scholarly journals A two-index generalization of conformable operators with potential applications in engineering and physics

2021 ◽  
Vol 67 (3 May-Jun) ◽  
pp. 429
Author(s):  
E. Reyes-Luis ◽  
G. Fernández Anaya ◽  
J. Chávez-Carlos ◽  
L. Diago-Cisneros ◽  
R. Muñoz Vega
Keyword(s):  
Differential Operator ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Lagrange Equation ◽  
Conformable Fractional Derivative ◽  
Fractional Order Calculus ◽  
Integer Order ◽  
Standard Scenario ◽  
Potential Applications ◽  
Sturm Liouville

We developed a somewhat novel fractional-order calculus workbench as a certain generalization of the Khalil’s conformable derivative. Although every integer-order derivate can naturally be consistent with fully physical-sense problem’s quotation, this is not the standard scenario of the non-integer-order derivatives, even aiming physics systems’s modelling, solely.We revisited a particular case of the generalized conformable fractional derivative and derived a differential operator, whose properties overcome those of the integer-order derivatives, though preserving its clue advantages.Worthwhile noting, that two-fractional indexes differential operator we are dealing, departs from the single-fractional index framework, which typifies the generalized conformable fractional derivative. This distinction leads to proper mathematical tools, useful in generalizing widely accepted results, with potential applications to fundamental Physics within fractional order calculus. The later seems to be especially appropriate for exercising the Sturm-Liouville eigenvalue problem, as well as the Euler-Lagrange equation and to clarify several operator algebra matters.

scholarly journals Novel hyperbolic and exponential ansatz methods to the fractional fifth-order Korteweg–de Vries equations

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Choonkil Park ◽  
R. I. Nuruddeen ◽  
Khalid K. Ali ◽  
Lawal Muhammad ◽  
M. S. Osman ◽  
...  
Keyword(s):  
Fractional Derivative ◽  
Fractional Order ◽  
Mathematical Physics ◽  
Order Derivative ◽  
Conformable Fractional Derivative ◽  
Fractional Order Derivative ◽  
De Vries ◽  
Fifth Order ◽  
2D And 3D

Abstract This paper aims to investigate the class of fifth-order Korteweg–de Vries equations by devising suitable novel hyperbolic and exponential ansatze. The class under consideration is endowed with a time-fractional order derivative defined in the conformable fractional derivative sense. We realize various solitons and solutions of these equations. The fractional behavior of the solutions is studied comprehensively by using 2D and 3D graphs. The results demonstrate that the methods mentioned here are more effective in solving problems in mathematical physics and other branches of science.

scholarly journals FTS and FTB of Conformable Fractional Order Linear Systems

2018 ◽  
Vol 2018 ◽  
pp. 1-5 ◽  
Author(s):  
Abdellatif Ben Makhlouf ◽  
Omar Naifar ◽  
Mohamed Ali Hammami ◽  
Bao-wei Wu
Keyword(s):  
Fractional Derivative ◽  
Linear Systems ◽  
Fractional Order ◽  
Finite Time ◽  
Conformable Fractional Derivative ◽  
Time Stability ◽  
Order Linear ◽  
Finite Time Stability ◽  
Finite Time Boundedness

In this paper, an extension of some existing results related to finite-time stability (FTS) and finite-time boundedness (FTB) into the conformable fractional derivative is presented. Illustrative example is presented at the end of the paper to show the effectiveness of the proposed result.

Formulation of Fractional Derivative-Based De-Hazing Algorithm and Implementation on Mobile-Embedded Devices

2019 ◽  
Vol 19 (01) ◽  
pp. 1950003
Author(s):  
Uche A. Nnolim
Keyword(s):  
Image Processing ◽  
Fractional Derivative ◽  
Mobile Devices ◽  
Fractional Order ◽  
Spatial Filter ◽  
Embedded Devices ◽  
Processing Application ◽  
Fractional Order Calculus ◽  
Image Processing Application ◽  
Logarithmic Image Processing

This paper presents the modification of a previously developed algorithm using fractional order calculus and its implementation on mobile-embedded devices such as smartphones. The system performs enhancement on three categories of images such as those exhibiting uneven illumination, faded features/colors and hazy appearance. The key contributions include the simplified scheme for illumination correction, contrast enhancement and de-hazing using fractional derivative-based spatial filter kernels. These are achieved without resorting to logarithmic image processing, histogram-based statistics and complex de-hazing techniques employed by conventional algorithms. The simplified structure enables ease of implementation of the algorithm on mobile devices as an image processing application. Results indicate that the fractional order version of the algorithm yields good results relative to the integer order version and other algorithms from the literature.

