Solvability and stability of nonlinear hybrid ∆-difference equations of fractional-order

Alzabut, Jehad; Selvam, A. George Maria; Vignesh, D.; Gholami, Yousef

doi:10.1515/ijnsns-2021-0005

Cited by 8 publications

(6 citation statements)

References 43 publications

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“…The asymptotic stability analysis of nonlinear discrete fractional equations were studied by Fulai Chen in [36,41]. Several authors have contributed on the stability analysis of various applications of fractional order discrete time equations as in [6,[42][43][44][45][46][47]. We devote this section to study the stability of the HDFGHPE (6).…”

Section: Existence Results For Hdfghpe (6)mentioning

confidence: 99%

A Study of Generalized Hybrid Discrete Pantograph Equation via Hilfer Fractional Operator

et al. 2022

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Pantograph, a device in which an electric current is collected from overhead contact wires, is introduced to increase the speed of trains or trams. The work aims to study the stability properties of the nonlinear fractional order generalized pantograph equation with discrete time, using the Hilfer operator. Hybrid fixed point theorem is considered to study the existence of solutions, and the uniqueness of the solution is proved using Banach contraction theorem. Stability results in the sense of Ulam and Hyers, and its generalized form of stability for the considered initial value problem are established and we depict numerical simulations to demonstrate the impact of the fractional order on stability.

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Section: Existence Results For Hdfghpe (6)mentioning

confidence: 99%

A Study of Generalized Hybrid Discrete Pantograph Equation via Hilfer Fractional Operator

et al. 2022

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“…The stability concept for the functional equations originated from Hyers' answer to Ulam's question was extended to differential and difference equations of integer and fractional order. Some significant articles on Hyers–Ulam stability of fractional order equations include previous works 40–46 . Motivation from the above mentioned works, this section is devoted to establish the Hyers–Ulam stability results for considered initial value problem () with tempered fractional derivative.…”

Section: Hyers–ulam Stabilitymentioning

confidence: 99%

“…Some significant articles on Hyers-Ulam stability of fractional order equations include previous works. [40][41][42][43][44][45][46] Motivation from the above mentioned works, this section is devoted to establish the Hyers-Ulam stability results for considered initial value problem (1.1) with tempered fractional derivative.…”

Section: Hyers-ulam Stabilitymentioning

confidence: 99%

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

Alzabut

Selvam

Vignesh

et al. 2023

Math Methods in App Sciences

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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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“…Non-linear differential equations with quadratic perturbations also known as hybrid type differential equations are of great interest to the mathematicians and engineers due to their ability to describe different dynamic models as special cases. The two types of fractional order perturbation equations are discussed in [38] and some recent contributions on hybrid type fractional order equations include [39][40][41][42][43][44][45][46][47][48][49]. The first work on Hyers-Ulam stability of fractional difference equation with boundary condition was carried out by Fulai Chen and Yong Zhou in 2013 [50].…”

Section: Introductionmentioning

confidence: 99%

Stability of Boundary Value Discrete Fractional Hybrid Equation of Second Type with Application to Heat Transfer with Fins

et al. 2022

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The development in the qualitative theory of fractional differential equations is accompanied by discrete analog which has been studied intensively in recent past. Suitable fixed point theorem is to be selected to study the boundary value discrete fractional equations due to the properties exhibited by fractional difference operators. This article aims at investigating the stability results in the sense of Hyers and Ulam with application of Mittag–Leffler function hybrid fractional order difference equation of second type. The symmetric structure of the operators defined in this article is vital in establishing the existence results by using Krasnoselkii’s fixed point theorem. Banach contraction mapping principle and Krasnoselkii’s fixed point theorem are employed to establish the uniqueness and existence results for solution of fractional order discrete equation. A problem on heat transfer with fins is provided as an application to considered hybrid type fractional order difference equation and the stability results are demonstrated with simulations.

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Solvability and stability of nonlinear hybrid ∆-difference equations of fractional-order

Cited by 8 publications

References 43 publications

A Study of Generalized Hybrid Discrete Pantograph Equation via Hilfer Fractional Operator

A Study of Generalized Hybrid Discrete Pantograph Equation via Hilfer Fractional Operator

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

Stability of Boundary Value Discrete Fractional Hybrid Equation of Second Type with Application to Heat Transfer with Fins

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