On fallacies in the decision between the Caputo and Riemann–Liouville fractional derivatives for the analysis of the dynamic response of a nonlinear viscoelastic oscillator |
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| Institution: | 1. Departamento de Fisica, Universidade Federal do Amazonas, 69077-000 Manaus, Brazil;2. Institut für Theoretische Physik, Technische Universität Dresden, 01062 Dresden, Germany;3. Theoretische Polymerphysik, Universität Freiburg, Hermann-Herder-Str. 3, 79104 Freiburg, Germany;1. Department of Mathematics, Lehigh University, Bethlehem, PA 18015, USA;2. Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA;3. Division of Applied Mathematics, Brown University, Providence, RI 02912, USA;1. Applied and Computational Mathematics Division, Beijing Computational Science Research Center, Beijing 100193, China;2. School of Mathematical Sciences, University College Cork, Ireland |
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| Abstract: | Recently Dal Dal, F., 2011. Multiple time scale solution of an equation with quadratic and cubic nonlinearities having fractional-order derivative. Mathematical and Computational Applications 16 (1), 301–308] presented ‘a new analytical scheme’ to calculate the dynamic response of a fractionally damped nonlinear oscillator possessing both quadratic and cubic nonlinearities via the method of multiple time scales. It has been claimed that damping features are modeled via the Caputo fractional derivative. In the present paper, it is shown that both the scheme and the object of investigation are not new, and moreover, the above mentioned author's statement is inconsistent, since under the assumptions made in the paper under consideration these two fractional-order derivatives coincide. Besides, the utilized procedure was inconsequential. It has been proved that the investigation of the dynamic response of a nonlinear viscoelastic oscillator presents the case that, with some minimal restrictions, the Riemann–Liouville and Caputo definitions produce completely equivalent mathematical models of the nonlinear viscoelastic phenomenon. |
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