Fourth-Order Runge-Kutta Method for Numerical Solution of First-Order Differential Equations

The fourth-order Runge-Kutta method is a numerical technique for solving first-order differential equations. It allows customization of iteration steps and precision thresholds, making it particularly suitable for beginners in computational methods. This method extends the Taylor series approach by calculating a set of intermediate approximations to progressively converge toward the true solution. Specifically, it employs four distinct slope calculations (k1-k4) at each step to determine the next function value with enhanced accuracy. Key implementation steps include: 1. Calculating the initial slope (k1) at the current point 2. Estimating slopes at intermediate points (k2, k3) using weighted averages 3. Computing the final slope (k4) to combine all estimates 4. Updating the solution using a weighted combination of all four slopes The method's advantages include high-order accuracy (O(h⁴)) and excellent numerical stability, making it widely practical for scientific computing. For computational method learners, mastering the fourth-order Runge-Kutta algorithm represents a crucial milestone in understanding numerical differential equation solvers.