Runge-Kutta (RK4) Method Calculator

This calculator is based on the standard Runge-Kutta method, a reliable numerical approach for solving differential equations. All calculations strictly adhere to this methodology.

The Formula Explained

\[ y_{n+1} = y_n + \frac{h}{6} (k_1 + 2k_2 + 2k_3 + k_4) \]
where:
\[ k_1 = f(x_n, y_n) \]
\[ k_2 = f(x_n + \frac{h}{2}, y_n + \frac{h}{2}k_1) \]
\[ k_3 = f(x_n + \frac{h}{2}, y_n + \frac{h}{2}k_2) \]
\[ k_4 = f(x_n + h, y_n + hk_3) \]

Glossary of Terms

How It Works: A Step-by-Step Example

Consider solving the differential equation \( \frac{dy}{dx} = xy \) with initial conditions x₀ = 0, y₀ = 1, step size h = 0.1, up to x = 0.5.

Frequently Asked Questions (FAQ)

What is the Runge-Kutta method?

The Runge-Kutta methods are a family of iterative methods used to approximate solutions to ordinary differential equations.

How does this calculator work?

This calculator implements the 4th-order Runge-Kutta method to compute the numerical solution of differential equations.

Why use the 4th-order method?

The 4th-order Runge-Kutta method provides a good balance between complexity and accuracy, making it a popular choice for practical applications.

Where is this method used?

This method is widely used in engineering, physics, and other fields requiring numerical solutions for differential equations.

Can I use this for any differential equation?

It is best suited for ordinary differential equations (ODEs). For partial differential equations (PDEs), other methods may be more appropriate.