A Lyapunov-type inequality for a fractional boundary value problem with Caputo-Fabrizio derivative

Kirane, Mokhtar; Torebek, Berikbol T.

doi:10.7153/jmi-2018-12-77

Cited by 10 publications

(20 citation statements)

References 14 publications

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“…Atangana‐Baleanu (ABC) fractional derivative developed this definition using Mittag‐Leffler function instead of exponential function in CF definition. These new nonsingular fractional derivatives have been analyzed in theoretical viewpoint in previous studies . Different approaches can be observed to the ABC fractional derivative in previous studies .…”

Section: Introductionmentioning

confidence: 99%

Marine system dynamical response to a changing climate in frame of power law, exponential decay, and Mittag‐Leffler kernel

Sekerci

Özarslan

2020

Math Methods in App Sciences

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The increase of sea surface temperature in ocean changes the photosynthetic production rate of phytoplankton. Therefore, it is crucial to understand the relation between temperature and phytoplanktons photosynthesis to deal the extinction caused by excessive increase in temperature. It is worth observing that temperature is one of the most principal limiting factors for phytoplanktons production due to photosynthetic enzymes work at their optimum temperature levels. In this study, the fractional oxygen-phytoplankton-zooplankton model is considered by singular and nonsingular fractional operators within Caputo, Caputo-Fabrizio, and Atangana-Baleanu in Caputo sense. The rate of oxygen production is considered by a function of temperature account for the sea surface warming. At first, the temperature function is constant and then it starts to increase, after a certain time of increase, before the oxygen depletion begins, the temperature is set to a higher secure value. With this temperature function choice, detailed numerical simulations are carried out to provide details of the internal structure of the system. We observe that the species are more sustainable in Caputo model than its corresponding integer-order model. KEYWORDSCaputo, existence and uniqueness, global warming, nonsingular fractional derivatives, oxygen-plankton system, piece-wise function MSC CLASSIFICATION26A33; 37N25; 92B05; 97M60

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Section: Introductionmentioning

confidence: 99%

Marine system dynamical response to a changing climate in frame of power law, exponential decay, and Mittag‐Leffler kernel

Sekerci

Özarslan

2020

Math Methods in App Sciences

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“…The extension of Lyapunov‐type inequalities appeared recently for fractional differential equations, see, for example, previous studies 1–3 and the references therein. Here, we derive Lyapunov‐ and Hartman–Wintner‐type inequalities for the fractional boundary value problem

^{H} 𝔻_{a}^{α, β, ψ} x false(t false) + q false(t false) f false(x false(t false) false) = 0, a < t < b,

x false(a false) = x false(b false) = 0,

where

false(a, b false) \in ℝ^{2}

^{H} 𝔻^{α, β, ψ}

is the ψ ‐Hilfer fractional derivative type of order 1 < α < 2, 0 ≤ β ≤ 1,

x, ψ \in C^{2} false(false[a, b false], ℝ false)

such that ψ is strictly increasing and

f, q : ℝ \to ℝ

are given functions that will be specified later.…”

Section: Introduction and Statement Of The Problemmentioning

confidence: 99%

Lyapunov‐ and Hartman–Wintner‐type inequalities for a nonlinear fractional BVP with generalized ψ‐Hilfer derivative

Zohra

Hedia

Kirane

2020

Math Methods in App Sciences

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“…In [7,8] the properties of the CF derivative and fractional integral associated with the CF derivative are studied. Boundary value problems with CF derivatives have been studied in [9,10]. In [11] a linear fuzzy model with CF operator is studied, and the (i, α) and (ii, α) differentiable solutions of the model are obtained.…”

Section: Introductionmentioning

confidence: 99%

Stability analysis of fractional-order linear system with time delay described by the Caputo–Fabrizio derivative

Zhong

Cheng

et al. 2019

Adv Differ Equ

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In this paper, we define a characteristic equation of fractional-order linear system with time delay described by the Caputo-Fabrizio derivative. At the same time, by applying the Laplace transform and matrix theory we give a necessary and sufficient stability condition and some brief sufficient stability conditions. The proposed method is quite different from the other in the literature. In addition, we provide some examples to demonstrate the effectiveness of our results.

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A Lyapunov-type inequality for a fractional boundary value problem with Caputo-Fabrizio derivative

Cited by 10 publications

References 14 publications

Marine system dynamical response to a changing climate in frame of power law, exponential decay, and Mittag‐Leffler kernel

Marine system dynamical response to a changing climate in frame of power law, exponential decay, and Mittag‐Leffler kernel

Lyapunov‐ and Hartman–Wintner‐type inequalities for a nonlinear fractional BVP with generalized ψ‐Hilfer derivative

Stability analysis of fractional-order linear system with time delay described by the Caputo–Fabrizio derivative

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