Comments on various extensions of the Riemann–Liouville fractional derivatives : About the Leibniz and chain rule properties

Cresson, Jacky; Szafrańska, Anna

doi:10.1016/j.cnsns.2019.104903

Cited by 34 publications

(18 citation statements)

References 23 publications

(44 reference statements)

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“…(25) (26) Substitute the expression of stresses due to deformations from the relations ( 4)- (9). Then, taking into account relation ( 3), the equilibrium conditions on the surface due to displacement were obtained:…”

Section: ) (24)mentioning

confidence: 99%

Finite Element Calculation of the Linear Elasticity Problem for Biomaterials with Fractal Structure

Shymanskyi¹,

Sokolovskyy²

2021

TOBIOIJ

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Aims: The aim of this study was to develop the mathematical models of the linear elasticity theory of biomaterials by taking into account their fractal structure. This study further aimed to construct a variational formulation of the problem, obtain the main relationships of the finite element method to calculate the rheological characteristics of a biomaterial with a fractal structure, and develop application software for calculating the components of the stress-strain state of biomaterials while considering their fractal structure. The obtained results were analyzed. Background: The development of adequate mathematical models of the linear elasticity theory for biomaterials with a fractal structure is an urgent scientific task. Finding its solution will make it possible to analyze the rheological behavior of biomaterials exposed to external loads by taking into account the existing effects of memory, spatial non-locality, self-organization, and deterministic chaos in the material. Objective: The objective of this study was the deformation process of biomaterials with a fractal structure under external load. Methods: The equations of the linear elasticity theory for the construction of the mathematical models of the deformation process of biomaterials under external load were used. Mathematical apparatus of integro-differentiation of fractional order to take into account the fractal structure of the biomaterial was used. A variational formulation of the linear elasticity problem while taking into account the fractal structure of the biomaterial was formulated. The finite element method with a piecewise linear basis for finding an approximate solution to the problem was used. Results: The main relations of the linear elasticity problem, which takes into account the fractal structure of the biomaterial, were obtained. A variational formulation of the problem was constructed. The main relations of the finite-element calculation of the linear elasticity problem of a biomaterial with a fractal structure using a piecewise-linear basis are found. The main components of the stress-strain state of the biomaterial exposed to external loads are found. Conclusion: Using the mathematical apparatus of integro-differentiation of fractional order in the construction of the mathematical models of the deformation process of biomaterials with a fractal structure makes it possible to take into account the existing effects of memory, spatial non-locality, self-organization, and deterministic chaos in the material. Also, this approach makes it possible to determine the residual stresses in the biomaterial, which play an important role in the appearance of stresses during repeated loads.

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Section: ) (24)mentioning

confidence: 99%

Finite Element Calculation of the Linear Elasticity Problem for Biomaterials with Fractal Structure

Shymanskyi¹,

Sokolovskyy²

2021

TOBIOIJ

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“…D α t (Ψ) is the conformable fraction derivative of Ψ of order α. Nowadays, the fiel of conformable fractional derivative become one of the most important and interesting fiel for scientists because of its uses nonlinear sciences suck as, flui mechanics, chemical and biological processes. In literature, there are so many definition which of them are, Riemann-Liouville [30,31], Atangana-Baleanu derivative in Caputo sense [32], Caputoa and Grunwald-Letnikov [33][34][35].…”

Section: Introductionmentioning

confidence: 99%

A variety of exact solutions for fractional (2+1)-dimensional Heisenberg ferromagnetic spin chain in the semi classical limit

Akhtar¹,

Tariq²,

Khater³

et al. 2021

Rev. Mex. Fís.

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This paper investigates exact voyaging (2 + 1) dimensional Heisenberg ferromagnetic spin chain solutions with conformable fractional derivatives, an important family of nonlinear equations from Schrödinger (NLSE) for the construction of hyperbolic, trigonometric and complex function solutions. The detailed rational sine-cosine system and rational sinh-cosh system were used to locate dim, special and periodic wave solutions successfully. These findings suggest that the proposed approaches may be useful to investigate a range of solutions inside a repository of applied sciences and engineering, with success, quality, and trust. In addition, graphical representations and physical expresses of such solutions are represented by a set of the required values of the parameters involved. The methods are essentially adequate and can be extended to different dynamic models that create the nonlinear processes in today’s research.

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“…It is important to emphasize the different assessments of researchers in the field of fractional calculus and their applications, regarding the criteria according to which such an operator is really a fractional operator or is the equivalent of an operator of integer order. Most of the discussions were clarified by Creson and Szafranska, [33], respectively, Tarasov and Tarasova [34] formulated criteria for this selection.…”

Section: Introductionmentioning

confidence: 99%

Analytical Solutions of the Diffusion–Wave Equation of Groundwater Flow with Distributed-Order of Atangana–Baleanu Fractional Derivative

2021

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A generalized mathematical model of the radial groundwater flow to or from a well is studied using the time-fractional derivative with Mittag-Lefler kernel. Two temporal orders of fractional derivatives which characterize small and large pores are considered in the fractional diffusion–wave equation. New analytical solutions to the distributed-order fractional diffusion–wave equation are determined using the Laplace and Dirichlet-Weber integral transforms. The influence of the fractional parameters on the radial groundwater flow is analyzed by numerical calculations and graphical illustrations are obtained with the software Mathcad.

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Comments on various extensions of the Riemann–Liouville fractional derivatives : About the Leibniz and chain rule properties

Cited by 34 publications

References 23 publications

Finite Element Calculation of the Linear Elasticity Problem for Biomaterials with Fractal Structure

Finite Element Calculation of the Linear Elasticity Problem for Biomaterials with Fractal Structure

A variety of exact solutions for fractional (2+1)-dimensional Heisenberg ferromagnetic spin chain in the semi classical limit

Analytical Solutions of the Diffusion–Wave Equation of Groundwater Flow with Distributed-Order of Atangana–Baleanu Fractional Derivative

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