Fractional operators with boundary points dependent kernels and integration by parts

Thabet Abdeljawad

doi:10.3934/dcdss.2020020

Article Contents

2020, Volume 13, Issue 3: 351-375. Doi: 10.3934/dcdss.2020020

This issue Previous Article Preface on "New trends of numerical and analytical methods" Next Article MHD flow of fractional Newtonian fluid embedded in a porous medium via Atangana-Baleanu fractional derivatives

Fractional operators with boundary points dependent kernels and integration by parts

Thabet Abdeljawad^1,2,3,

1.
Department of Mathematics and General Sciences, Prince Sultan University, P. O. Box 66833, 11586 Riyadh, Saudi Arabia
2.
Department of Medical Research, China Medical University, Taichung 40402, Taiwan
3.
Department of Computer Science and Information Engineering, Asia University, Taichung, Taiwan

Received: March 31, 2018

Revised: April 30, 2018

Published: December 2019

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Abstract

Recently, U. N. Katugampola presented some generalized fractional integrals and derivatives by iterating a $ t^{\rho-1}- $weighted integral, $ \rho>0 $. The case $ \rho = 1 $ produces Riemann and Caputo fractional derivatives and the limiting case $ \rho\rightarrow 0^+ $ results in Hadamard type fractional operators. In this article, we discuss the differences between a new class of nonlocal generalized fractional derivatives generated by iterating left and right type conformable integrals weighted by $ (t-a)^{\rho-1} $ and $ (b-t)^{\rho-1} $ and the ones introduced by Katugampola. In fact, we will present very different integration by parts formulas by presenting new mixed left and right generalized fractional operators with boundary points dependent kernels. The properties of this new class of mixed fractional operators are analyzed in newly defined function spaces as well.

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Mathematics Subject Classification: 26A33, 34H05, 34K35, 31A10, 33E30.

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References

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