Runge-Kutta 4th Order Method to Solve Differential Equation

09 Jun, 2022
Given following inputs, 
An ordinary differential equation that defines value of dy/dx in the form x and y.
Initial value of y, i.e., y(0)
Thus we are given below.The task is to find value of unknown function y at a given point x.The Runge-Kutta method finds approximate value of y for a given x. Only first order ordinary differential equations can be solved by using the Runge Kutta 4th order method.Below is the formula used to compute next value yn+1 from previous value yn. The value of n are 0, 1, 2, 3, ....(x – x0)/h. Here h is step height and xn+1 = x0 + h. Lower step size means more accuracy.
The formula basically computes next value yn+1 using current yn plus weighted average of four increments. 
k1 is the increment based on the slope at the beginning of the interval, using y
k2 is the increment based on the slope at the midpoint of the interval, using y + hk1/2.
k3 is again the increment based on the slope at the midpoint, using  y + hk2/2.
k4 is the increment based on the slope at the end of the interval, using y + hk3.
The method is a fourth-order method, meaning that the local truncation error is on the order of O(h5), while the total accumulated error is order O(h4).Source: https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods
Below is implementation for the above formula. 
C++
C
Java
Python3
C#
PHP
Javascript
// C++ program of th above approach#include <bits/stdc++.h>using namespace std; // A sample differential equation "dy/dx = (x - y)/2"float dydx(float x, float y){    return((x - y)/2);}  // Finds value of y for a given x using step size h// and initial value y0 at x0.float rungeKutta(float x0, float y0, float x, float h){    // Count number of iterations using step size or    // step height h    int n = (int)((x - x0) / h);      float k1, k2, k3, k4, k5;      // Iterate for number of iterations    float y = y0;    for (int i=1; i<=n; i++)    {        // Apply Runge Kutta Formulas to find        // next value of y        k1 = h*dydx(x0, y);        k2 = h*dydx(x0 + 0.5*h, y + 0.5*k1);        k3 = h*dydx(x0 + 0.5*h, y + 0.5*k2);        k4 = h*dydx(x0 + h, y + k3);          // Update next value of y        y = y + (1.0/6.0)*(k1 + 2*k2 + 2*k3 + k4);;          // Update next value of x        x0 = x0 + h;    }      return y;} // Driver Codeint main(){    float x0 = 0, y = 1, x = 2, h = 0.2;    cout << "The value of y at x is : " <<            rungeKutta(x0, y, x, h);     return 0;} // This code is contributed by code_hunt.
// C program to implement Runge Kutta method#include<stdio.h> // A sample differential equation "dy/dx = (x - y)/2"float dydx(float x, float y){    return((x - y)/2);} // Finds value of y for a given x using step size h// and initial value y0 at x0.float rungeKutta(float x0, float y0, float x, float h){    // Count number of iterations using step size or    // step height h    int n = (int)((x - x0) / h);     float k1, k2, k3, k4, k5;     // Iterate for number of iterations    float y = y0;    for (int i=1; i<=n; i++)    {        // Apply Runge Kutta Formulas to find        // next value of y        k1 = h*dydx(x0, y);        k2 = h*dydx(x0 + 0.5*h, y + 0.5*k1);        k3 = h*dydx(x0 + 0.5*h, y + 0.5*k2);        k4 = h*dydx(x0 + h, y + k3);         // Update next value of y        y = y + (1.0/6.0)*(k1 + 2*k2 + 2*k3 + k4);;         // Update next value of x        x0 = x0 + h;    }     return y;} // Driver methodint main(){    float x0 = 0, y = 1, x = 2, h = 0.2;    printf("\nThe value of y at x is : %f",            rungeKutta(x0, y, x, h));    return 0;}
// Java program to implement Runge Kutta methodimport java.io.*;class differential{    double dydx(double x, double y)    {        return ((x - y) / 2);    }         // Finds value of y for a given x using step size h    // and initial value y0 at x0.    double rungeKutta(double x0, double y0, double x, double h)    {        differential d1 = new differential();        // Count number of iterations using step size or        // step height h        int n = (int)((x - x0) / h);         double k1, k2, k3, k4, k5;         // Iterate for number of iterations        double y = y0;        for (int i = 1; i <= n; i++)        {            // Apply Runge Kutta Formulas to find            // next value of y            k1 = h * (d1.dydx(x0, y));            k2 = h * (d1.dydx(x0 + 0.5 * h, y + 0.5 * k1));            k3 = h * (d1.dydx(x0 + 0.5 * h, y + 0.5 * k2));            k4 = h * (d1.dydx(x0 + h, y + k3));             // Update next value of y            y = y + (1.0 / 6.0) * (k1 + 2 * k2 + 2 * k3 + k4);                         // Update next value of x            x0 = x0 + h;        }        return y;    }         public static void main(String args[])    {        differential d2 = new differential();        double x0 = 0, y = 1, x = 2, h = 0.2;                 System.out.println("\nThe value of y at x is : "                    + d2.rungeKutta(x0, y, x, h));    }} // This code is contributed by Prateek Bhindwar
