Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus

By Ji-Huan He, S. Elagan, Zheng-Biao Li. 2012

388 citations recorded by the source corpus.

Abstract The fractional complex transform is suggested to convert a fractional differential equation with Jumarieʼs modification of Riemann–Liouville derivative into its classical differential partner. Understanding the fractional complex transform and the chain rule for fractional calculus are elucidated geometrically.

Subfields: Modeling and Simulation, Numerical Analysis, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/0d965fcf8b053334099a4ad7f2bf71b769fceeb2

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Citations in the corpus

On two-scale dimension and its applications (2019, neutral): “In [13, 14] the following transform is used:”
A note on exp-function method combined with complex transform method applied to fractional differential equations (2015, methodology): “The fractional complex transform [25, 33, 34] is the simplest approach to convert fractional di erential equations into ordinary di erential equations.”
Reduced differential transform method for partial differential equations within local fractional derivative operators (2016, methodology): “u(x, 0)= 1+sina(x ) , ∂u(x, 0) ∂ta = 0: ð26Þ In view of equations (11) and (15), the local fractional iteration algorithms can be written as follows G 1+(k + 2)a ð Þ G(1+ ka) Fk + 2(x) ∂Fk(x) ∂x2a Fk(x)= 0 ð27Þ or”
Exact solutions of nonlinear time fractional partial differential equations by sub‐equation method (2015, methodology): “With this kind of fractional derivative and some useful formulas, we can convert FDEs into integer-order differential equations by fractional complex transformation [20, 21].”
Fractional Complex Transform and exp-Function Methods for Fractional Differential Equations (2013, background): “The fractional complex transform [12, 13] is the simplest approach; it is to convert the fractional differential equations into ordinary differential equations, making the solution procedure extremely simple.”
Advanced Local Fractional Calculus and Its Applications (2012, neutral): “[76] J.”
New promises and future challenges of fractal calculus: From two-scale thermodynamics to fractal variational principle (2020, methodology): “Hereby we will introduce the basic properties of fractional derivatives by the variational iteration method[27-30].”