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.

scholarly journals A remark on local fractional calculus and ordinary derivatives

2016 ◽  
Vol 14 (1) ◽  
pp. 1122-1124 ◽  
Author(s):  
Ricardo Almeida ◽  
Małgorzata Guzowska ◽  
Tatiana Odzijewicz
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivative ◽  
Short Note ◽  
General Definition ◽  
Local Fractional Derivative ◽  
Fundamental Properties ◽  
Local Fractional Calculus ◽  
Definition Of

AbstractIn this short note we present a new general definition of local fractional derivative, that depends on an unknown kernel. For some appropriate choices of the kernel we obtain some known cases. We establish a relation between this new concept and ordinary differentiation. Using such formula, most of the fundamental properties of the fractional derivative can be derived directly.

Half-lives of spherical proton emitters within the framework of fractional calculus

2014 ◽  
Vol 23 (09) ◽  
pp. 1450044 ◽  
Author(s):  
Abdullah Engin Çalik ◽  
Hüseyin Şirin ◽  
Hüseyin Ertik ◽  
Buket Öder ◽  
Mürsel Şen
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivative ◽  
Nuclear Structure ◽  
Caputo Fractional Derivative ◽  
Half Life ◽  
Nuclear Decay ◽  
Life Values ◽  
Derivative Order

In this paper, the half-life values of spherical proton emitters such as Sb , Tm , Lu , Ta , Re , Ir , Au , Tl and Bi have been calculated within the framework of fractional calculus. Nuclear decay equation, related to this phenomenon, has been resolved by using Caputo fractional derivative. The order of fractional derivative μ being considered is 0 < μ ≤ 1, and characterizes the fractality of time. Half-life values have been calculated equivalent with empirical ones. The dependence of fractional derivative order μ on the nuclear structure has also been investigated.

The bouncing ball and the Grünwald-Letnikov definition of fractional derivative

2021 ◽  
Vol 24 (4) ◽  
pp. 1003-1014
Author(s):  
J. A. Tenreiro Machado
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Restitution Coefficient ◽  
Classical Physics ◽  
Bouncing Ball ◽  
Mechanical Experiment ◽  
Fractional Models ◽  
Definition Of

Abstract This paper proposes a conceptual experiment embedding the model of a bouncing ball and the Grünwald-Letnikov (GL) formulation for derivative of fractional order. The impacts of the ball with the surface are modeled by means of a restitution coefficient related to the coefficients of the GL fractional derivative. The results are straightforward to interpret under the light of the classical physics. The mechanical experiment leads to a physical perspective and allows a straightforward visualization. This strategy provides not only a motivational introduction to students of the fractional calculus, but also triggers possible discussion with regard to the use of fractional models in mechanics.

scholarly journals Mathematical Models Arising in the Fractal Forest Gap via Local Fractional Calculus

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Chun-Ying Long ◽  
Yang Zhao ◽  
Hossein Jafari
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivative ◽  
Mathematical Models ◽  
Cantor Sets ◽  
Forest Gap ◽  
Local Fractional Derivative ◽  
Local Fractional Calculus ◽  
Growth Equations ◽  
Gap Models ◽  
Individual Trees

The forest new gap models via local fractional calculus are investigated. The JABOWA and FORSKA models are extended to deal with the growth of individual trees defined on Cantor sets. The local fractional growth equations with local fractional derivative and difference are discussed. Our results are first attempted to show the key roles for the nondifferentiable growth of individual trees.

scholarly journals Some New Fractional-Calculus Connections between Mittag–Leffler Functions

Mathematics ◽  
2019 ◽  
Vol 7 (6) ◽  
pp. 485 ◽  
Author(s):  
Hari M. Srivastava ◽  
Arran Fernandez ◽  
Dumitru Baleanu
Keyword(s):  
Experimental Data ◽  
Fractional Calculus ◽  
Fractional Derivative ◽  
Fractional Integral ◽  
Integral Expression ◽  
Potential Applications ◽  
The One ◽  
Two Parameter

We consider the well-known Mittag–Leffler functions of one, two and three parameters, and establish some new connections between them using fractional calculus. In particular, we express the three-parameter Mittag–Leffler function as a fractional derivative of the two-parameter Mittag–Leffler function, which is in turn a fractional integral of the one-parameter Mittag–Leffler function. Hence, we derive an integral expression for the three-parameter one in terms of the one-parameter one. We discuss the importance and applications of all three Mittag–Leffler functions, with a view to potential applications of our results in making certain types of experimental data much easier to analyse.

