Counterexamples on Jumarie’s two basic fractional calculus formulae

Liu, Cheng-shi

doi:10.1016/j.cnsns.2014.07.022

Cited by 80 publications

(37 citation statements)

References 9 publications

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“…We first briefly recall the concept of fractional derivative ( [33][34][35][36] and references therein). In particular, the Riemann-Liouville fractional derivative is defined by…”

Section: Lie Symmetry Analysis Of the Class Of The Time-fractional Nomentioning

confidence: 99%

Symmetry analysis and conservation laws for the class of time-fractional nonlinear dispersive equation

2015

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“…We first briefly recall the concept of fractional derivative ( [33][34][35][36] and references therein). In particular, the Riemann-Liouville fractional derivative is defined by…”

Section: Lie Symmetry Analysis Of the Class Of The Time-fractional Nomentioning

confidence: 99%

Symmetry analysis and conservation laws for the class of time-fractional nonlinear dispersive equation

2015

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“…In the framework of fractional calculus we cannot expect analogous formulas to (1) and (2), mainly because fractional derivatives have a non-local behavior, sometimes called "memory property", that is not compatible with these identities (see [38] for more details on the concept that underlies this notion). Despite of the fact that some authors proclaim that their fractional versions of derivative satisfy these equalities (see for instance [8,23,27,56] and several others), Tarasov and Liu have already constructed sufficiently convincing arguments that invalidate such claim, as can be seen in [32,48,49,50].…”

Section: Introductionmentioning

confidence: 99%

On the fractional version of Leibniz rule

Neto

Júnior

2020

Mathematische Nachrichten

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This manuscript is dedicated to prove a new inequality that involves an important case of Leibniz rule regarding Riemann–Liouville and Caputo fractional derivatives of order α∈(0,1). In the context of partial differential equations, the aforesaid inequality allows us to address the Faedo–Galerkin method to study several kinds of partial differential equations with fractional derivative in the time variable; particularly, we apply these ideas to prove the existence and uniqueness of solution to the fractional version of the 2D unsteady Stokes equations in bounded domains.

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“…But, I neglected the conditions of the Jumarie's formulae. Indeed, for example, the formula (2) requires that the functions u and v are non-differentiable and continuous, and the formula (3) requires that f is differentiable while u is non-differentiable and continuous, at the point t. Therefore, the examples in [10] cannot be considered as suitable direct counterexamples to Jumarie's formulae under the conditions of non-differentiable functions. However, I will show that these counterexamples do hold indirectly.…”

Section: Introductionmentioning

confidence: 99%

Counterexamples on Jumarie’s three basic fractional calculus formulae for non-differentiable continuous functions

Liu

2018

Chaos, Solitons & Fractals

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Juamrie proposed a modified Riemann-Liouville derivative definition and gave three so-called basic fractional calculus formulae (u(t)and v are required to be non-differentiable and continuous for the first formula, f is assumed to be differentiable for the second formula, while in the third formula f is non-differentiable and u is differentiable, at the point t. I once gave three counterexamples to show that Jumarie's three formulae are not true for differentiable functions(Cheng-shi Liu. Counterexamples on Jumaries two basic fractional calculus formulae. Communications in Nonlinear Science and Numerical Simulation, 2015, 22(1): 92-94.). However, these examples cannot directly become the suitable counterexamples for the case of non-differentiable continuous functions. In the present paper, I first provide five counterexamples to show directly the Jumarie's formulae are also not true for non-differentiable continuous functions. Then I prove that essentially non-differentiable cases can be transformed to the differentiable cases. Therefore, those counterexamples in the above paper are indirectly right. In summary, the Jumarie's formulae are not true. This paper can be considered as the corrigendum and supplement to the above paper.

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Counterexamples on Jumarie’s two basic fractional calculus formulae

Cited by 80 publications

References 9 publications

Symmetry analysis and conservation laws for the class of time-fractional nonlinear dispersive equation

Symmetry analysis and conservation laws for the class of time-fractional nonlinear dispersive equation

On the fractional version of Leibniz rule

Counterexamples on Jumarie’s three basic fractional calculus formulae for non-differentiable continuous functions

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