A Novel Semi-Analytical Approach for High-Order Delay Differential Equations Based on History Functions: Application to Nonlinear Vibration of Delayed Systems

Abstract

In this study, a novel semi-analytical approach based on the Perturbation-Iteration Algorithm is proposed for solving high-order delayed differential equations using history functions. By employing the method of steps to transform the delayed problem into a system of ordinary differential equations defined over sub-intervals, this approach provides a systematic solution framework distinct from existing methods in the literature. Another significant contribution of this study is the development of an algorithmic procedure for determining the initial function that starts the iteration process. By integrating the history function and continuity conditions between consecutive intervals directly into the equation through matrix operations, this procedure enables the algorithm to produce a smooth and high-precision solution within each sub-interval. The proposed method is applied to the dynamic analysis of the Delayed Mathieu and Delayed Damped Mathieu equations, which are critically important in nonlinear vibration theory. The findings show that the developed method is an effective and reliable tool for modelling and analysing complex engineering problems involving delay terms.