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Stability analysis of tempered fractional nonlinear Mathieu type equation model of an ion motion with octopole‐only imperfections

Authors:
Jehad Alzabut at Prince Sultan University/OSTIM Technical Unversity
Jehad Alzabut
  • Prince Sultan University/OSTIM Technical Unversity
George Maria Selvam A at Sacred Heart College
George Maria Selvam A
Vignesh D. at CMR University
Vignesh D.
Sina Etemad at Azarbaijan Shahid Madani University
Sina Etemad
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Abstract and Figures

The development of laser‐based cooling and spectroscopic methods has produced unprecedented growth in the ion trapping industry. Mathieu equation, a differential equation with periodic coefficients, is employed to develop models of ion motions under the influence of fields. Ion traps with octopole field is described with nonlinear Mathieu equation with cubic term. This article aims at considering motion of ions under the electric potential with negative octopole field with damping caused by the collision of the ions with Helium buffer gas modeled with tempered fractional derivative. Schaefer's fixed point theorem and Banach's contraction principle are employed to establish the existence of unique solution for the considered tempered fractional nonlinear Mathieu equation model of an ion motion. Further, the analysis of stability is performed in the sense of Hyers and Ulam. The feasibility of the obtained theoretical results are numerically confirmed for suitable parametric values, and simulations are performed supporting them.
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Received: 18 September 2022 Revised: 25 December 2022 Accepted: 11 January 2023
DOI: 10.1002/mma.9073
R E S E A R C H A R T I C L E
Stability analysis of tempered fractional nonlinear Mathieu
type equation model of an ion motion with octopole-only
imperfections
Jehad Alzabut
1,2
A. George Maria Selvam
3
Dhakshinamoorthy Vignesh
4
Sina Etemad
5
Shahram Rezapour
5,6,7
1Department of Mathematics and
Sciences, Prince Sultan University,
Riyadh, Saudi Arabia
2Department of Industrial Engineering,
OST˙
IM Technical University, Ankara,
Turkey
3Department of Mathematics, Sacred
Heart College (Autonomous), Tirupattur,
Tamil Nadu, India
4Cyber Security and Digital Industrial
Revolution Center, Universiti Pertahanan
Nasional Malaysia, Sungai Besi, Malaysia
5Department of Mathematics, Azarbaijan
Shahid Madani University, Tabriz, Iran
6Department of Mathematics, Kyung Hee
University, Seoul, Republic of Korea
7Department of Medical Research, China
Medical University Hospital, China
Medical University, Taichung, Taiwan
Correspondence
Jehad Alzabut, Department of
Mathematics and Sciences, Prince Sultan
University, 11586 Riyadh, Saudi Arabia.
Email: jalzabut@psu.edu.sa
Shahram Rezapour, Department of
Mathematics, Kyung Hee University,
Seoul, Republic of Korea.
Email: rezapourshahram@yahoo.ca
Communicated by: S. Trostorff
Funding information
No funds were received.
The development of laser-based cooling and spectroscopic methods has pro-
duced unprecedented growth in the ion trapping industry. Mathieu equation,
a differential equation with periodic coefficients, is employed to develop mod-
els of ion motions under the influence of fields. Ion traps with octopole field is
described with nonlinear Mathieu equation with cubic term. This article aims
at considering motion of ions under the electric potential with negative octopole
field with damping caused by the collision of the ions with Helium buffer gas
modeled with tempered fractional derivative. Schaefer's fixed point theorem and
Banach's contraction principle are employed to establish the existence of unique
solution for the considered tempered fractional nonlinear Mathieu equation
model of an ion motion. Further, the analysis of stability is performed in the
sense of Hyers and Ulam. The feasibility of the obtained theoretical results
are numerically confirmed for suitable parametric values, and simulations are
performed supporting them.
KEYW ORDS
Banach's contraction principle, HU stability, Schaefer's fixed point theorem, tempered fractional
differential equation
M SC C L AS S I FI C AT I ON
26A33, 34A08, 39A12, 39A13, 39A21
1 INTRO DUCT IO N
Fractional calculus has originated during the same time period as that of classical integer order calculus. Initially, the
subject of modeling real-life problems with differential equations of real or complex order was not popular among the
engineers and scientific research community. After the development of the super computers and advancement in tech-
nology to perform higher level simulations, the study of fractional calculus has been a field of attraction and is applied in
Math Meth Appl Sci. 2023;1–13. wileyonlinelibrary.com/journal/mma © 2023 John Wiley & Sons, Lt d. 1
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