# Python program to implement Runge Kutta method# A sample differential equation "dy / dx = (x - y)/2"def dydx(x, y):    return ((x - y)/2) # Finds value of y for a given x using step size h# and initial value y0 at x0.def rungeKutta(x0, y0, x, h):    # Count number of iterations using step size or    # step height h    n = (int)((x - x0)/h)    # Iterate for number of iterations    y = y0    for i in range(1, n + 1):        "Apply Runge Kutta Formulas to find next value of y"        k1 = h * dydx(x0, y)        k2 = h * dydx(x0 + 0.5 * h, y + 0.5 * k1)        k3 = h * dydx(x0 + 0.5 * h, y + 0.5 * k2)        k4 = h * dydx(x0 + h, y + k3)         # Update next value of y        y = y + (1.0 / 6.0)*(k1 + 2 * k2 + 2 * k3 + k4)         # Update next value of x        x0 = x0 + h    return y # Driver methodx0 = 0y = 1x = 2h = 0.2print ('The value of y at x is:', rungeKutta(x0, y, x, h)) # This code is contributed by Prateek Bhindwar
// C# program to implement Runge// Kutta methodusing System; class GFG {         static double dydx(double x, double y)    {        return ((x - y) / 2);    }         // Finds value of y for a given x    // using step size h and initial    // value y0 at x0.    static double rungeKutta(double x0,                double y0, double x, double h)    {             // Count number of iterations using        // step size or step height h        int n = (int)((x - x0) / h);         double k1, k2, k3, k4;         // Iterate for number of iterations        double y = y0;                 for (int i = 1; i <= n; i++)        {                         // Apply Runge Kutta Formulas            // to find next value of y            k1 = h * (dydx(x0, y));                         k2 = h * (dydx(x0 + 0.5 * h,                             y + 0.5 * k1));                                          k3 = h * (dydx(x0 + 0.5 * h,                            y + 0.5 * k2));                                         k4 = h * (dydx(x0 + h, y + k3));             // Update next value of y            y = y + (1.0 / 6.0) * (k1 + 2                       * k2 + 2 * k3 + k4);                         // Update next value of x            x0 = x0 + h;        }                 return y;    }         // Driver code    public static void Main()    {                 double x0 = 0, y = 1, x = 2, h = 0.2;                 Console.WriteLine("\nThe value of y"                             + " at x is : "                 + rungeKutta(x0, y, x, h));    }} // This code is contributed by Sam007.
<?php// PHP program to implement// Runge Kutta method // A sample differential equation// "dy/dx = (x - y)/2"function dydx($x, $y){    return(($x - $y) / 2);} // Finds value of y for a// given x using step size h// and initial value y0 at x0.function rungeKutta($x0, $y0, $x, $h){         // Count number of iterations    // using step size or step    // height h    $n = (($x - $x0) / $h);     $k1; $k2; $k3; $k4; $k5;     // Iterate for number    // of iterations    $y = $y0;    for($i = 1; $i <= $n; $i++)    {                 // Apply Runge Kutta        // Formulas to find        // next value of y        $k1 = $h * dydx($x0, $y);        $k2 = $h * dydx($x0 + 0.5 * $h,                        $y + 0.5 * $k1);        $k3 = $h * dydx($x0 + 0.5 * $h,                        $y + 0.5 * $k2);        $k4 = $h * dydx($x0 + $h, $y + $k3);         // Update next value of y        $y = $y + (1.0 / 6.0) * ($k1 + 2 *                    $k2 + 2 * $k3 + $k4);;         // Update next value of x        $x0 = $x0 + $h;    }     return $y;}     // Driver method    $x0 = 0;    $y = 1;    $x = 2;    $h = 0.2;    echo "The value of y at x is : ",          rungeKutta($x0, $y, $x, $h); // This code is contributed by anuj_67.?>
<script> // Javascript program to implement Runge Kutta method // A sample differential equation "dy/dx = (x - y)/2"function dydx(x, y){    return((x - y) / 2);} // Finds value of y for a given x using step size h// and initial value y0 at x0.function rungeKutta(x0, y0, x, h){         // Count number of iterations using    // step size or step height h    let n = parseInt((x - x0) / h, 10);     let k1, k2, k3, k4, k5;     // Iterate for number of iterations    let y = y0;    for(let i = 1; i <= n; i++)    {                 // Apply Runge Kutta Formulas to find        // next value of y        k1 = h * dydx(x0, y);        k2 = h * dydx(x0 + 0.5 * h, y + 0.5 * k1);        k3 = h * dydx(x0 + 0.5 * h, y + 0.5 * k2);        k4 = h * dydx(x0 + h, y + k3);         // Update next value of y        y = y + (1 / 6) * (k1 + 2 * k2 +                            2 * k3 + k4);;         // Update next value of x        x0 = x0 + h;    }    return y.toFixed(6);} // Driver codelet x0 = 0, y = 1, x = 2, h = 0.2; document.write("The value of y at x is : " +               rungeKutta(x0, y, x, h));                   // This code is contributed by divyesh072019 </script>
Output: 
The value of y at x is : 1.103639
Time Complexity of above solution is O(n) where n is (x-x0)/h.Some useful resources for detailed examples and more explanation. http://w3.gazi.edu.tr/~balbasi/mws_gen_ode_txt_runge4th.pdf https://www.youtube.com/watch?v=kUcc8vAgoQ0  
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