2026-10-15T11:46:06Z https://data.mendeley.com/oai

oai:data.mendeley.com/yfyj89tkwf.1 2026-09-24T23:09:32Z

Laniel Sousa Gonçalves , Thalisson Dynamics of Systems with Memory: A Fractional Calculus Approach to the Harmonic Oscillator Mendeley Data Fractional calculus has emerged as a fundamental mathematical tool for modeling complex physical systems in which the instantaneous rate of change depends not only on the current state but also on the entire temporal history (memory effect). This work presents a pedagogical yet mathematically rigorous approach to Fractional Differential Equations (FDEs), focusing on the Caputo fractional derivative. By extending the classical second-order differential equation of the harmonic oscillator to a non-integer order ($1 < \alpha \le 2$), the paper formally demonstrates how the non-local nature of the fractional operator introduces damping and intrinsic energy dissipation. The main result lies in demonstrating that this damping occurs without the need to add explicit phenomenological friction terms proportional to velocity. Using the Laplace transform, we derive the exact analytical solution of the system, expressed in terms of Mittag-Leffler functions. Mathematics Dataset doi:10.17632/yfyj89tkwf.1 oai:data.mendeley.com/yfyj89tkwf.1 Creative Commons Attribution 4.0 International http://creativecommons.org/licenses/by/4.0 https://data.mendeley.com/datasets/yfyj89tkwf 2026-09-24T23:09:32Z