Existence and stability of nonlinear discrete fractional initial value problems with application to vibrating eardrum

Selvam, A. George Maria; Alzabut, Jehad; Vignesh, D.; Jonnalagadda, Jagan Mohan; Abodayeh, Kamaleldin

doi:10.3934/mbe.2021195

Cited by 16 publications

(6 citation statements)

References 35 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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“…Many important processes and phenomena in real-world situations can be mathematically modeled by autonomous dynamical systems described by differential equations associated with the classical and fractional derivative operators [1][2][3][4][5][6][7][8]. While differential equation models with the classical derivatives have been formed and studied for a long time [1,3,5,6,8], mathematical models based on fractional differential equations have been strongly developed in recent years (see, for example, [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28]). The stability analysis of differential equation models has been a central and prominent problem with many useful applications.…”

Section: Introductionmentioning

confidence: 99%

A simple method for studying asymptotic stability of discrete dynamical systems and its applications

Hoang

Ngo²,

Truong

2023

Int. J. Optim. Control, Theor. Appl. (IJOCTA)

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In this work, we introduce a simple method for investigating the asymptotic stability of discrete dynamical systems, which can be considered as an extension of the classical Lyapunov's indirect method. This method is constructed based on the classical Lyapunov's indirect method and the idea proposed by Ghaffari and Lasemi in a recent work. The new method can be applicable even when equilibia of dynamical systems are non-hyperbolic. Hence, in many cases, the classical Lyapunov's indirect method fails but the new one can be used simply. In addition, by combining the new stability method with the Mickens' methodology, we formulate some nonstandard finite difference (NSFD) methods which are able to preserve the asymptotic stability of some classes of differential equation models even when they have non-hyperbolic equilibrium points. As an important consequence, some well-known results on stability-preserving NSFD schemes for autonomous dynamical systems are improved and extended. Finally, a set of numerical examples are performed to illustrate and support the theoretical findings.

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“…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 technology to perform higher level simulations, the study of fractional calculus has been a field of attraction and is applied in several fields of research like physics, circuit theory, chemistry, geology, biology, and sociology 1–11 . Existence results for nonlinear integral equations was presented in Deep et al 12 and Chauhan et al 13 and qualitative analysis of fractional order equations with delay was carried out in Tunc and Tunc 14 .…”

Section: Introductionmentioning

confidence: 99%

“…several fields of research like physics, circuit theory, chemistry, geology, biology, and sociology. [1][2][3][4][5][6][7][8][9][10][11] Existence results for nonlinear integral equations was presented in Deep et al 12 and Chauhan et al 13 and qualitative analysis of fractional order equations with delay was carried out in Tunc and Tunc. 14 Delay integro-differential equations with proportional fractional order Caputo type derivative was considered and solution was estimated in Tunc and Tunc.…”

mentioning

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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Existence and stability of nonlinear discrete fractional initial value problems with application to vibrating eardrum

Cited by 16 publications

References 35 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

A simple method for studying asymptotic stability of discrete dynamical systems and its applications

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

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