Abstract
In this manuscript we propose the discrete versions for the recently introduced fractional derivatives with nonsingular Mittag-Leffler function. The properties of such fractional differences are studied and the discrete integration by parts formulas are proved. Then a discrete variational problem is considered with an illustrative example. Finally, some more tools for these derivatives and their discrete versions have been obtained.
Highlights
1 Introduction and preliminaries Fractional calculus has become an important mathematical tool used in several branches of science and engineering in order to describe better the properties of non-local complex systems [ – ]
Finding the discrete counterparts of these new fractional operators is an important step to apply them to model the dynamics of complex systems
For the rest of this section, we summarize some facts as regards the discrete Laplace transform of Mittag-Leffler type and convolution type functions
Summary
Lemma Let f be a function defined on Na. The following result holds: Na∇ f (t) (z) = z(Naf )(z) – ( – z)af (a). For the following lemma we will present an alternative proof without using convolutions as was done in [ ]. The result follows by Lemma with a = and (∇ –( –α)f )( ) =. Remark Lemma can be generalized as follows. NaC∇aαf (z) = zα(Naf )(z) – ( – z)azα– f (a). This can be proved by making use of Remark. Lemma [ ] Let < α ≤ and f be defined on N. Since dividing by balls of Gamma function leads to zero, we have ∇Eα(λ, t) =
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