Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results

Abstract

The paper gives some results and improves the derivation of the fractional Taylor's series of nondifferentiable functions obtained recently in the form fwhere E~ is the Mittag-Leffler function. Here, one defines fractional derivative as the limit of fractional difference, and by this way one can circumvent tile problem which arises with the definition of the fractional derivative of constant using Riemann-Liouville definition. As a result, a modified Riemann-Liouville definition is proposed, which is fully consistent with the fractional difference definition and avoids any reference to the derivative of order greater than the considered one's. In order to support this F-Taylor series, one shows how its first term can be obtained directly in tile form of a mean value formula. The fractional derivative of the Dirac delta function is obtained together with the fractional Taylor's series of multivariate functions. The relation with irreversibility of time and symmetry breaking is exhibited, and to some extent, this F-Taylor's series generalizes the fractional mean value formula obtained a few years ago by Kolwantar.

Exact solutions for fractional DDEs via auxiliary equation method coupled with the fractional complex transform

“…Several local versions of fractional derivatives have been proposed for the investigation of local behavior of fractional models. For example, Cresson's derivative [38], Kolwankar-Gangal local fractional derivative [39], and Jumarie's modified RiemannLiouville derivative [40]. Indeed, Liouville-Caputo derivatives are defined only for differentiable functions, while f can be a continuous (but not necessarily differentiable) function.…”

New Exact Solutions of the Time-Fractional Nonlinear Dispersive KdV Equation

“…For an introduction to the classical fractional calculus we indicate the reader to [1]- [3]. Here, we shortly review the modified Riemann-Liouville derivative from the recent fractional calculus proposed by Jumarie [8]- [10]. Let   f : 0, 1  be a continuous function and…”

Modified Trial Equation Method to the Nonlinear Fractional Sharma–Tasso–Olever Equation

“…For an introduction to the classical fractional calculus we indicate the reader to [1]- [3]. Here, we shortly review the modified Riemann-Liouville derivative from the recent fractional calculus proposed by Jumarie [19]- [21]. Let [22] as follows:…”