Results on generalized neutral fractional impulsive dynamic equation over time scales using nonlocal initial condition

Ahmed Morsy; C. Anusha; Kottakkaran Sooppy Nisar; C. Ravichandran; Ahmed Morsy; C. Anusha; Kottakkaran Sooppy Nisar; C. Ravichandran

doi:10.3934/math.2024403

AIMS Mathematics

2024, Volume 9, Issue 4: 8292-8310. doi: 10.3934/math.2024403

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Research article

Results on generalized neutral fractional impulsive dynamic equation over time scales using nonlocal initial condition

1.
Department of Mathematics, College of Arts and Sciences, Prince Sattam bin Abdulaziz University, Wadi Aldawaser 11991, Saudi Arabia
2.
Department of Mathematics, Kongunadu Arts and Science College, Coimbatore 641029, India
3.
Department of Mathematics, College of Science and Humanities in Alkharj, Prince Sattam bin Abdulaziz University, Alkharj 11942, Saudi Arabia
4.
Saveetha School of Engineering, SIMATS, Chennai, India

Received: 27 December 2023 Revised: 19 February 2024 Accepted: 20 February 2024 Published: 27 February 2024
MSC : 26E70, 34K40, 34N05, 37C25

This paper explored the existence and uniqueness of a neutral fractional impulsive dynamic equation over time scales that included nonlocal initial conditions and employed the Caputo-nabla derivative (C$ \nabla $D). The establishment of existence and uniqueness relies on the fine fixed point theorem. Furthermore, a comparison was conducted between the fractional order C$ \nabla $D and the Riemann-Liouville nabla derivative (RL$ \nabla $D) over time scales. Theoretical findings were substantiated through a numerical methodology, and an illustrative graph using MATLAB was presented for the provided example.
- neutral differential equations,
- C$ \nabla $D,
- RL$ \nabla $D,
- time scales
Citation: Ahmed Morsy, C. Anusha, Kottakkaran Sooppy Nisar, C. Ravichandran. Results on generalized neutral fractional impulsive dynamic equation over time scales using nonlocal initial condition[J]. AIMS Mathematics, 2024, 9(4): 8292-8310. doi: 10.3934/math.2024403

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Abstract

This paper explored the existence and uniqueness of a neutral fractional impulsive dynamic equation over time scales that included nonlocal initial conditions and employed the Caputo-nabla derivative (C$ \nabla $D). The establishment of existence and uniqueness relies on the fine fixed point theorem. Furthermore, a comparison was conducted between the fractional order C$ \nabla $D and the Riemann-Liouville nabla derivative (RL$ \nabla $D) over time scales. Theoretical findings were substantiated through a numerical methodology, and an illustrative graph using MATLAB was presented for the provided example.

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