Abstract
This research explores Caputo k-fractional integral inequalities for functions whose nth order derivatives are absolutely continuous and possess Grüss type variable bounds. Using Chebyshev inequality (Waheed et al. in IEEE Access 7:32137–32145, 2019) for Caputo k-fractional derivatives, several integral inequalities are derived. Further, Laplace transform of Caputo k-fractional derivative is presented and Caputo k-fractional derivative and Riemann–Liouville k-fractional integral of an extended generalized Mittag-Leffler function are calculated. Moreover, using the extended generalized Mittag-Leffler function, Caputo k-fractional differential equations are presented and their solutions are proposed by applying the Laplace transform technique.
Highlights
1 Introduction Fractional calculus is the study of fractional order derivatives and integrals
Riemann–Liouville fractional integrals and fractional derivatives are the basis of fractional calculus [7]
Caputo gave an improved fractional derivative formula known as Caputo fractional derivative [2]
Summary
Fractional calculus is the study of fractional order derivatives and integrals. It has gained extensive attention of the researchers in the last few decades. The left-sided and right-sided Caputo k-fractional derivatives of order α are defined by. In [4], the following definition of Caputo k-fractional derivatives for a convolution of two functions is studied and some interesting results have been established. This behaves as a generalization of Caputo k-fractional derivatives [12]. Definition 6 The right-sided and the left-sided Caputo k-fractional derivatives of convolution g ∗ h of two functions g and h are defined by. The aim of this paper is to explore new Caputo k-fractional integral inequalities by using Grüss type variable bounds of functions having n-time derivatives absolutely continuous. M ∈ R satisfying m ≤ g(n)(x) ≤ M, ∀x ∈ [a, b] the following inequality for the Caputo k-fractional derivatives holds: M(x
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