Bifurcations and Exact Solutions for a Class of MKdV Equations with the Conformable Fractional Derivative via Dynamical System Method

2020 ◽  
Vol 30 (01) ◽  
pp. 2050004 ◽  
Author(s):  
Jianli Liang ◽  
Longkun Tang ◽  
Yonghui Xia ◽  
Yi Zhang
Keyword(s):  
Fractional Derivative ◽  
Exact Solutions ◽  
Fractional Order ◽  
Traveling Wave ◽  
Nonlinear Differential Equations ◽  
Traveling Wave Solutions ◽  
Leibniz Rule ◽  
Wave Transformation ◽  
Conformable Fractional Derivative ◽  
Simplest Equation Method

In 2014, Khalil et al. [2014] proposed the conformable fractional derivative, which obeys chain rule and the Leibniz rule. In this paper, motivated by the monograph of Jibin Li [Li, 2013], we study the exact traveling wave solutions for a class of third-order MKdV equations with the conformable fractional derivative. Our approach is based on the bifurcation theory of planar dynamical systems, which is much different from the simplest equation method proposed in [Chen & Jiang, 2018]. By employing the traveling wave transformation [Formula: see text] [Formula: see text], we reduce the PDE to an ODE which depends on the fractional order [Formula: see text], then the analysis depends on the order [Formula: see text]. Moreover, as [Formula: see text], the exact solutions are consistent with the integer PDE. However, in all the existing papers, the reduced ODE is independent of the fractional order [Formula: see text]. It is believed that this method can be applicable to solve the other nonlinear differential equations with the conformable fractional derivative.

A Comparative Study of Integer and Noninteger Order Wavelets for Fractional Nonlinear Fredholm Integro-Differential Equations

2018 ◽  
Vol 13 (8) ◽  
Author(s):  
F. Mohammadi ◽  
J. A. Tenreiro Machado
Keyword(s):  
Differential Equations ◽  
Comparative Study ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Efficient Algorithm ◽  
Operational Matrix ◽  
Tau Method ◽  
Legendre Wavelets ◽  
Numerical Examples ◽  
Integer Order

This paper compares the performance of Legendre wavelets (LWs) with integer and noninteger orders for solving fractional nonlinear Fredholm integro-differential equations (FNFIDEs). The generalized fractional-order Legendre wavelets (FLWs) are formulated and the operational matrix of fractional derivative in the Caputo sense is obtained. Based on the FLWs, the operational matrix and the Tau method an efficient algorithm is developed for FNFIDEs. The FLWs basis leads to more efficient and accurate solutions of the FNFIDE than the integer-order Legendre wavelets. Numerical examples confirm the superior accuracy of the proposed method.

scholarly journals Fractional Dynamics of Computer Virus Propagation

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Carla M. A. Pinto ◽  
J. A. Tenreiro Machado
Keyword(s):  
Steady State ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Initial Conditions ◽  
Computer Virus ◽  
Fractional Dynamics ◽  
Fractional Order Systems ◽  
Virus Propagation ◽  
Integer Order ◽  
Fast Transients

We propose a fractional model for computer virus propagation. The model includes the interaction between computers and removable devices. We simulate numerically the model for distinct values of the order of the fractional derivative and for two sets of initial conditions adopted in the literature. We conclude that fractional order systems reveal richer dynamics than the classical integer order counterpart. Therefore, fractional dynamics leads to time responses with super-fast transients and super-slow evolutions towards the steady-state, effects not easily captured by the integer order models.

Dynamic response of Euler–Bernoulli beam resting on fractionally damped viscoelastic foundation subjected to a moving point load

2020 ◽  
Vol 234 (24) ◽  
pp. 4801-4812 ◽  
Author(s):  
Rajendra K Praharaj ◽  
Nabanita Datta
Keyword(s):  
Dynamic Response ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Point Load ◽  
Order Derivative ◽  
Viscoelastic Foundation ◽  
Bernoulli Beam ◽  
Integer Order ◽  
Foundation Model ◽  
Euler Bernoulli Beam

The dynamic behaviour of an Euler–Bernoulli beam resting on the fractionally damped viscoelastic foundation subjected to a moving point load is investigated. The fractional-order derivative-based Kelvin–Voigt model describes the rheological properties of the viscoelastic foundation. The Riemann–Liouville fractional derivative model is applied for a fractional derivative order. The modal superposition method and Triangular strip matrix approach are applied to solve the fractional differential equation of motion. The dependence of the modal convergence on the system parameters is studied. The influences of (a) the fractional order of derivative, (b) the speed of the moving point load and (c) the foundation parameters on the dynamic response of the system are studied and conclusions are drawn. The damping of the beam-foundation system increases with increasing the order of derivative, leading to a decrease in the dynamic amplification factor. The results are compared with those using the classical integer-order derivative-based foundation model. The classical foundation model over-predicts the damping and under-predicts the dynamic deflections and stresses. The results of the classical (integer-order) foundation model are verified with literature.

Synthesis of Fractional-Order Biquadratic Immittance Functions

2019 ◽  
Vol 28 (11) ◽  
pp. 1950187
Author(s):  
Guishu Liang ◽  
Xiaoyan Huo
Keyword(s):  
System Theory ◽  
Fractional Order ◽  
Network Synthesis ◽  
Fractional Order Calculus ◽  
Integer Order ◽  
Passive Network ◽  
Passive Networks ◽  
Circuit And System ◽  
Impedance Functions

Passive network synthesis, as an important part of circuit and system theory, has been well developed in integer-order circuits. With the development of fractional-order calculus and fractional-order elements, the problem of using fractional-order passive networks to realize fractional-order immittance functions has drawn much attention. In this paper, the realization of a fractional-order biquadratic immittance function is considered. First, the form of a fractional-order biquadratic function and some theorems that could promote later research are introduced. Second, a detailed study for the realization of a fractional-order biquadratic immittance function is shown. Finally, through summarizing the realizability conditions of each network, we have obtained the scope of fractional biquadratic impedance functions that can be realized by this paper.

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