A note on new unified fractional derivative

Abstract

Recently, a new and unified definition of fractional derivative of the Caputo type is introduced Zheng et al.(Int. J. Non-Linear Mechanics, 2019). Indeed, the proposed formula is very interesting and unified one as it generalizes the notion of fractional derivative of Caputo type in global sense adopting full memory effect and also is capable to capture the short memory effect through local fractional derivatives. Some basic properties of the proposed fractional differential operator such as linearity, Backward compatibility, Identity, consistency, semigroup, etc. are discussed in detail. Integral transforms such as Fourier transform and Laplace transform of the proposed derivatives are also determined. Upon further investigation, it is found that some of the properties of this operator lack validity and consistency. The prime objective of this note is to point out these properties and study deeply by comparing the results of fractional derivatives of the Caputo type.