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.

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.

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.

A Local Fractional Derivative

2003 ◽  
Author(s):  
Xiaorang Li ◽  
Christopher Essex ◽  
Matt Davison
Keyword(s):  
Fractional Derivative ◽  
Fractional Order ◽  
Local Property ◽  
Order Derivative ◽  
Fractional Order Derivative ◽  
Local Fractional Derivative ◽  
Basic Properties ◽  
Differentiable Functions ◽  
Definition Of

A new definition of fractional order derivative is given and its basic properties are investigated. This definition is based on the Weyl derivative and is a local property of functions. It can be applied to non-differentiable functions and may be useful for studying fractal curves.

scholarly journals Solution of Fractional Order Equations in the Domain of the Mellin Transform

2019 ◽  
pp. 138-142
Author(s):  
Sunday Emmanuel Fadugba
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Caputo Fractional Derivative ◽  
Fractional Integral ◽  
Mellin Transform ◽  
Applied Mathematics ◽  
Fundamental Properties ◽  
Sequential Fractional Derivative ◽  
Liouville Fractional Derivative

This paper presents the Mellin transform for the solution of the fractional order equations. The Mellin transform approach occurs in many areas of applied mathematics and technology. The Mellin transform of fractional calculus of different flavours; namely the Riemann-Liouville fractional derivative, Riemann-Liouville fractional integral, Caputo fractional derivative and the Miller-Ross sequential fractional derivative were obtained. Three illustrative examples were considered to discuss the applications of the Mellin transform and its fundamental properties. The results show that the Mellin transform is a good analytical method for the solution of fractional order equations.

Comparative analysis of suitability of fractional derivatives in modelling the practical capacitor

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Rawid Banchuin
Keyword(s):  
Comparative Analysis ◽  
Fractional Derivative ◽  
Fractional Derivatives ◽  
Fractional Power ◽  
Electrolytic Capacitor ◽  
Content Type ◽  
Local Fractional Derivative ◽  
Definition Of ◽  
Double Layer Capacitors ◽  
Derivatives Of

Purpose The purpose of this paper is to compare the suitability of fractional derivatives in the modelling of practical capacitors. Such suitability refers to ability to provide the analytical capacitance function that matches the experimental ones of each fractional derivative. Design/methodology/approach The analytical capacitance functions based on various fractional derivatives of both local and nonlocal types including the author’s have been derived. The derived capacitance functions have been simulated and compared with the experimental ones of aluminium electrolytic and electrical double layer capacitors (EDLCs). Findings This paper has found that any local fractional derivative with fractional power law-based relationship with the conventional one is suitable for modelling the aluminium electrolytic capacitor (AEC) by incorporating with the conventional capacitance definition. On the other hand, the author’s nonlocal fractional derivatives have been found to be more suitable than the others for modelling the EDLC by incorporating with the revisited definition of capacitance. Originality/value The proposed comparative analysis has been originally presented in this work. The criterion for local fractional derivative, to be suitable for modelling the AEC, has been found. The nonlocal fractional operators which are most suitable for modelling the EDLC have been derived where the unsuitable one has been pointed out.

scholarly journals On the origin of space

Open Physics ◽  
2013 ◽  
Vol 11 (10) ◽  
Author(s):  
Richard Herrmann
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivative ◽  
Harmonic Oscillator ◽  
First Principles ◽  
Space Dimension ◽  
Variable Order ◽  
Quantum Harmonic Oscillator ◽  
Minimum Threshold ◽  
Fractional Quantum ◽  
Definition Of

AbstractWithin the framework of fractional calculus with variable order the evolution of space in the adiabatic limit is investigated. Based on the Caputo definition of a fractional derivative using the fractional quantum harmonic oscillator a model is presented, which describes space generation as a dynamic process, where the dimension d of space evolves smoothly with time in the range 0 ≤ d(t) ≤ 3, where the lower and upper boundaries of dimension are derived from first principles. It is demonstrated, that a minimum threshold for the space dimension is necessary to establish an interaction with external probe particles. A possible application in cosmology is suggested.

scholarly journals A New Generalized Definition of Fractional Derivative with Non-Singular Kernel

Computation ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 49 ◽  
Author(s):  
Khalid Hattaf
Keyword(s):  
Fractional Derivative ◽  
Fractional Derivatives ◽  
Singular Kernel ◽  
Fundamental Properties ◽  
Generalized Fractional Derivatives ◽  
Definition Of

This paper proposes a new definition of fractional derivative with non-singular kernel in the sense of Caputo which generalizes various forms existing in the literature. Furthermore, the version in the sense of Riemann–Liouville is defined. Moreover, fundamental properties of the new generalized fractional derivatives in the sense of Caputo and Riemann–Liouville are rigorously studied. Finally, an application in epidemiology as well as in virology is presented.

scholarly journals INFRARED SPECTROSCOPY OF DIATOMIC MOLECULES — A FRACTIONAL CALCULUS APPROACH

2013 ◽  
Vol 27 (06) ◽  
pp. 1350019 ◽  
Author(s):  
RICHARD HERRMANN
Keyword(s):  
Infrared Spectroscopy ◽  
Fractional Calculus ◽  
Fractional Derivative ◽  
Harmonic Oscillator ◽  
Diatomic Molecules ◽  
Smooth Transition ◽  
Quantum Harmonic Oscillator ◽  
Eigenvalue Spectrum ◽  
Fractional Quantum ◽  
Definition Of

The eigenvalue spectrum of the fractional quantum harmonic oscillator is calculated numerically, solving the fractional Schrödinger equation based on the Riemann and Caputo definition of a fractional derivative. The fractional approach allows a smooth transition between vibrational and rotational type spectra, which is shown to be an appropriate tool to analyze IR spectra of diatomic molecules.

scholarly journals Local Fractional Sumudu Transform with Application to IVPs on Cantor Sets

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
H. M. Srivastava ◽  
Alireza Khalili Golmankhaneh ◽  
Dumitru Baleanu ◽  
Xiao-Jun Yang
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivative ◽  
Analytical Solutions ◽  
Initial Value Problems ◽  
Fractional Derivatives ◽  
Cantor Sets ◽  
Coupling Method ◽  
Initial Value ◽  
Local Fractional Calculus ◽  
Sumudu Transform

Local fractional derivatives were investigated intensively during the last few years. The coupling method of Sumudu transform and local fractional calculus (called as the local fractional Sumudu transform) was suggested in this paper. The presented method is applied to find the nondifferentiable analytical solutions for initial value problems with local fractional derivative. The obtained results are given to show the advantages.

Hadamard’s inequality and its extensions for conformable fractional integrals of any order α>0

2018 ◽  
Vol 27 (2) ◽  
pp. 197-206
Author(s):  
ERHAN SET ◽  
◽  
AHMET OCAK AKDEMIR ◽  
I. MUMCU ◽  
◽  
...  
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivative ◽  
Fractional Integral ◽  
Fractional Integrals ◽  
Hadamard’S Inequality ◽  
Conformable Fractional Integral ◽  
Definition Of ◽  
Appl Math

Recently the authors Abdeljawad [Abdeljawad, T., On conformable fractional calculus, J. Comput. Appl. Math., 279 (2015), 57-66] and Khalil et al. [Khalil, R., Horani, M. Al., Yousef, A. and Sababheh, M., A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65-70] introduced a new and simple well-behaved concept of fractional integral called conformable fractional integral. In this article, we establish Hermite-Hadamard’s inequalities for conformable fractional integral. We also give extensions of Hermite-Hadamard type inequalities for conformable fractional integrals.

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