Mellin transform for fractional integrals with general analytic kernel

Maliha Rashid; Amna Kalsoom; Maria Sager; Mustafa Inc; Dumitru Baleanu; Ali S. Alshomrani; Maliha Rashid; Amna Kalsoom; Maria Sager; Mustafa Inc; Dumitru Baleanu; Ali S. Alshomrani

doi:10.3934/math.2022524

AIMS Mathematics

2022, Volume 7, Issue 5: 9443-9462. doi: 10.3934/math.2022524

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Mellin transform for fractional integrals with general analytic kernel

1.
Department of Mathematics and Statistics, International Islamic University, Islamabad, Pakistan
2.
Department of Computer Engineering, Biruni University, Istanbul, Turkey
3.
Department of Mathematics, Firat University, Elazig 23119, Turkey
4.
Department of Medical Research, China Medical University, Taichung, Taiwan
5.
Department of Mathematics, Çankaya University, Balgat 06530, Ankara, Turkey
6.
Institute of Space Sciences, P.O. Box, MG-23, R 76900, Magurele-Bucharest, Romania
7.
Faculty of Science, Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia

Received: 27 November 2021 Revised: 21 January 2022 Accepted: 10 February 2022 Published: 14 March 2022
MSC : Primary: 58F15, 58F17; Secondary: 53C35

Many different operators of fractional calculus have been proposed, which can be organized in some general classes of operators. According to this study, the class of fractional integrals and derivatives can be classified into two main categories, that is, with and without general analytical kernel (introduced in 2019). In this article, we define the Mellin transform for fractional differential operator with general analytic kernel in both Riemann-Liouville and Caputo derivatives of order $ \varsigma\ge0 $ and $ \varrho $ be a fixed parameter. We will also establish relation between Mellin transform with Laplace and Fourier transforms.
- Mellin transform,
- fractional integrals,
- Caputo fractional derivative,
- Laplace and Fourier transforms
Citation: Maliha Rashid, Amna Kalsoom, Maria Sager, Mustafa Inc, Dumitru Baleanu, Ali S. Alshomrani. Mellin transform for fractional integrals with general analytic kernel[J]. AIMS Mathematics, 2022, 7(5): 9443-9462. doi: 10.3934/math.2022524

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Abstract

Many different operators of fractional calculus have been proposed, which can be organized in some general classes of operators. According to this study, the class of fractional integrals and derivatives can be classified into two main categories, that is, with and without general analytical kernel (introduced in 2019). In this article, we define the Mellin transform for fractional differential operator with general analytic kernel in both Riemann-Liouville and Caputo derivatives of order $ \varsigma\ge0 $ and $ \varrho $ be a fixed parameter. We will also establish relation between Mellin transform with Laplace and Fourier transforms.

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