Modeling of a Mass-Spring-Damper System by Fractional Derivatives with and without a Singular Kernel

Gómez-Aguilar, J. F.; Yépez-Martínez, H.; Calderón-Ramón, C.; Cruz-Orduña, Ines; Escobar-Jiménez, R.F.; Olivares-Peregrino, Víctor Hugo

doi:10.3390/e17096289

Cited by 134 publications

(82 citation statements)

References 34 publications

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“…We used the local derivative in one model, the Caputo derivative with power function kernel with singularity and the Caputo-Fabrizio derivative with exponential kernel, which has no singularity [21][22][23][24]. We presented the existence and uniqueness of the model with local derivative and then we derived approximate solutions via iterative method.…”

Section: Discussionmentioning

confidence: 99%

Chaos on the Vallis Model for El Niño with Fractional Operators

Alkahtani

Atangana

2016

Entropy

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Abstract:The Vallis model for El Niño is an important model describing a very interesting physical problem. The aim of this paper is to investigate and compare the models using both integer and non-integer order derivatives. We first studied the model with the local derivative by presenting for the first time the exact solution for equilibrium points, and then we presented the exact solutions with the numerical simulations. We further examined the model within the scope of fractional order derivatives. The fractional derivatives used here are the Caputo derivative and Caputo-Fabrizio type. Within the scope of fractional derivatives, we presented the existence and unique solutions of the model. We derive special solutions of both models with Caputo and Caputo-Fabrizio derivatives. Some numerical simulations are presented to compare the models. We obtained more chaotic behavior from the model with Caputo-Fabrizio derivative than other one with local and Caputo derivative. When compare the three models, we realized that, the Caputo derivative plays a role of low band filter when the Caputo-Fabrizio presents more information that were not revealed in the model with local derivative.

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Section: Discussionmentioning

confidence: 99%

Chaos on the Vallis Model for El Niño with Fractional Operators

Alkahtani

Atangana

2016

Entropy

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“…Another utilization is scrutinizing the macroscopic behaviours of some materials, identified with nonlocal communications between atoms. Other applications of the CF fractional derivative can be achieved, for example, in [19][20][21].…”

Section: Introductionmentioning

confidence: 99%

Approximating Solution of Fabrizio-Caputo Volterra’s Model for Population Growth in a Closed System by Homotopy Analysis Method

Bashiri

Vaezpour

Nieto

2018

Journal of Function Spaces

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Volterra's model for population growth in a closed system consists in an integral term to indicate accumulated toxicity besides the usual terms of the logistic equation. Scudo in 1971 suggested the Volterra model for a population ( ) of identical individuals to show crowding and sensitivity to "total metabolism": / = ( ) − 2 ( ) − ( ) ∫ 0 ( ) s. In this paper our target is studying the existence and uniqueness as well as approximating the following Caputo-Fabrizio Volterra's model for population growth in a closed system:, subject to the initial condition (0) = 0. The mechanism for approximating the solution is Homotopy Analysis Method which is a semianalytical technique to solve nonlinear ordinary and partial differential equations. Furthermore, we use the same method to analyze a similar closed system by considering classical Caputo's fractional derivative. Comparison between the results for these two factional derivatives is also included.

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“…From the standpoint of engineering, fractional-order control has drawn the attention since it not only provides an alternate control method but also shows a rich frequency response; see the simple mechanical plants [19] up to quite advanced applications in aeronautics [20]. A restrictive class of nonlinear systems with a model-free output feedback and a model-based observer is studied in [21], while [22] studies the case of fractional parametric uncertainties and [23] studies singular linear fractional-order systems.…”

Section: Introductionmentioning

confidence: 99%

Output Feedback Finite‐Time Stabilization of Systems Subject to Hölder Disturbances via Continuous Fractional Sliding Modes

Muñoz‐Vázquez

Parra-Vega²,

Sánchez‐Orta³

et al. 2017

Mathematical Problems in Engineering

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The problem of designing a continuous control to guarantee finite-time tracking based on output feedback for a system subject to a Hölder disturbance has remained elusive. The main difficulty stems from the fact that such disturbance stands for a function that is continuous but not necessarily differentiable in any integer-order sense, yet it is fractional-order differentiable. This problem imposes a formidable challenge of practical interest in engineering because (i) it is common that only partial access to the state is available and, then, output feedback is needed; (ii) such disturbances are present in more realistic applications, suggesting a fractional-order controller; and (iii) continuous robust control is a must in several control applications. Consequently, these stringent requirements demand a sound mathematical framework for designing a solution to this control problem. To estimate the full state in finite-time, a high-order sliding mode-based differentiator is considered. Then, a continuous fractional differintegral sliding mode is proposed to reject Hölder disturbances, as well as for uncertainties and unmodeled dynamics. Finally, a homogeneous closed-loop system is enforced by means of a continuous nominal control, assuring finite-time convergence. Numerical simulations are presented to show the reliability of the proposed method.

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Modeling of a Mass-Spring-Damper System by Fractional Derivatives with and without a Singular Kernel

Cited by 134 publications

References 34 publications

Chaos on the Vallis Model for El Niño with Fractional Operators

Chaos on the Vallis Model for El Niño with Fractional Operators

Approximating Solution of Fabrizio-Caputo Volterra’s Model for Population Growth in a Closed System by Homotopy Analysis Method

Output Feedback Finite‐Time Stabilization of Systems Subject to Hölder Disturbances via Continuous Fractional Sliding Modes

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