A new definition of fractional derivative

Authors

Roshdi Khalil, Mohammed Al Horani, A. Yousef, Mohammad Sababheh

Type: Article

Publication Date: 2014-01-16

Citations: 2996

DOI: https://doi.org/10.1016/j.cam.2014.01.002

View

Abstract

We give a new definition of fractional derivative and fractional integral. The form of the definition shows that it is the most natural definition, and the most fruitful one. The definition for 0≤α<1 coincides with the classical definitions on polynomials (up to a constant). Further, if α=1, the definition coincides with the classical definition of first derivative. We give some applications to fractional differential equations.

Topics

Locations

Journal of Computational and Applied Mathematics
View

A new definition of fractional derivative

2018-10-20

Zhibao Zheng , Wei Zhao , Hongzhe Dai

Technical Note on a New Definition of Fractional Derivative

2017-07-01

Vincenzo Ciancio , Bruno Felice Filippo Flora

A New Fractional Derivative with Classical Properties

2014-01-01

Udita N. Katugampola

We introduce a new fractional derivative which obeys classical properties including: linearity, product rule, quotient rule, power rule, chain rule, vanishing derivatives for constant functions, the Rolle's Theorem and the … We introduce a new fractional derivative which obeys classical properties including: linearity, product rule, quotient rule, power rule, chain rule, vanishing derivatives for constant functions, the Rolle's Theorem and the Mean Value Theorem. The definition, \[ D^α(f)(t) = \lim_{ε\rightarrow 0} \frac{f(te^{εt^{-α}}) - f(t)}ε, \] is the most natural generalization that uses the limit approach. For $0\leq α< 1$, it generalizes the classical calculus properties of polynomials. Furthermore, if $α= 1$, the definition is equivalent to the classical definition of the first order derivative of the function $f$. Furthermore, it is noted that there are $α-$differentiable functions which are not differentiable.

A New Definition of Fractional Calculus

2023-09-04

Chii-Huei Yu

View PDF Chat

Fractional Calculus

2018-06-29

Ricardo Almeida , Dina Tavares , Delfim F. M. Torres

Introduction of Derivatives and Integrals of Fractional Order and Its Applications

2013-01-23

Mehdi Delkhosh

Fractional calculus is a branch of classical mathematics, which deals with the generalization of operations of differentiation and integration to fractional order. Such a generalization is not merely a mathematical … Fractional calculus is a branch of classical mathematics, which deals with the generalization of operations of differentiation and integration to fractional order. Such a generalization is not merely a mathematical curiosity but has found applications in various fields of physical sciences. In this paper, we review the definitions and properties of fractional derivatives and integrals, and we express the prove some of them.

The Fractional Derivative of Some Functions

2024-11-30

Louie Resti Rellon , Cymber Orvie Quevedo , Ruben D. Cordero +2 more

A Review of Definitions for Fractional Derivatives and Integral

2014-01-01

E. Capelas de Oliveira , J. A. Tenreiro Machado

This paper presents a review of definitions of fractional order derivatives and integrals that appear in mathematics, physics, and engineering. This paper presents a review of definitions of fractional order derivatives and integrals that appear in mathematics, physics, and engineering.

View PDF Chat

Fractional Calculus of Variations: A Novel Way to Look At It

2019-08-01

Rui A. C. Ferreira

View PDF Chat

Generalized Fractional Derivative, Fractional differential ring

2021-01-01

Zeinab Toghani , Luis Gaggero

There are many possible definitions of derivatives, here we present some and present one that we have called generalized that allows us to put some of the others as a … There are many possible definitions of derivatives, here we present some and present one that we have called generalized that allows us to put some of the others as a particular case of this but, what interests us is to determine that there is an infinite number of possible definitions of fractional derivatives, all are correct as differential operators each of which must be properly defined in its algebra. We introduce a generalized version of the fractional derivative that extends the existing ones in the literature. To those extensions, it is associated with a differentiable operator and a differential ring and applications that show the advantages of the generalization. We also review the different definitions of fractional derivatives proposed by Michele Caputo in \cite{GJI:GJI529}, Khalil, Al Horani, Yousef, Sababheh in \cite{khalil2014new}, Anderson and Ulness in \cite{anderson2015newly}, Guebbai and Ghiat in \cite{guebbai2016new}, Udita N. Katugampola in \cite{katugampola2014new}, Camrud in \cite{camrud2016conformable} and it is shown how the generalized version contains the previous ones as a particular case.

