Implementing Fourth-Order Runge-Kutta Method for Delay Differential Equations in MATLAB

In MATLAB, the fourth-order Runge-Kutta method can be employed to numerically solve delay differential equations (DDEs). This widely-used numerical analysis technique provides an effective approach for understanding differential equation solutions computationally. The implementation requires discretizing the time-delay equations first, followed by applying the Runge-Kutta algorithm for iterative solving. The key implementation steps involve: defining the derivative function that accounts for delayed states, using MATLAB's built-in functions or custom implementations to handle historical values, and applying the four-stage Runge-Kutta calculation (k1, k2, k3, k4 coefficients) at each time step. A typical MATLAB implementation would utilize array indexing to manage delayed variables and employ vectorization for efficient computation. This method yields highly accurate numerical solutions and enhances the understanding of the underlying physical phenomena described by the equations. The fourth-order Runge-Kutta method serves as a powerful numerical tool with applications across various scientific and engineering domains, particularly useful for systems where current rates of change depend on past states.