Editing Equations, Formulas and Algorithms (section)

== Runge-Kutta Methods ==
Runge-Kutta methods are numerical techniques for solving ordinary differential equations. They involve iterative procedures to approximate the solution trajectory of the differential equation over a specified time interval. Runge-Kutta methods are widely used in numerical simulations and scientific computing.

'''Example: Runge-Kutta Method'''
Consider the following ordinary differential equation:
<math mode="display"> \frac{dy}{dt} = f(t, y) </math>
where <math> y(t) </math> is the solution and <math> f(t, y) </math> is the derivative function.

'''Source Code Example (Python):'''
<syntaxhighlight lang="python">
import numpy as np

# Define derivative function
def f(t, y):
    return -y

# Define initial conditions
y0 = 1
t0 = 0
t_end = 10
dt = 0.1

# Initialize arrays to store solution
t_values = [t0]
y_values = [y0]

# Runge-Kutta integration loop
while t_values[-1] < t_end:
    t = t_values[-1]
    y = y_values[-1]
    k1 = f(t, y)
    k2 = f(t + dt/2, y + dt/2 * k1)
    k3 = f(t + dt/2, y + dt/2 * k2)
    k4 = f(t + dt, y + dt * k3)
    y_new = y + dt / 6 * (k1 + 2*k2 + 2*k3 + k4)
    t_values.append(t + dt)
    y_values.append(y_new)

# Plot solution
import matplotlib.pyplot as plt
plt.plot(t_values, y_values)
plt.xlabel('Time')
plt.ylabel('Solution')
plt.title('Runge-Kutta Method')
plt.show()
</syntaxhighlight>