Some Basic Definitions Of Fractional Calculus

2015-10-01

Ramesh Chandra Bagadi

In this research monograph, the author presents some basic definitions of Fractional Derivative, Fractional integration and Functional Derivative and Functional Integral. In this research monograph, the author presents some basic definitions of Fractional Derivative, Fractional integration and Functional Derivative and Functional Integral.

Fractional Calculus of Variations: a novel way to look at it

2018-09-06

Rui A. C. Ferreira

In this work we look at the original fractional calculus of variations problem in a somewhat different way. As a simple consequence, we show that a fractional generalization of a … In this work we look at the original fractional calculus of variations problem in a somewhat different way. As a simple consequence, we show that a fractional generalization of a classical problem has a solution without any restrictions on the derivative-order $\alpha$.

Fractional Calculus of Variations: a novel way to look at it

2018-01-01

Rui A. C. Ferreira

View PDF Chat

On a Unified Fractional Derivative

2011-01-01

Manuel Duarte Ortigueira , Juan J. Trujillo

Fractional Integrals and Derivatives

2023-01-01

K. Balachandran

Chapter 2 Fractional integrals and fractional derivatives

2006-01-01

Novel Definitions of α-Fractional Integral and Derivative of the Functions

2023-11-30

Mohammed S. Mechee , Zahrah I. Salman , Shaymaa Y. Alkufi

An α-fractional integral and derivative of real function have been introduced in new definitions and then, they compared with the existing definitions. According to the properties of these definitions, the … An α-fractional integral and derivative of real function have been introduced in new definitions and then, they compared with the existing definitions. According to the properties of these definitions, the formulas demonstrate that they are most significant and suitable in fractional integrals and derivatives. The definitions of α-fractional derivative and integral coincide with the existing definitions for the polynomials for 0 ≤ α < 1. Furthermore, if α = 1, the proposed definitions and the usual definition of integer derivative and integral are identical. Some of the properties of the new definitions are discussed and proved, as well, we have introduced some applications in the α- fractional derivatives and integrals. Moreover, α-power series and α–rule of integration by parts have been proposed and implemented in this study.

View PDF Chat

A new fractional derivative extending classical concepts: Theory and applications

2024-08-24

Mutaz Mohammad , Mohamed Saadaoui

In this paper, a novel general definition for the fractional derivative and fractional integral based on an undefined kernel function is introduced. For 0 < α ≤ 1 , this … In this paper, a novel general definition for the fractional derivative and fractional integral based on an undefined kernel function is introduced. For 0 < α ≤ 1 , this definition aligns with classical interpretations and is applicable for calculating the derivative in an open negative interval I ⊆ [ a , + ∞ ) , a ∈ R . Additionally, when α = 1 , the definition coincides with the classical derivative. Fundamental properties of the fractional integral and derivative, including the product rule, quotient rule, chain rule, Rolle's theorem, and the mean value theorem, are derived. These properties are illustrated through various applications to demonstrate their applicability. Furthermore, some applications of solving fractional nonlinear systems of integro-differential equations using framelets are presented.

A unified approach to fractional derivatives

2012-05-23

Manuel Duarte Ortigueira , Juan J. Trujillo

A review of definitions of fractional derivatives and other operators

2019-03-18

Grazielle Sales Teodoro , J. A. Tenreiro Machado , E. Capelas de Oliveira

DISPERSION EQUATION OF ATMOSPHERIC POLLUTANTS WITH FRACTIONAL PARAMETER IN THE DIFFUSIVE TERM USING CONFORMABLE DERIVATIVE AN ANALYTICAL SOLUTION

2021-12-01

André Luiz Santos da Soledade , Paulo Henrique Farias Xavier , JOSÉ ROBERTO DANTAS DA SILVA +2 more

