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

By Cheng-shi Liu. 2016

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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.

Subfields: Mathematical Physics, Modeling and Simulation, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/0ee5060a6640a5639b5b24a45b3e5af01f935912