Fundamental calculus of the fractional derivative defined with Rabotnov exponential kernel and application to nonlinear dispersive wave model

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.