Gill’s 4th Order Method to solve Differential Equations

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Given the following inputs:

An ordinary differential equation that defines the value of dy/dx in the form x and y.
$\LARGE \frac{dy}{dx} = f(x, y)$
Initial value of y, i.e., y(0).
$\LARGE y(0)= y_o$

The task is to find the value of the unknown function y at a given point x, i.e. y(x).
Examples:

Input: x0 = 0, y = 3.0, x = 5.0, h = 0.2
Output: y(x) = 3.410426
Input: x0 = 0, y = 1, x = 3, h = 0.3
Output: y(x) = 1.669395

Approach:
Gill’s method is used to find an approximate value of y for a given x. Below is the formula used to compute the next value y_n+1 from the previous value y_n.
Therefore:

y_n+1 = value of y at (x = n + 1)
y_n = value of y at (x = n)
where
  0 ? n ? (x - x₀)/h
  h is step height
  x_n+1 = x₀ + h

The essential formula to compute the value of y(n+1):
$\LARGE K_{1} = h*f(x_{n}, y_{n})$
$\LARGE K_{2} = h*f(x_n + \frac{1}{2}h, y_n + \frac{1}{2}k_1)$
$\LARGE K_{3} = h*f[x_n + \frac{1}{2}h, y_n + \frac{1}{2}(-1 + \sqrt{2})k_1 + (1 - \frac{1}{2}\sqrt{2})k_2]$
$\LARGE K_{4} = h*f[x_n + h, y_n - \frac{1}{2}\sqrt{2}k_2 + (1 + \frac{1}{2}\sqrt{2})k_3]$
$\LARGE y_{n+1} = y_{n} + \frac{1}/{6}[K_1 + (2 - \sqrt{2})K_{2} + (2 + \sqrt{2})K_3 + K_4 ] + Error Terms$
The formula basically computes the next value y_n+1 using current y_n:

K₁ is the increment based on the slope at the beginning of the interval, using y.
K₂ is the increment based on the slope, using
$\LARGE (y + \frac{1}{2}(K_1*h))$
K₃ is the increment based on the slope, using
$\LARGE (y + \frac{1}{2}(-1 + \sqrt{2})K_1 + (1 - \frac{1}{2}\sqrt{2})K_2)$
K₄ is the increment based on the slope, using
$(y - \frac{1}{2}\sqrt{2}K_2 + (1 + \frac{1}{2}\sqrt{2})K_3)$

The method is a fourth-order method, meaning that the local truncation error is of the order of O(h⁵).
Below is the implementation of the above approach:

C++

// C++ program to implement Gill's method
 
#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 Gill(float x0, float y0,
        float x, float h)
{
    // Count number of iterations
    // using step size or height h
    int n = (int)((x - x0) / h);
 
    // Value of K_i
    float k1, k2, k3, k4;
 
    // Initial value of y(0)
    float y = y0;
 
    // Iterate for number of iteration
    for (int i = 1; i <= n; i++) {
 
        // Apply Gill's Formulas to
        // find next value of y
 
        // Value of K1
        k1 = h * dydx(x0, y);
 
        // Value of K2
        k2 = h * dydx(x0 + 0.5 * h,
                    y + 0.5 * k1);
 
        // Value of K3
        k3 = h * dydx(x0 + 0.5 * h,
                    y + 0.5 * (-1 + sqrt(2)) * k1
                        + k2 * (1 - 0.5 * sqrt(2)));
 
        // Value of K4
        k4 = h * dydx(x0 + h,
                    y - (0.5 * sqrt(2)) * k2
                        + k3 * (1 + 0.5 * sqrt(2)));
 
        // Find the next value of y(n+1)
        // using y(n) and values of K in
        // the above steps
        y = y + (1.0 / 6) * (k1 + (2 - sqrt(2)) * k2 
                            + (2 + sqrt(2)) * k3 + k4);
 
        // Update next value of x
        x0 = x0 + h;
    }
 
    // Return the final value of dy/dx
    return y;
}
 
// Driver Code
int main()
{
    float x0 = 0, y = 3.0,
        x = 5.0, h = 0.2;
 
    printf("y(x) = %.6f",
        Gill(x0, y, x, h));
    return 0;
}

Java

// Java program to implement Gill's method
class GFG{
 
// A sample differential equation
// "dy/dx = (x - y)/2"
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 Gill(double x0, double y0,
                   double x, double h)
{
     
    // Count number of iterations
    // using step size or height h
    int n = (int)((x - x0) / h);
 
    // Value of K_i
    double k1, k2, k3, k4;
 
    // Initial value of y(0)
    double y = y0;
 
    // Iterate for number of iteration
    for(int i = 1; i <= n; i++)
    {
         
       // Apply Gill's Formulas to
       // find next value of y
        
       // Value of K1
       k1 = h * dydx(x0, y);
        
       // Value of K2
       k2 = h * dydx(x0 + 0.5 * h,
                      y + 0.5 * k1);
        
       // Value of K3
       k3 = h * dydx(x0 + 0.5 * h,
                      y + 0.5 * (-1 + Math.sqrt(2)) *
                     k1 + k2 * (1 - 0.5 * Math.sqrt(2)));
        
       // Value of K4
       k4 = h * dydx(x0 + h,
                      y - (0.5 * Math.sqrt(2)) *
                     k2 + k3 * (1 + 0.5 * Math.sqrt(2)));
        
       // Find the next value of y(n+1)
       // using y(n) and values of K in
       // the above steps
       y = y + (1.0 / 6) * (k1 + (2 - Math.sqrt(2)) * 
                            k2 + (2 + Math.sqrt(2)) *
                            k3 + k4);
        