Este estudo tem como objetivo investigar o potencial de derivados fracionários na modelagem de dispersão atmosférica. Portanto, uma solução analítica da equação bidimensional de advecção-difusão fracionada é proposta usando métodos … Este estudo tem como objetivo investigar o potencial de derivados fracionários na modelagem de dispersão atmosférica. Portanto, uma solução analítica da equação bidimensional de advecção-difusão fracionada é proposta usando métodos GILTT e derivados conformáveis. A novidade deste estudo é a inserção de um parâmetro fracionário no termo difusivo considerando a derivada conformável, levando em consideração o comportamento anômalo no processo de difusão, resultando em uma nova metodologia aqui denominada método α-GILTT. As simulações foram comparadas com os dados moderadamente instáveis do experimento de Copenhagen e os melhores resultados são para o parâmetro fracionário α = 0.99.

View PDF Chat

On Leibniz type rule for generalized fractional derivatives

2024-08-26

Wael Abdelhedi

Optical solitons for the concatenation model with fractional temporal evolution

2024-12-27

Ahmed H. Arnous , Muhammad Amin S. Murad , Anjan Biswas +5 more

Conformable Fractional Nikiforov—Uvarov Method

2016-07-01

H. Karayer , D. Demirhan , Fevzi Büyükkılıç

We introduce conformable fractional Nikiforov—Uvarov (NU) method by means of conformable fractional derivative which is the most natural definition in non-integer calculus. Since, NU method gives exact eigenstate solutions of … We introduce conformable fractional Nikiforov—Uvarov (NU) method by means of conformable fractional derivative which is the most natural definition in non-integer calculus. Since, NU method gives exact eigenstate solutions of Schrödinger equation (SE) for certain potentials in quantum mechanics, this method is carried into the domain of fractional calculus to obtain the solutions of fractional SE. In order to demonstrate the applicability of the conformable fractional NU method, we solve fractional SE for harmonic oscillator potential, Woods—Saxon potential, and Hulthen potential.

New abundant wave solutions of the conformable space–time fractional (4+1)-dimensional Fokas equation in water waves

2019-04-11

Shoukry El‐Ganaini , Mohammed O. Al‐Amr

Experimental Study of Fractional-Order RC Circuit Model Using the Caputo and Caputo-Fabrizio Derivatives

2021-01-14

Da Lin , Xiaozhong Liao , Lei Dong +5 more

This study employs the Caputo-Fabrizio fractional derivative to determine the model of fractional-order RC circuits with arbitrary voltage input which can be widely used in a variety of electrical systems. … This study employs the Caputo-Fabrizio fractional derivative to determine the model of fractional-order RC circuits with arbitrary voltage input which can be widely used in a variety of electrical systems. Analog circuit implementation of fractional-order RC circuits defined by Caputo-Fabrizio fractional derivative is presented and verified by comparing with the model proposed in this work. For the purpose of judging whether the fractional-order model defined by the Caputo-Fabrizio derivative is practical, the comparison experiments are carried out. By using Laplace transform, the analytical solutions of fractional-order RC circuits based on the Caputo-Fabrizio derivatives with constant and periodic voltage sources are deduced. Fractional-order model of RC circuits with arbitrary input are also calculated using the convolution formula. The correctness of the derivation of the model using the Caputo-Fabrizio derivative is verified. Through discussing the impedance model of capacitor in frequency domain, the analog realization of fractional capacitor based on the Caputo-Fabrizio derivative is derived. The fractional-orders of the RC circuits models defined by the Caputo and Caputo-Fabrizio fractional derivatives are fitted respectively through repeated charging and discharging experiment data. The fractional-order models based on the Caputo and Caputo-Fabrizio derivatives, and the integer-order model are all compared with the experiment data.