scholarly journals Investigation Progresses and Applications of Fractional Derivative Model in Geotechnical Engineering

2016 ◽  
Vol 2016 ◽  
pp. 1-15 ◽  
Author(s):  
Jinxing Lai ◽  
Sheng Mao ◽  
Junling Qiu ◽  
Haobo Fan ◽  
Qian Zhang ◽  
...  
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivative ◽  
Simulation Analysis ◽  
Geotechnical Engineering ◽  
Constitutive Models ◽  
Parameter Determination ◽  
Deformation Analysis ◽  
Mathematical Tool ◽  
Circular Tunnel ◽  
Fractional Derivative Model

Over the past couple of decades, as a new mathematical tool for addressing a number of tough problems, fractional calculus has been gaining a continually increasing interest in diverse scientific fields, including geotechnical engineering due primarily to geotechnical rheology phenomenon. Unlike the classical constitutive models in which simulation analysis gradually fails to meet the reasonable accuracy of requirement, the fractional derivative models have shown the merits of hereditary phenomena with long memory. Additionally, it is traced that the fractional derivative model is one of the most effective and accurate approaches to describe the rheology phenomenon. In relation to this, an overview aimed first at model structure and parameter determination in combination with application cases based on fractional calculus was provided. Furthermore, this review paper shed light on the practical application aspects of deformation analysis of circular tunnel, rheological settlement of subgrade, and relevant loess researches subjected to the achievements acquired in geotechnical engineering. Finally, concluding remarks and important future investigation directions were pointed out.

THE INCREMENTAL RATIO BASED CAUSAL FRACTIONAL CALCULUS

2012 ◽  
Vol 22 (04) ◽  
pp. 1250078 ◽  
Author(s):  
MANUEL D. ORTIGUEIRA ◽  
MARGARITA RIVERO ◽  
JUAN J. TRUJILLO
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivative ◽  
Analytic Functions ◽  
Group Structure ◽  
Semi Group ◽  
Set Up

The generalized incremental ratio fractional derivative is revised and its main properties deduced. It is shown that in the case of analytic functions, it enjoys some interesting properties like: linearity and causality and has a semi-group structure. Some simple examples are presented. The enlargement of the set of functions for which the group properties of the fractional derivative are valid is done. With this, it is shown that some well-known results are valid in a more general set-up. Some examples are presented.

scholarly journals On Opial-Type Inequalities for Superquadratic Functions and Applications in Fractional Calculus

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Waqas Nazeer ◽  
Ghulam Farid ◽  
Zabidin Salleh ◽  
Ayesha Bibi
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivative ◽  
Fractional Integral ◽  
Mean Value ◽  
Mean Value Theorems ◽  
Superquadratic Functions

We have studied the Opial-type inequalities for superquadratic functions proved for arbitrary kernels. These are estimated by applying mean value theorems. Furthermore, by analyzing specific functions, the fractional integral and fractional derivative inequalities are obtained.

scholarly journals The Influence of Magnetohydrodynamic Flow and Slip Condition on Generalized Burgers’ Fluid with Fractional Derivative

2020 ◽  
Vol 17 (1) ◽  
pp. 0150
Author(s):  
Nassief Et al.
Keyword(s):  
Shear Stress ◽  
Fractional Calculus ◽  
Fractional Derivative ◽  
Velocity Distribution ◽  
Constant Pressure ◽  
Fluid Model ◽  
Magnetohydrodynamic Flow ◽  
Slip Condition ◽  
Constitutive Relationship ◽  
Burgers Fluid

This paper investigates the effect of magnetohydrodynamic (MHD) of an incompressible generalized burgers’ fluid including a gradient constant pressure and an exponentially accelerate plate where no slip hypothesis between the burgers’ fluid and an exponential plate is no longer valid. The constitutive relationship can establish of the fluid model process by fractional calculus, by using Laplace and Finite Fourier sine transforms. We obtain a solution for shear stress and velocity distribution. Furthermore, 3D figures are drawn to exhibit the effect of magneto hydrodynamic and different parameters for the velocity distribution.

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