l3, l4;
	real h, tn, xn, yn, xn1, yn1;
	
	h = (t1 - t0) / nSteps;
	tn = t0;
	xn = x0;
	yn = z0;
	tv.push_back(tn);
	xv.push_back(xn);
	yv.push_back(yn);

	for (unsigned int n = 1; n <= nSteps; n++)
	{
		tn = t0 + n * h;
		k1 = h * f(tn, xn, yn);
		l1 = h * g(tn, xn, yn);
		k2 = h * f(tn + 0.5 * h, xn + 0.5 * k1, yn + 0.5 * l1);
		l2 = h * g(tn + 0.5 * h, xn + 0.5 * k1, yn + 0.5 * l1);
		k3 = h * f(tn + 0.5 * h, xn + 0.5 * k2, yn + 0.5 * l2);
		l3 = h * g(tn + 0.5 * h, xn + 0.5 * k2, yn + 0.5 * l2);
		k4 = h * f(tn + h, xn + k3, yn + l3);
		l4 = h * g(tn + h, xn + k3, yn + l3);
		xn1 = xn + (k1 + 2 * k2 + 2 * k3 + k4) / 6.0;
		yn1 = yn + (l1 + 2 * l2 + 2 * l3 + l4) / 6.0;
		xn = xn1;
		yn = yn1;
		tv.push_back(tn);
		xv.push_back(xn);
		yv.push_back(yn);
	}
}

#pragma once
// Solves the following system of first order
// ordinary differential equations. Formulas
// are from "Elementary Numerical Analysis:
// An Algorithmic Approach" by S. D. Conte &
// Carl de Boor (c) 1980 8.12 page 398.
// x' = f(t, x, y)
// y' = g(t, x, y)

#include <vector>
using namespace std;

typedef long double real;

class RK4System
{
public:
	static void Solve(
		real t0, real t1,
		real x0, real  y0,
		real(*f)(real, real, real),
		real(*g)(real, real, real),
		int nSteps, vector<real>& tv,
		vector<real>& xv, vector<real>& yv);
};
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Author: jamespatewilliamsjr

My whole legal name is James Pate Williams, Jr. I was born in LaGrange, Georgia approximately 70 years ago. I barely graduated from LaGrange High School with low marks in June 1971. Later in June 1979, I graduated from LaGrange College with a Bachelor of Arts in Chemistry with a little over a 3 out 4 Grade Point Average (GPA). In the Spring Quarter of 1978, I taught myself how to program a Texas Instruments desktop programmable calculator and in the Summer Quarter of 1978 I taught myself Dayton BASIC (Beginner's All-purpose Symbolic Instruction Code) on LaGrange College's Data General Eclipse minicomputer. I took courses in BASIC in the Fall Quarter of 1978 and FORTRAN IV (Formula Translator IV) in the Winter Quarter of 1979. Professor Kenneth Cooper, a genius poly-scientist taught me a course in the Intel 8085 microprocessor architecture and assembly and machine language. We would hand assemble our programs and insert the resulting machine code into our crude wooden box computer which was designed and built by Professor Cooper. From 1990 to 1994 I earned a Bachelor of Science in Computer Science from LaGrange College. I had a 4 out of 4 GPA in the period 1990 to 1994. I took courses in C, COBOL, and Pascal during my BS work. After graduating from LaGrange College a second time in May 1994, I taught myself C++. In December 1995, I started using the Internet and taught myself client-server programming. I created a website in 1997 which had C and C# implementations of algorithms from the "Handbook of Applied Cryptography" by Alfred J. Menezes, et. al., and some other cryptography and number theory textbooks and treatises.

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