Ordinary Differential Equation with Left and Right Fractional Derivatives and Modeling of Oscillatory Systems

Liana Eneeva, Arsen Pskhu and Sergo Rekhviashvili
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Liana Eneeva: Institute of Applied Mathematics and Automation, Kabardino-Balkarian Scientific Center of Russian Academy of Sciences, 89-A Shortanov Street, 360000 Nalchik, Russia
Arsen Pskhu: Institute of Applied Mathematics and Automation, Kabardino-Balkarian Scientific Center of Russian Academy of Sciences, 89-A Shortanov Street, 360000 Nalchik, Russia
Sergo Rekhviashvili: Institute of Applied Mathematics and Automation, Kabardino-Balkarian Scientific Center of Russian Academy of Sciences, 89-A Shortanov Street, 360000 Nalchik, Russia

Mathematics, 2020, vol. 8, issue 12, 1-7

Abstract: We consider the principle of least action in the context of fractional calculus. Namely, we derive the fractional Euler–Lagrange equation and the general equation of motion with the composition of the left and right fractional derivatives defined on infinite intervals. In addition, we construct an explicit representation of solutions to a model fractional oscillator equation containing the left and right Gerasimov–Caputo fractional derivatives with origins at plus and minus infinity. We derive a representation for the composition of the left and right derivatives with origins at plus and minus infinity in terms of the Riesz potential, and introduce special functions with which we give solutions to the model fractional oscillator equation with a complex coefficient. This approach can be useful for describing dissipative dynamical systems with the property of heredity.

Keywords: principle of least action; fractional oscillator; equation with left and right fractional derivative; Gerasimov–Caputo fractional derivative; Riesz potential (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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