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ARTICLE

On Riemann-Type Weighted Fractional Operators and Solutions to Cauchy Problems

Muhammad Samraiz¹, Muhammad Umer¹, Thabet Abdeljawad^2,3,*, Saima Naheed¹, Gauhar Rahman⁴, Kamal Shah^2,5

1 Department of Mathematics, University of Sargodha, P.O. Box 40100, Sargodha, 40100, Pakistan
2 Department of Mathematics and Sciences, Prince Sultan University, P.O. Box 66833, Riyadh, 11586, Saudi Arabia
3 Department of Medical Research, China Medical University, Taichung, 40402, Taiwan
4 Department of Mathematics and Statistics, Hazara University Mansehra, Mansehra, 21300, Pakistan
5 Department of Mathematics, University of Malakand, Chakdara Dir (L), KPK, 18000, Pakistan

* Corresponding Author: Thabet Abdeljawad. Email:

(This article belongs to this Special Issue: Applications of Fractional Operators in Modeling Real-world Problems: Theory, Computation, and Applications)

Computer Modeling in Engineering & Sciences 2023, 136(1), 901-919. https://doi.org/10.32604/cmes.2023.024029

Received 22 May 2022; Accepted 02 September 2022; Issue published 05 January 2023

Abstract

In this paper, we establish the new forms of Riemann-type fractional integral and derivative operators. The novel fractional integral operator is proved to be bounded in Lebesgue space and some classical fractional integral and differential operators are obtained as special cases. The properties of new operators like semi-group, inverse and certain others are discussed and its weighted Laplace transform is evaluated. Fractional integro-differential free-electron laser (FEL) and kinetic equations are established. The solutions to these new equations are obtained by using the modified weighted Laplace transform. The Cauchy problem and a growth model are designed as applications along with graphical representation. Finally, the conclusion section indicates future directions to the readers.

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Keywords

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