Fractional Complex Transform for Fractional Differential Equations

Abstract

Fractional complex transform is proposed to convert fractional differential equations into ordinary differential equations, so that all analytical methods devoted to advanced calculus can be easily applied to fractional calculus. Two examples are given.

An analytic approach to a class of fractional differential‐difference equations of rational type via symbolic computation

“…In a recent study, Li and He [38] proved that FDEs in the sense of Jumarie's modified Riemann-Liouville derivative can be easily turned into ODEs so that anyone can deal with fractional calculus with ease. The authors proposed a fractional complex transformation in the form…”

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

“…The fractional complex transform was first proposed in 2010 by Li and He [41] to convert FDEs into ODEs so that all analytical methods for advanced calculus can be easily applied to fractional calculus. This approach requires no special knowledge of fractional calculus.…”

Exact Solutions for a Local Fractional DDE Associated with a Nonlinear Transmission Line

“…To make mention a few; the Laplace transform, the Fourier transform, the wave transformation, the Backlund transformation, the integral transform, etc. The fractional complex transform [3,42] was introduced to convert FDEs in the sense of the Jumaire's modified Riemann-Liouville derivative to integer order partners. Though many applications of the fractional complex transform appeared in the literature, a counter-example making the approach much skeptical was found.…”

“…By applying the variable-coefficient discrete (G ′ /G)-expansion method, Abdoulkary et al [35] investigated exact solutions of the nonlinear DDEs associated with the network. In order to complement the existing literature, in this study, our strategy is to construct exact solutions for the local time-fractional DDE model (1) using the discrete tanh method [27] coupled with the fractional complex transform which was first proposed in 2010 by Li and He [42] to convert FDEs into ODEs.…”