Hyers-Ulam-Mittag-Leffler stability of fractional differential equations with two caputo derivative using fractional fourier transform

Anumanthappa Ganesh; Swaminathan Deepa; Dumitru Baleanu; Shyam Sundar Santra; Osama Moaaz; Vediyappan Govindan; Rifaqat Ali; Anumanthappa Ganesh; Swaminathan Deepa; Dumitru Baleanu; Shyam Sundar Santra; Osama Moaaz; Vediyappan Govindan; Rifaqat Ali

doi:10.3934/math.2022103

AIMS Mathematics

2022, Volume 7, Issue 2: 1791-1810. doi: 10.3934/math.2022103

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Hyers-Ulam-Mittag-Leffler stability of fractional differential equations with two caputo derivative using fractional fourier transform

1.
Department of Mathematics, Government Arts and Science College, Hosur, 635 110, Tamilnadu, India
2.
Department of Mathematics, Adhiyamaan college of engineering, Hosur, 635 109, Tamilnadu, India
3.
Department of Mathematics and Computer Science, Faculty of Arts and Sciences, Çankaya University Ankara, 06790 Etimesgut, Turkey
4.
Instiute of Space Sciences, Magurele-Bucharest, 077125 Magurele, Romania
5.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, 40402, Taiwan, China
6.
Department of Mathematics, JIS College of Engineering, Kalyani, West Bengal-741 235, India
7.
Department of Mathematics, Faculty of Science, Mansoura University, 35516 Mansoura, Egypt
8.
Department of Mathematics, Phuket Rajabhat University, 83000, Thailand
9.
Department of Mathematics, College of Science and Arts, Muhayil, King Khalid University, Abha 9004, Saudi Arabia

Received: 13 July 2021 Accepted: 22 October 2021 Published: 03 November 2021
MSC : 34A08, 34B10, 34B15

In this paper, we discuss standard approaches to the Hyers-Ulam Mittag Leffler problem of fractional derivatives and nonlinear fractional integrals (simply called nonlinear fractional differential equation), namely two Caputo fractional derivatives using a fractional Fourier transform. We prove the basic properties of derivatives including the rules for their properties and the conditions for the equivalence of various definitions. Further, we give a brief basic Hyers-Ulam Mittag Leffler problem method for the solving of linear fractional differential equations using fractional Fourier transform and mention the limits of their usability. In particular, we formulate the theorem describing the structure of the Hyers-Ulam Mittag Leffler problem for linear two-term equations. In particular, we derive the two Caputo fractional derivative step response functions of those generalized systems. Finally, we consider some physical examples, in the particular fractional differential equation and the fractional Fourier transform.
- fractional fourier transform,
- fractional differential equation,
- Hyers-Ulam-Mittag-Leffler stability,
- Mittag-Leffler function,
- Caputo derivative
Citation: Anumanthappa Ganesh, Swaminathan Deepa, Dumitru Baleanu, Shyam Sundar Santra, Osama Moaaz, Vediyappan Govindan, Rifaqat Ali. Hyers-Ulam-Mittag-Leffler stability of fractional differential equations with two caputo derivative using fractional fourier transform[J]. AIMS Mathematics, 2022, 7(2): 1791-1810. doi: 10.3934/math.2022103

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Abstract

In this paper, we discuss standard approaches to the Hyers-Ulam Mittag Leffler problem of fractional derivatives and nonlinear fractional integrals (simply called nonlinear fractional differential equation), namely two Caputo fractional derivatives using a fractional Fourier transform. We prove the basic properties of derivatives including the rules for their properties and the conditions for the equivalence of various definitions. Further, we give a brief basic Hyers-Ulam Mittag Leffler problem method for the solving of linear fractional differential equations using fractional Fourier transform and mention the limits of their usability. In particular, we formulate the theorem describing the structure of the Hyers-Ulam Mittag Leffler problem for linear two-term equations. In particular, we derive the two Caputo fractional derivative step response functions of those generalized systems. Finally, we consider some physical examples, in the particular fractional differential equation and the fractional Fourier transform.

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