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Multivariate Mittag-Leffler Solution for a Forced Fractional-Order Harmonic Oscillator

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  • Jessica Mendiola-Fuentes

    (Departamento de Ciencias Básicas e Ingenierías, Universidad del Caribe, L-1. Mz 1, Esq. Fracc. Tabachines SM 78, Cancún 77528, Quintana Roo, Mexico
    These authors contributed equally to this work.)

  • Eugenio Guerrero-Ruiz

    (Department of Mathematics, Faculty of Natural Sciences, University of Puerto Rico, Rio Piedras Campus, 17 Ave. Universidad Suite 1701, San Juan 00925-2537, Puerto Rico
    These authors contributed equally to this work.)

  • Juan Rosales-García

    (Departamento de Ingeniería Eléctrica, División de Ingenierías, Campus Irapuato-Salamanca, Universidad de Guanajuato, Carretera Salamanca-Valle de Santiago, km. 3.5 + 1.8 km, Comunidad Palo Blanco, Salamanca 36885, Guanajuato, Mexico)

Abstract

The harmonic oscillator is a fundamental physical–mathematical system that allows for the description of a variety of models in many fields of physics. Utilizing fractional derivatives instead of traditional derivatives enables the modeling of a more diverse array of behaviors. Furthermore, if the effect of the fractional derivative is applied to each of the terms of the differential equation, this will involve greater complexity in the description of the analytical solutions of the fractional differential equation. In this work, by using the Laplace method, the solutions to the multiple-term forced fractional harmonic oscillator are presented, described through multivariate Mittag-Leffler functions. Additionally, the cases of damped and undamped free fractional harmonic oscillators are addressed. Finally, through simulations, the effect of the fractional non-integer derivative is demonstrated, and the consistency of the result is verified when recovering the integer case.

Suggested Citation

  • Jessica Mendiola-Fuentes & Eugenio Guerrero-Ruiz & Juan Rosales-García, 2024. "Multivariate Mittag-Leffler Solution for a Forced Fractional-Order Harmonic Oscillator," Mathematics, MDPI, vol. 12(10), pages 1-11, May.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:10:p:1502-:d:1392784
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    References listed on IDEAS

    as
    1. Mark Naber, 2010. "Linear Fractionally Damped Oscillator," International Journal of Differential Equations, Hindawi, vol. 2010, pages 1-12, October.
    2. R. K. Saxena & S. L. Kalla, 2005. "Solution of Volterra-type integro-differential equations with a generalized Lauricella confluent hypergeometric function in the kernels," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2005, pages 1-16, January.
    3. H. J. Haubold & A. M. Mathai & R. K. Saxena, 2011. "Mittag-Leffler Functions and Their Applications," Journal of Applied Mathematics, Hindawi, vol. 2011, pages 1-51, May.
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