A Semi-Analytical Framework for higher-Order Delay Differential Equations: Utilizing Optimal Auxiliary Functions (Global Journal of Sciences, 2024)
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.
Keywords: Delay differential equations; Least Square Method; Collocation Method; Optimal Auxiliary Functions Method