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

2012 ◽  
Vol 376 (4) ◽  
pp. 257-259 ◽  
Author(s):  
Ji-Huan He ◽  
S.K. Elagan ◽  
Z.B. Li
Keyword(s):  
Fractional Calculus ◽  
Chain Rule ◽  
Fractional Complex Transform ◽  
Derivative Chain ◽  
Geometrical Explanation

scholarly journals New Expansion Formulas for a Family of theλ-Generalized Hurwitz-Lerch Zeta Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
H. M. Srivastava ◽  
Sébastien Gaboury
Keyword(s):  
Fractional Calculus ◽  
Zeta Functions ◽  
Chain Rule ◽  
New Family ◽  
Special Cases ◽  
Leibniz Rules

We derive several new expansion formulas for a new family of theλ-generalized Hurwitz-Lerch zeta functions which were introduced by Srivastava (2014). These expansion formulas are obtained by making use of some important fractional calculus theorems such as the generalized Leibniz rules, the Taylor-like expansions in terms of different functions, and the generalized chain rule. Several (known or new) special cases are also considered.

scholarly journals New Relations Involving an Extended Multiparameter Hurwitz-Lerch Zeta Function with Applications

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
H. M. Srivastava ◽  
Sébastien Gaboury ◽  
Richard Tremblay
Keyword(s):  
Fractional Calculus ◽  
Zeta Function ◽  
Chain Rule ◽  
Special Cases ◽  
Lerch Zeta Function ◽  
Leibniz Rules

We derive several new expansion formulas involving an extended multiparameter Hurwitz-Lerch zeta function introduced and studied recently by Srivastava et al. (2011). These expansions are obtained by using some fractional calculus methods such as the generalized Leibniz rules, the Taylor-like expansions in terms of different functions, and the generalized chain rule. Several (known or new) special cases are also given.

scholarly journals An analytical approach to fractional Bousinesq-Burges equations

2020 ◽  
Vol 24 (4) ◽  
pp. 2581-2588
Author(s):  
Feng Lu
Keyword(s):  
Fractional Calculus ◽  
Iteration Method ◽  
Analytical Approach ◽  
Variational Iteration Method ◽  
Fractional Complex Transform ◽  
Variational Iteration ◽  
Modified Variational Iteration Method

This paper proposes an analytical approach to fractional calculus by the fractional complex transform and the modified variational iteration method. The fractional Bousinesq-Burges equations are used as an example to reveal the main merits of the present technology.

scholarly journals On fractal space-time and fractional calculus

2016 ◽  
Vol 20 (3) ◽  
pp. 773-777 ◽  
Author(s):  
Yue Hu ◽  
Ji-Huan He
Keyword(s):  
Fractional Calculus ◽  
Fractional Order ◽  
Physical Meaning ◽  
Solution Process ◽  
Space Time ◽  
Continuum Models ◽  
Nano Structure ◽  
Fractional Complex Transform ◽  
Fractal Space ◽  
Whitham Equation

This paper gives an explanation of fractional calculus in fractal space-time. On observable scales, continuum models can be used, however, when the scale tends to a smaller threshold, a fractional model has to be adopted to describe phenomena in micro/nano structure. A time-fractional Fornberg-Whitham equation is used as an example to elucidate the physical meaning of the fractional order, and its solution process is given by the fractional complex transform.

scholarly journals Extension of Matched Asymptotic Method to Fractional Boundary Layers Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Abdon Atangana ◽  
Emile Franc Doungmo Goufo
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivative ◽  
Boundary Layers ◽  
Asymptotic Method ◽  
Chain Rule

We were concerned with the description of the boundary layers problems within the scope of fractional calculus. However, we will note that one of the main methods used to solve these problems is the matched asymptotic method. We should mention that this was not achievable via the existing fractional derivative definitions, because they do not obey the chain rule. In order to accommodate the matched asymptotic method to the scope of fractional derivative, we proposed a relatively new derivative called thebeta-derivative. We presented some useful information for this operator. With the reward of this operator, we presented the idea of matched asymptotic method in finding solutions of the fractional boundary layers problems. The method was illustrated with an example.

scholarly journals On chain rule in fractional calculus

2016 ◽  
Vol 20 (3) ◽  
pp. 803-806
Author(s):  
Jun Wang ◽  
Yue Hu
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivative ◽  
Chain Rule

Chain rule plays an important role in fractional calculus. There are many definitions of fractional derivative, and this paper shows that the chain rule is invalid for Jumarie's modification of Riemann-Liouville definition.

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

Open Physics ◽  
2013 ◽  
Vol 11 (6) ◽  
Author(s):  
Guy Jumarie
Keyword(s):  
Fractional Calculus ◽  
Short Note ◽  
Order System ◽  
Fractional Difference ◽  
First Order ◽  
Fractional Models ◽  
Derivative Chain ◽  
Systems Modelling ◽  
Fractional Differential ◽  
The One

AbstractIt 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.

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