       // Update next value of x
       x0 = x0 + h;
    }
 
    // Return the final value of dy/dx
    return y;
}
 
// Driver Code
public static void main(String[] args)
{
    double x0 = 0, y = 3.0,
            x = 5.0, h = 0.2;
 
    System.out.printf("y(x) = %.6f", Gill(x0, y, x, h));
}
}
 
// This code is contributed by Amit Katiyar

Python3

# Python3 program to implement Gill's method
from math import sqrt
 
# 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 Gill(x0, y0, x, h):
     
    # Count number of iterations
    # using step size or height h
    n = ((x - x0) / h)
 
    # Initial value of y(0)
    y = y0
 
    # Iterate for number of iteration
    for i in range(1, int(n + 1), 1):
         
        # Apply Gill's Formulas to
        # find next value of y
 
        # Value of K1
        k1 = h * dydx(x0, y)
 
        # Value of K2
        k2 = h * dydx(x0 + 0.5 * h, 
                       y + 0.5 * k1)
 
        # Value of K3
        k3 = h * dydx(x0 + 0.5 * h,
                       y + 0.5 * (-1 + sqrt(2)) *
                      k1 + k2 * (1 - 0.5 * sqrt(2)))
 
        # Value of K4
        k4 = h * dydx(x0 + h, y - (0.5 * sqrt(2)) *
                    k2 + k3 * (1 + 0.5 * sqrt(2)))
 
        # Find the next value of y(n+1)
        # using y(n) and values of K in
        # the above steps
        y = y + (1 / 6) * (k1 + (2 - sqrt(2)) *
                           k2 + (2 + sqrt(2)) *
                           k3 + k4)
 
        # Update next value of x
        x0 = x0 + h
 
    # Return the final value of dy/dx
    return y
 
# Driver Code
if __name__ == '__main__':
     
    x0 = 0
    y = 3.0
    x = 5.0
    h = 0.2
 
    print("y(x) =", round(Gill(x0, y, x, h), 6))
 
# This code is contributed by Surendra_Gangwar

C#

// C# program to implement Gill's method
using System;
 
class GFG{
  
// A sample differential equation
// "dy/dx = (x - y)/2"
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 Gill(double x0, double y0,
                   double x, double h)
{
      
    // Count number of iterations
    // using step size or height h
    int n = (int)((x - x0) / h);
  
    // Value of K_i
    double k1, k2, k3, k4;
  
    // Initial value of y(0)
    double y = y0;
  
    // Iterate for number of iteration
    for(int i = 1; i <= n; i++)
    {
          
       // Apply Gill's Formulas to
       // find next value of y
         
       // Value of K1
       k1 = h * dydx(x0, y);
         
       // Value of K2
       k2 = h * dydx(x0 + 0.5 * h,
                      y + 0.5 * k1);
         
       // Value of K3
       k3 = h * dydx(x0 + 0.5 * h,
                      y + 0.5 * (-1 + Math.Sqrt(2)) *
                     k1 + k2 * (1 - 0.5 * Math.Sqrt(2)));
         
       // Value of K4
       k4 = h * dydx(x0 + h,
                      y - (0.5 * Math.Sqrt(2)) *
                     k2 + k3 * (1 + 0.5 * Math.Sqrt(2)));
         
       // Find the next value of y(n+1)
       // using y(n) and values of K in
       // the above steps
       y = y + (1.0 / 6) * (k1 + (2 - Math.Sqrt(2)) * 
                            k2 + (2 + Math.Sqrt(2)) *
                            k3 + k4);
         
       // Update next value of x
       x0 = x0 + h;
    }
  
    // Return the final value of dy/dx
    return y;
}
  
// Driver Code
public static void Main(String[] args)
{
    double x0 = 0, y = 3.0,
            x = 5.0, h = 0.2;
  
    Console.Write("y(x) = {0:F6}", Gill(x0, y, x, h));
}
}
 
// This code is contributed by Amit Katiyar

Javascript

<script>
 
// Javascript program to implement Gill's 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 Gill(x0, y0, x, h)
{
       
    // Count number of iterations
    // using step size or height h
    let n = ((x - x0) / h);
   
    // Value of K_i
    let k1, k2, k3, k4;
   
    // Initial value of y(0)
    let y = y0;
   
    // Iterate for number of iteration
    for(let i = 1; i <= n; i++)
    {
           
       // Apply Gill's Formulas to
       // find next value of y
          
       // Value of K1
       k1 = h * dydx(x0, y);
          
       // Value of K2
       k2 = h * dydx(x0 + 0.5 * h,
                      y + 0.5 * k1);
          
       // Value of K3
       k3 = h * dydx(x0 + 0.5 * h,
                      y + 0.5 * (-1 + Math.sqrt(2)) *
                     k1 + k2 * (1 - 0.5 * Math.sqrt(2)));
          
       // Value of K4
       k4 = h * dydx(x0 + h,
                      y - (0.5 * Math.sqrt(2)) *
                     k2 + k3 * (1 + 0.5 * Math.sqrt(2)));
          
       // Find the next value of y(n+1)
       // using y(n) and values of K in
       // the above steps
       y = y + (1.0 / 6) * (k1 + (2 - Math.sqrt(2)) * 
                            k2 + (2 + Math.sqrt(2)) *
                            k3 + k4);
          
       // Update next value of x
       x0 = x0 + h;
    }
   
    // Return the final value of dy/dx
    return y;
}
 
// Driver Code
     
    let x0 = 0, y = 3.0,
            x = 5.0, h = 0.2;
   
    document.write("y(x) = ", Gill(x0, y, x, h).toFixed(6));
         
</script>

Output:

y(x) = 3.410426

Time Complexity: O(n^3/2)

Auxiliary Space: O(1)

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