Solving Damped Harmonic Oscillator as a Second-Order Differential Equations and Caputo Fractional Differential Equation in Bipolar Menger Probabilistic b-Metric Spaces

In this study, we explore the newly proposed bipolar Menger probabilistic b-metric spaces and present several novel fixed-point theorems within this framework. We also provide a range of complex examples and apply our main results to the analysis of the damped harmonic oscillator, modeled by second-order differential equations. Furthermore, we demonstrate the applicability of our theoretical results to significant problem: Caputo fractional differential equations with integral boundary conditions. The proposed methods and results contribute to the broader understanding of probabilistic metric spaces and their utility in advanced mathematical modeling and analysis.