An approach to differential geometry of fractional order via modified Riemann-Liouville derivative

Jumarie, Guy

doi:10.1007/s10114-012-0507-3

Cited by 50 publications

(50 citation statements)

References 17 publications

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“…In addition, Kolwankar obtained the same formula (4) by using an approach on Cantor space [30]. In addition, Jumarie in [31] gave detailed proofs of the above formulas (see Proposition 3.1 page 1746 and Section 4 (Some Basic Formulae for Fractional Derivative and Integral) page 1748). The above properties play an important role in the fractional mapping method.…”

Section: Mathematical Problems In Engineeringmentioning

confidence: 99%

Analytic Solutions of the Space-Time Fractional Combined KdV-mKdV Equation

Abdel-Salam

Al-Muhiameed

2015

Mathematical Problems in Engineering

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The fractional mapping method is proposed to solve fractional differential equations. To illustrate the effectiveness of the method, we discuss the space-time fractional combined KdV-mKdV equation. Many types of exact analytical solutions are obtained. The solutions include generalized trigonometric and hyperbolic functions solutions. These solutions are useful to understand the mechanisms of the complicated nonlinear physical phenomena and fractional differential equations. Among these solutions, some are found for the first time.

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Section: Mathematical Problems In Engineeringmentioning

confidence: 99%

Analytic Solutions of the Space-Time Fractional Combined KdV-mKdV Equation

Abdel-Salam

Al-Muhiameed

2015

Mathematical Problems in Engineering

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“…We have proposed recently a fractional calculus based on fractional difference [9][10][11][12][13][14][15][16] which is slightly different from the classical Riemann-Liouville framework [1][2][3][4][5][6][20][21][22][23][24][25][26][27][28], and results in a useful fractional Taylor series [12] providing (2) as the first term. The fractional calculus so obtained is quite parallel to the classical calculus, and it involves non-commutative derivatives, what seems to be quite consistent with non commutative geometry.…”

Section: Purpose and Organization Of The Articlementioning

confidence: 99%

On the derivative chain-rules in fractional calculus via fractional difference and their application to systems modelling

Jumarie

2013

Open Physics

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Abstract:It has been pointed out that the derivative chains rules in fractional differential calculus via fractional calculus are not quite satisfactory as far as they can yield different results which depend upon how the formula is applied, that is to say depending upon where is the considered function and where is the function of function. The purpose of the present short note is to display some comments (which might be clarifying to some readers) on the matter. This feature is basically related to the non-commutativity of fractional derivative on the one hand, and furthermore, it is very close to the physical significance of the systems under consideration on the other hand, in such a manner that everything is right so. As an example, it is shown that the trivial first order system may have several fractional modelling depending upon the way by which it is observed. This suggests some rules to construct the fractional models of standard dynamical systems, in as meaningful a model as possible. It might happen that this pitfall comes from the feature that a function which is continuous everywhere, but is nowhere differentiable, exhibits random-like features. PACS

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“…In the sequence we propose an alternative approach by considering fractional space-time instead of fractional space functions, that is, we consider that a coarse-grained space-time, which means that space and time are non-differentiable and considering the chain rule as [2] …”

Section: Introductionmentioning

confidence: 99%

“…It can be shown that the ansatz φ = φ(x α + λt β ) is a solution of the FWE in a coarse-grained space-time [2], subject to the condition λ α,β = ±v β Γ(α+1) Γ(β+1) . The result above gives the insight to redefine the light-cone variables ξ, η as ξ = x α − λt β and η = x α + λt β .…”

Section: Introductionmentioning

confidence: 99%

Modified Riemann-Liouville Approach to Field Theory and Particle Physics

Weberszpil

Godinho

Helayël-Neto

2015

Proceeding Series of the Brazilian Society of Computational and Applied Mathematics

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In this contribution we present our recent results where we have applied a modified form of Riemann-Liouville fractional derivative to build up a generalized fractional D'Alembertian and by means of a transformation of variables to light-cone coordinates, an explicit analytical solution was obtained. We have also set up the coarse-grained formulation of a fractional Schrödinger equation that incorporates a higher (spatial) derivative term which accounts for relativistic effects up to the lowest order in momentum.

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An approach to differential geometry of fractional order via modified Riemann-Liouville derivative

Cited by 50 publications

References 17 publications

Analytic Solutions of the Space-Time Fractional Combined KdV-mKdV Equation

Analytic Solutions of the Space-Time Fractional Combined KdV-mKdV Equation

On the derivative chain-rules in fractional calculus via fractional difference and their application to systems modelling

Modified Riemann-Liouville Approach to Field Theory and Particle Physics

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