LAPLACE TRANSFORMATION AND MITTAG-LEFFLER FUNCTION FOR THE SOLUTION OF DAMPED OSCILLATOR EQUATION WITH FRACTIONAL ORDER
Abstract
Fractional calculus has emerged as an active area of research due to its ability to model complex dynamical systems with memory effects and anomalous diffusion. In particular, the Mittag–Leffler function plays a fundamental role in solving fractional differential equations. This study aims to derive the analytical solution of the Linear Fractionally Damped Oscillator using the Laplace transform and the Mittag–Leffler function, where the derivative is of Caputo type with order 0<α<1. We further extend the analysis to both homogeneous and nonhomogeneous models, the latter corresponding to the presence of an external forcing term. The results indicate that the oscillatory behavior exhibits algebraic decay and eventual convergence due to damping or dissipation effects. The decay rate is directly influenced by the asymptotic properties of the Mittag–Leffler function, which depend on the fractional order α. These findings provide a deeper understanding of fractional-order damped oscillatory systems and offer a more generalized framework for analyzing dissipative processes in engineering, physics, and control systems.
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References
I. Podlubny, Fractional Differential Equations: AN INTRODUCTION TO FRACTIONAL DERIVATIVES, FRACTIONAL DIFFERENTIAL EQUATIONS, TO METHODS OF THEIR SOLUTION AND SOME OF THEIR APPLICATIONS, San Diego: Academic Press, 1999.
S. Das, KINDERGARTEN OF FRACTIONAL CALCULUS, 1st ed., Newcastle upon Tyne: Cambridge Scholars Publishing, 2020.
D. V. Singh and S. Singh, "FRACTIONAL CALCULUS AND ITS APPLICATIONS: A COMPREHENSIVE REVIEW," International Journal on Science and Technology (IJSAT), vol. 16, no. 1, 2025.doi : https://doi.org/10.71097/IJSAT.v16.i1.2012
C. Milici, G. Drăgănescu and J. T. Machado, INTRODUCTION TO FRACTIONAL DIFFERENTIAL EQUATIONS (NONLINEAR SYSTEMS AND COMPLEXITY), Switzerland: Springer, 2019. doi : https://doi.org/10.1007/978-3-030-00895-6
Y. Yang and H. H. Zhang, "FRACTIONAL CALCULUS WITH ITS APPLICATIONS IN ENGINEERING AND TECHNOLOGY," in Chapter Introduction, Springer International Publishing, 2019. doi : https://doi.org/10.1007/978-3-031-79625-8_1
A. Kochubei and Y. Luchko, "VOLUME 1: BASIC THEORY," in Handbook of Fractional Calculus with Applications, Berlin, De Gruyter, 2019. doi : https://doi.org/10.1515/9783110571622
D. Kumar, "CHAPTER 8 - SPECIAL FUNCTIONS IN FRACTIONAL CALCULUS," IN REcent Developments in Theory and Applications of Fractional Order Systems, Cambridge, Elsevier, 2025, pp. 117-128.doi : https://doi.org/10.1016/B978-0-44-323952-6.00013-4
B. Sikora, "REMARKS ON THE CAPUTO FRACTIONAL DERIVATIVE," MINUT, vol. 5, pp. 76-84, 2023.
K. M. Owolabi and A. Atangana, NUMERICAL METHODS FOR FRACTIONAL DIFFERENTIATION (Springer Series in Computational Mathematics), Singapore: Springer, 2019.doi : https://doi.org/10.1007/978-981-15-0098-5
R. Gorenflo, A. A. Kilbas and F. Main, MITTAG-LEFFLER FUNCTIONS, RELATED TOPICS AND APPLICATIONS, Berlin: Springer Berlin, Heidelberg, 2020. doi : https://doi.org/10.1007/978-3-662-61550-8
K. Diethelm, THE ANALYSIS OF FRACTIONAL DIFFERENTIAL EQUATIONS: AN APPLICATION-ORIENTED EXPOSITION USING DIFFERENTIAL OPERATORS OF CAPUTO TYPE, 2nd ed., Berlin: Springer, 2010. doi : https://doi.org/10.1007/978-3-642-14574-2
F. Mainardi, "WHY THE MITTAG-LEFFLER FUNCTION CAN BE CONSIDERED THE QUEEN FUNCTION OF THE FRACTIONAL CALCULUS?," Entropy, vol. 22, no. 12, p. 1359, November 2020. doi : https://doi.org/10.3390/e22121359
G. E. H. Ali, A. A. Asaad, S. K. Elagan, E. A. Mawaheb and M. S. AlDien, "USING LAPLACE TRANSFORM METHOD FOR OBTAINING THE EXACT ANALYTIC SOLUTIONS OF SOME ORDINARY FRACTIONAL DIFFERENTIAL EQUATIONS," Global Journal of Pure and Applied Mathematics, vol. 13, no. 9, pp. 5021-5035, 2017.
D. G. Zill, A FIRST COURSE IN DIFFERENTIAL EQUATIONS WITH MODELING APPLICATIONS, INTERNATIONAL METRIC EDITION, California: Blue Kingfisher, 2023.
E.-M. Jesús, J. F. Gómez-Aguilar, C. M. Calderon-Ramon, L. Morales-Mendoza, I. Cruz-Orduña and J. Laguna-Camacho, "EXPERIMENTAL EVALUATION OF VISCOUS DAMPING COEFFICIENT IN THE FRACTIONAL UNDERDAMPED OSCILLATOR," Advances in Mechanical Engineering, vol. 8, 2016. doi : https://doi.org/10.1177/1687814016643068
M. Caputo and M. Fabrizio, "A NEW DEFINITION OF FRACTIONAL DERIVATIVE WITHOUT SINGULAR KERNEL," Progress in Fractional Differentiation and Applications, vol. 1, no. 2, pp. 73-85, 2015.
Z. Odibat and N. Shawagfeh, "GENERALIZED TAYLOR’S FORMULA," Applied Mathematics and Computation, vol. 186, pp. 286-293, 2007. doi : https://doi.org/10.1016/j.amc.2006.07.102
R. Almeida, "A CAPUTO FRACTIONAL DERIVATIVE OF A FUNCTION WITH RESPECT TO ANOTHER FUNCTION," Communications in Nonlinear Science and Numerical Simulation, vol. 44, pp. 406-481, 2017. doi : https://doi.org/10.1016/j.cnsns.2016.09.006
W. E. Boyce, R. C. Diprima and B. D. Meade, ELEMENTARY DIFFERENTIAL EQUATIONS AND BOUNDARY VALUE PROBLEMS, 11th ed., Hoboken, NJ: John Wiley & Sons, 2017.
S. Momani, S. Jain, R. Goyal and P. Agarwal, "A GENERALIZATION OF Q-MITTAG-LEFFLER FUNCTION WITH FOUR PARAMETERS," Iraqi Journal of Science, vol. 66, no. 1, pp. 239-252, 2025. doi : https://doi.org/10.24996/ijs.2025.66.1.20
H. J. Haubold, A. M. Mathai and R. K. Saena, "MITTAG-LEFFLER FUNCTIONS AND THEIR APPLICATIONS," Journal of Applied Mathematics, vol. 2011, no. 1, p. 298628, 2018. doi : https://doi.org/10.1155/2011/298628
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