A Semi-Analytical Framework for higher-Order Delay Differential Equations: Utilizing Optimal Auxiliary Functions

Delay differential equations (DDEs) are extensively utilized in fields such as control systems, biology, and engineering to model processes where current states depend on past states, effectively accounting for time lags. Key applications include population dynamics, epidemic modeling, and economic systems, where delayed responses significantly influence system behavior. This paper presents the first extension of the Optimal Auxiliary Functions Method (OAFM) to second-order and third-order DDEs. The strength of this method lies in its convergence control parameters and auxiliary functions. Notably, the OAFM guarantees the convergence of approximate solutions after just one iteration, without requiring assumptions about small or large parameters. The method demonstrates both effectiveness and efficiency, with its accuracy validated through graphical and numerical results. Additionally, the results obtained are compared with those from the least squares method. Auxiliary functions and convergence control parameters are employed to further manage the convergence of the OAFM.