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

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  • Liu, Cheng-shi

Abstract

Jumarie proposed a modified Riemann–Liouville derivative definition and gave three so-called basic fractional calculus formulae such as Leibniz rule (u(t)v(t))(α)=u(α)(t)v(t)+u(t)v(α)(t), where u and v are required to be non-differentiable and continuous at the point t. We once gave the counterexamples to show that Jumarie’s formulae are not true for differentiable functions. In the paper, we give further counterexamples to prove that in non-differentiable cases these Jumarie’s formulae are also not true. Therefore, we proved that Jumarie’s formulae are not true for both cases of differentiable and non-differentiable functions, and then those results on fractional soliton equations obtained by using Jumarie’s formulae are not right.

Suggested Citation

  • Liu, Cheng-shi, 2018. "Counterexamples on Jumarie’s three basic fractional calculus formulae for non-differentiable continuous functions," Chaos, Solitons & Fractals, Elsevier, vol. 109(C), pages 219-222.
    • Handle: RePEc:eee:chsofr:v:109:y:2018:i:c:p:219-222
    DOI: 10.1016/j.chaos.2018.02.036
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    References listed on IDEAS

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    1. Hasan Bulut & Haci Mehmet Baskonus & Yusuf Pandir, 2013. "The Modified Trial Equation Method for Fractional Wave Equation and Time Fractional Generalized Burgers Equation," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-8, September.
    2. Yusuf Pandir & Yusuf Gurefe & Emine Misirli, 2013. "The Extended Trial Equation Method for Some Time Fractional Differential Equations," Discrete Dynamics in Nature and Society, Hindawi, vol. 2013, pages 1-13, June.
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    Cited by:

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    2. Nazir, Aqsa & Ahmed, Naveed & Khan, Umar & Mohyud-din, Syed Tauseef, 2020. "On stability of improved conformable model for studying the dynamics of a malnutrition community," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 537(C).
    3. Lianguang Jia & Li Tang, 2021. "The Classification of Traveling Wave Solutions to Space‐Time Fractional Resonance Nonlinear Schrödinger Type Equation with a Special Kind of Fractional Derivative," Advances in Mathematical Physics, John Wiley & Sons, vol. 2021(1).

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