Lab 6: Solution of ODEs using Runge-Kutta 4th Order Method

import numpy as np
import matplotlib.pyplot as plt

def f(x, y):
    """Differential equation: dy/dx = f(x,y)"""
    return x**2 - y  # Example function

def rk4(f, x0, y0, xn, n):
    """
    Runge-Kutta 4th order method
    f: function dy/dx = f(x,y)
    x0: initial x value
    y0: initial y value at x0
    xn: final x value
    n: number of steps
    Returns array of y values
    """
    h = (xn - x0) / n
    x = np.linspace(x0, xn, n+1)
    y = np.zeros(n+1)
    y[0] = y0
    
    for i in range(n):
        k1 = h * f(x[i], y[i])
        k2 = h * f(x[i] + h/2, y[i] + k1/2)
        k3 = h * f(x[i] + h/2, y[i] + k2/2)
        k4 = h * f(x[i] + h, y[i] + k3)
        
        y[i+1] = y[i] + (k1 + 2*k2 + 2*k3 + k4)/6
    
    return x, y

# Example usage
x0, y0 = 0, 1  # Initial condition
xn = 2.0       # Final x value
n = 10         # Number of steps

x, y = rk4(f, x0, y0, xn, n)

# Print results
print("Solution using RK4 method:")
for xi, yi in zip(x, y):
    print(f"x = {xi:.4f}, y = {yi:.6f}")

# Plot solution
plt.plot(x, y, 'b-', label='RK4 Solution')
plt.xlabel('x')
plt.ylabel('y')
plt.title('Solution of ODE using RK4 Method')
plt.grid(True)
plt.legend()
plt.show()
          

import numpy as np
import matplotlib.pyplot as plt

def rk4_dataset(f, x_data, y0):
    """
    Runge-Kutta 4th order for given x data points
    f: function dy/dx = f(x,y)
    x_data: array of x values (can be non-uniform)
    y0: initial y value at x_data[0]
    Returns array of y values
    """
    n = len(x_data) - 1
    y = np.zeros(n+1)
    y[0] = y0
    
    for i in range(n):
        h = x_data[i+1] - x_data[i]
        k1 = h * f(x_data[i], y[i])
        k2 = h * f(x_data[i] + h/2, y[i] + k1/2)
        k3 = h * f(x_data[i] + h/2, y[i] + k2/2)
        k4 = h * f(x_data[i] + h, y[i] + k3)
        
        y[i+1] = y[i] + (k1 + 2*k2 + 2*k3 + k4)/6
    
    return y

# Example usage with non-uniform x data
def f(x, y):
    return np.sin(x) - y  # Example function

# Given x data points (can be non-uniform)
x_data = np.array([0, 0.5, 1.2, 1.8, 2.5, 3.0])
y0 = 0  # Initial condition

y = rk4_dataset(f, x_data, y0)

# Print results
print("Solution using RK4 method for given data points:")
for xi, yi in zip(x_data, y):
    print(f"x = {xi:.4f}, y = {yi:.6f}")

# Plot solution
plt.plot(x_data, y, 'ro-', label='RK4 Solution')
plt.xlabel('x')
plt.ylabel('y')
plt.title('Solution of ODE at Given Points using RK4')
plt.grid(True)
plt.legend()
plt.show()