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

• Published in: AIMS mathematics Vol. 7; no. 2; pp. 1791 - 1810
• Main Authors: Ganesh, Anumanthappa, Deepa, Swaminathan, Baleanu, Dumitru, Santra, Shyam Sundar, Moaaz, Osama, Govindan, Vediyappan, Ali, Rifaqat
• Format: Journal Article
• Language: English
• Published: AIMS Press 01.01.2022
• Subjects: caputo derivative; fractional differential equation; fractional fourier transform; hyers-ulam-mittag-leffler stability; mittag-leffler function
• ISSN: 2473-6988; 2473-6988
• DOI: 10.3934/math.2022103

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.