Abstract
In this work, the study of the fractional behavior of the Bateman–Feshbach–Tikochinsky and Caldirola–Kanai oscillators by using different fractional derivatives is presented. We obtained the Euler–Lagrange and the Hamiltonian formalisms in order to represent the dynamic models based on the Liouville–Caputo, Caputo–Fabrizio–Caputo and the new fractional derivative based on the Mittag–Leffler kernel with arbitrary order α. Simulation results are presented in order to show the fractional behavior of the oscillators, and the classical behavior is recovered when α is equal to 1.
Highlights
Several phenomenological models of dissipative systems have been proposed, such as the Bateman–Feshbach–Tikochinsky (BFT) or Caldirola–Kanai (CK) oscillators, the first model consists of a damped and an amplified oscillator, and this one-dimensional system exhibits an exponentially increasing mass with a Lagrangian given by Bateman [1–5]
We obtain alternative representations of the BFT and CK oscillators by using the Liouville–Caputo, Caputo–Fabrizio–Caputo and the new fractional derivative based in Mittag–Leffler kernel with arbitrary order α
For the fractional derivative based on the Mittag–Leffler kernel, we used the numerical approximation scheme developed in [20]
Summary
Several phenomenological models of dissipative systems have been proposed, such as the Bateman–Feshbach–Tikochinsky (BFT) or Caldirola–Kanai (CK) oscillators, the first model consists of a damped and an amplified oscillator, and this one-dimensional system exhibits an exponentially increasing mass with a Lagrangian given by Bateman [1–5]. Both quantum damped oscillators have been studied as a model to understand dissipation in quantum theory [6]. We obtain alternative representations of the BFT and CK oscillators by using the Liouville–Caputo, Caputo–Fabrizio–Caputo and the new fractional derivative based in Mittag–Leffler kernel with arbitrary order α.
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