Solving Delay Differential Equations Using the Fourth-Order Runge-Kutta Method

Delay differential equation systems are widely used in engineering and mathematical biology, characterized by their dependency on past states at specific time points. The classical Runge-Kutta method requires special modifications to handle these delay characteristics effectively. The core implementation approach consists of three key phases: Historical Function Management Delay terms require storage of historical state values, typically implemented through a dynamic array or interpolation function that records the system's trajectory within the [t-τ, t] time window. In code implementation, this often involves maintaining a circular buffer data structure for efficient memory management and quick access to past values. Fourth-Order Algorithm Modification All four slope calculation stages in the standard Runge-Kutta method require additional handling of delay terms. During each slope computation: - If the delay time τ falls within the known historical interval, directly retrieve stored values from the history buffer - If τ extends beyond the current step size, employ numerical interpolation methods like Hermite interpolation for accurate prediction The algorithm modification requires implementing separate functions for delay term evaluation at each Runge-Kutta stage. Stability Control Delay systems are more sensitive to step sizes, recommending adaptive step size control strategies. The implementation should include automatic step reduction when state mutations occur, preventing divergence caused by interpolation errors in delay terms. This typically involves embedding error estimation mechanisms within the Runge-Kutta framework. Practical implementation considerations: Storage efficiency of historical data significantly impacts computational performance, where circular buffers serve as common optimization solutions. For complex systems, initial validation through simple delay equations is recommended to verify algorithm correctness before scaling to more complicated scenarios.