A remark on local fractional calculus and ordinary derivatives

Almeida, Ricardo; Guzowska, Małgorzata; Odzijewicz, Tatiana

doi:10.1515/math-2016-0104

Cited by 66 publications

(50 citation statements)

References 13 publications

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“…Although these fractional definitions display desired advantages, such as the description of memory and hereditary effects in natural phenomena, they unfortunately lead to computational complexities, requiring an improvement of numerical methods, because of their non-local descriptions with weakly singular kernels. Due to these complications, fractional researchers have shown increasing interest for the new local fractional definitions [16][17][18][19][20]. One of these definitions is the limit-based conformable fractional derivative, which is defined as follows.…”

Section: Preliminariesmentioning

confidence: 99%

See 1 more Smart Citation

Optimal Control Problem for a Conformable Fractional Heat Conduction Equation

2017

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This paper presents an optimal boundary temperature control of thermal stresses in a plate, based on timeconformable fractional heat conduction equation. The aim is to find the boundary temperature that takes thermal stress under control. The fractional Laplace and finite Fourier sine transforms are used to obtain the fundamental solution. Then the optimal control is held by successive iterations. Numerical results are depicted by plots produced by MATLAB codes.

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Section: Preliminariesmentioning

confidence: 99%

“…In the present work, we consider a centrally symmetric temperature distribution T (x, t) on a line segment 0 ≤ x ≤ L at a time t. In this case, the thermoelastic stress σ (x, t) is proportional to the deviation from the average temperature [19]:…”

Section: Problem Formulationmentioning

confidence: 99%

Optimal Control Problem for a Conformable Fractional Heat Conduction Equation

2017

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“…Few years ago, in Abdeljawad, the idea proposed by Khalil is continued and a set of properties in that direction are obtained. One year later, Almeida et al, based on the idea of Khalil, present a new definition of local fractional derivative, that depends on an unknown kernel function. For some appropriate choices of the kernel function, the authors obtain some known cases (see, for instance, other studies).…”

Section: Introductionmentioning

confidence: 99%

“…More specifically: If

α \in false(0, 1 false]

and

T false(t, α false) = t^{1 - α}

, then the conformable fractional derivative defined in Khalil et al is obtained. If

α \in false(0, 1 false]

and

T false(t, α false) = k {false(t false)}^{1 - α}

, then the general conformable fractional derivative defined in Almeida et al is derived. If

α \in false(0, 1 false]

and

T false(t, α false) = e^{t^{- α}}

, then the non‐conformable fractional derivative defined in Guzman et al is obtained. …”

Section: Introductionmentioning

confidence: 99%

On the conformable fractional logistic models

et al. 2020

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In this paper, we use a conformable fractional derivative GTα, with kernel Tfalse(t,αfalse)=efalse(α−1false)t, in order to study the fractional differential equation associated to a logistic growth model. As a practical application, we estimate the order of the derivative of the fractional logistic models, by solving an inverse problem involving real data. In the same direction, we show the feasibility of our approach with respect to the Ordinary, Khalil et al and Caputo approaches.

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“…In [17], Almeida introduced a similar limit definition of fractional derivative of a function if we do not know the kernel as follows…”

Section: Introductionmentioning

confidence: 99%

Lyapunov Characterization of the Fractional Nonlinear Systems with Exogenous Input

Sene

2018

Fractal Fract

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This paper deals with a Lyapunov characterization of the conditional Mittag-Leffler stability and conditional asymptotic stability of the fractional nonlinear systems with exogenous input. A particular class of the fractional nonlinear systems is studied. The paper contributes to giving in particular the Lyapunov characterization of fractional linear systems and fractional bilinear systems with exogenous input.

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A remark on local fractional calculus and ordinary derivatives

Cited by 66 publications

References 13 publications

Optimal Control Problem for a Conformable Fractional Heat Conduction Equation

Optimal Control Problem for a Conformable Fractional Heat Conduction Equation

On the conformable fractional logistic models

Lyapunov Characterization of the Fractional Nonlinear Systems with Exogenous Input

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