(7)Fundamental calculus of the fractional derivative defined with Rabotnov exponential kernel and application to nonlinear dispersive wave model
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Mehmet Yavuza,b, Ndolane Senec,∗
a Department of Mathematics and Computer Sciences, Faculty of Science, Necmettin Erbakan University, 42090 Konya, Turkey
b Department of Mathematics, College of Engineering, Mathematics and Physical Sciences, University of Exeter, Cornwall TR10, UK
c Laboratoire Lmdan, Département de Mathématiques de la Décision, Université Cheikh Anta Diop de Dakar, Faculté des Sciences Economiques et Gestion,
BP 5683 Dakar Fann, Senegal
Received 23 May 2020; received in revised form 21 October 2020; accepted 28 October 2020
Available online 2 November 2020
Abstract
Before going further with fractional derivative which is constructed by Rabotnov exponential kernel, there exist many questions that are
not addressed. In this paper, we try to recapitulate all the fundamental calculus, which we can obtain with this new fractional operator. The
problems in this paper are to determine the solutions of the fractional differential equations where the second members are constant functions,
polynomial functions, exponential functions, trigonometric functions, or Mittag-Leffler functions. For all the fractional differential equations,
the obtained solutions are represented graphically. The Laplace transform of the fractional derivative with Rabotnov exponential kernel is
the primary tool in the investigations. Finally, we give the fundamental solution to the nonlinear time-fractional modified Degasperis–Procesi
equation by considering the fractional operator with Rabotnov exponential kernel.
© 2020 Shanghai Jiaotong University. Published by Elsevier B.V.
This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/)
Keywords: Fractional differential equation; Nonlinear dispersive wave model; Rabotnov exponential kernel; Mittag-Leffler function; Laplace transformation.