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

• Published in: Mathematical methods in the applied sciences 2023-05, Vol.46 (8), p.9542-9554
• Main Authors: Alzabut, Jehad, Selvam, A. George Maria, Vignesh, Dhakshinamoorthy, Etemad, Sina, Rezapour, Shahram
• Format: Article
• Language: English
• Subjects: Banach's contraction principle; Damping; Differential equations; Existence theorems; Fixed points (mathematics); HU stability; Ion motion; Ion traps (instrumentation); Laser cooling; Motion stability; Schaefer's fixed point theorem; Stability analysis; tempered fractional differential equation
• ISSN: 0170-4214; 1099-1476
• DOI: 10.1002/mma.9073

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.