MATLAB Implementation of Fourth-Order Runge-Kutta Method for Numerical Integration

The Fourth-Order Runge-Kutta Method (RK4) is a widely-used numerical integration algorithm primarily applied to solve initial value problems of ordinary differential equations (ODEs). Its core principle involves using multiple weighted averages to enhance computational accuracy. Compared to simpler numerical integration methods like Euler's method, RK4 delivers superior precision and stability. Implementing the Fourth-Order Runge-Kutta method in MATLAB typically follows these key steps: Defining the differential equation: First, clearly define the target ODE and represent it as a MATLAB function using proper function handle syntax (e.g., `f = @(t,y) expression`). Parameter initialization: Set integration parameters including step size (h), initial conditions (t0, y0), and integration interval. These parameters directly impact both solution accuracy and computational load. Iterative computation: The algorithm calculates four intermediate slopes (k1, k2, k3, k4) at each step using specific weighted combinations. The solution updates through the formula: y_{n+1} = y_n + (h/6)*(k1 + 2k2 + 2k3 + k4), where each k-value represents differently weighted derivative evaluations. Result output: The final numerical solution can be visualized through plotting functions like `plot()` or exported as numerical tables using `table()` or matrix operations for further analysis and validation. The key advantage of RK4 lies in its high precision (with local truncation error of O(h^5)), making it suitable for most engineering and scientific computing scenarios. In MATLAB implementation, programmers can optimize computational efficiency through loop structures or vectorized programming techniques, particularly beneficial for large-scale numerical simulations. The algorithm's stability and accuracy make it preferable for systems requiring reliable long-term integration.