Solutions behavior of mechanical oscillator equations with impulsive effects under Power Caputo fractional operator and its symmetric cases.

This study investigates the qualitative behavior of mechanical oscillator equations with impulsive effects using the generalized Power Caputo fractional operator, encompassing several known derivatives (such as Caputo-Fabrizio and Atangana-Baleanu) as special cases. The operator's parameter 'p' offers enhanced flexibility in modeling memory effects. Key findings, derived through integral equation formulations and fixed-point theory (including Banach's contraction principle), establish rigorous conditions for the existence, uniqueness, and Ulam-Hyers stability of solutions under specific assumptions on the nonlinear and impulsive terms. The numerical scheme is developed by the Lagrange interpolation polynomial to obtain approximate solutions, and the analysis of symmetric cases connects the model to established fractional derivatives. This work offers a rigorous mathematical framework and numerical tools for analyzing systems, like mechanical oscillators, that exhibit both fractional-order memory and impulsive behavior, significantly enhancing modeling accuracy in engineering and control systems.