Abstract
In this paper, we introduce the (k; s)-Hilfer Prabhakar fractional derivative. We discuss its properties and find a generalized Laplace transform of this operator. Furthermore, we show the applications of the (k; s)-Hilfer Prabhakar derivative in mathematical physics such as heat equation, free-electron laser equation, and kinetic differ-integral equation.
Highlights
Fractional calculus is the area of mathematical analysis that deals with the study and application of integrals and derivatives of arbitrary order
Fractional calculus has become of increasing significance due to its applications in many fields of science and engineering [1,2,3,4,5]
The first application of fractional calculus was given by Abel [6] and includes the solution to the tautocrone problem
Summary
Fractional calculus is the area of mathematical analysis that deals with the study and application of integrals and derivatives of arbitrary order. R. Hilfer introduced the Hilfer fractional derivative in [9], which is a generalization of the RiemannLiouville and Caputo fractional derivatives.The Prabhakar integral and derivative operators are obtained from the RiemannLiouville integral operator by extending its kernel to involve the three-parameter Mittag-Leffler function [19]. The modified (k, s)-fractional integral operator involving the kMittag-Leffler function given in [Samraiz et al, accepted] is defined as follows. The modified (k, s)-fractional integral operator involving the k-Mittag-Leffler function is given by The (k, s)-Prabhakar fractional derivative operator with the k-Mittag-Leffler function as its kernel is given by (skDω0+,γ;ρ,μ )(θ ) =. 2. THE (k,s)-HILFER-PRABHAKAR FRACTIONAL DERIVATIVE AND GENERALIZED LAPLACE TRANSFORMS. By using Definition 2.1 and the semigroup property of the modified (k, s)-fractional integral operator with the k-MittagLeffler function, we obtain s k skJω0+,σ;ρ,λ.
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