Deep neural network methods for solving forward and inverse problems of time fractional diffusion equations with conformable derivative

2022-08-08

Yinlin Ye , Hongtao Fan , Yajing Li +2 more

View PDF Chat

New Explicit Solutions to the Fractional-Order Burgers’ Equation

2021-06-11

M. Hafiz Uddin , Mohammad Asif Arefin , M. Ali Akbar +1 more

The closed-form wave solutions to the time-fractional Burgers’ equation have been investigated by the use of the two variables <math xmlns="http://www.w3.org/1998/Math/MathML" id="M1"> <mfenced open="(" close=")" separators="|"> <mrow> <mfenced open="(" close=")" … The closed-form wave solutions to the time-fractional Burgers’ equation have been investigated by the use of the two variables <math xmlns="http://www.w3.org/1998/Math/MathML" id="M1"> <mfenced open="(" close=")" separators="|"> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <mrow> <mrow> <msup> <mrow> <mi>G</mi> </mrow> <mrow> <mo>′</mo> </mrow> </msup> </mrow> <mo>/</mo> <mi>G</mi> </mrow> </mrow> </mfenced> <mo>,</mo> <mfenced open="(" close=")" separators="|"> <mrow> <mrow> <mn>1</mn> <mo>/</mo> <mi>G</mi> </mrow> </mrow> </mfenced> </mrow> </mfenced> </math> -expansion, the extended tanh function, and the exp-function methods translating the nonlinear fractional differential equations (NLFDEs) into ordinary differential equations. In this article, we ascertain the solutions in terms of <math xmlns="http://www.w3.org/1998/Math/MathML" id="M2"> <mtext>tanh</mtext> </math> , <math xmlns="http://www.w3.org/1998/Math/MathML" id="M3"> <mtext>sech</mtext> </math> , <math xmlns="http://www.w3.org/1998/Math/MathML" id="M4"> <mtext>sinh</mtext> </math> , rational function, hyperbolic rational function, exponential function, and their integration with parameters. Advanced and standard solutions can be found by setting definite values of the parameters in the general solutions. Mathematical analysis of the solutions confirms the existence of different soliton forms, namely, kink, single soliton, periodic soliton, singular kink soliton, and some other types of solitons which are shown in three-dimensional plots. The attained solutions may be functional to examine unidirectional propagation of weakly nonlinear acoustic waves, the memory effect of the wall friction through the boundary layer, bubbly liquids, etc. The methods suggested are direct, compatible, and speedy to simulate using algebraic computation schemes, such as Maple, and can be used to verify the accuracy of results.

View PDF Chat

A reflection on the fractional order nuclear reactor dynamics

2019-04-16

M. Zarei

Solitary Wave Solution of a Generalized Fractional–Stochastic Nonlinear Wave Equation for a Liquid with Gas Bubbles

2023-04-01

Wael W. Mohammed , Farah M. Al‐Askar , Clemente Cesarano +1 more

In the sense of a conformable fractional operator, we consider a generalized fractional–stochastic nonlinear wave equation (GFSNWE). This equation may be used to depict several nonlinear physical phenomena occurring in … In the sense of a conformable fractional operator, we consider a generalized fractional–stochastic nonlinear wave equation (GFSNWE). This equation may be used to depict several nonlinear physical phenomena occurring in a liquid containing gas bubbles. The analytical solutions of the GFSNWE are obtained by using the F-expansion and the Jacobi elliptic function methods with the Riccati equation. Due to the presence of noise and the conformable derivative, some solutions that were achieved are shown together with their physical interpretations.

View PDF Chat

Modelling disease spread with spatio-temporal fractional derivative equations and saturated incidence rate

2023-04-08

Chouaib Bounkaicha , Karam Allali

View PDF Chat

Construction of Analytical Solutions to the Conformable New (3+1)-Dimensional Shallow Water Wave Equation

2023-06-30

Mehmet Şenol , Mehmet Gençyiğit

This study investigates the new (3+1)-dimensional shallow water wave equation. To do so, the definitions of conformable derivatives and their descriptions are given. Using the Riccati equation and modified Kudryashov … This study investigates the new (3+1)-dimensional shallow water wave equation. To do so, the definitions of conformable derivatives and their descriptions are given. Using the Riccati equation and modified Kudryashov methods, exact solutions to this problem are discovered. The gathered data's contour plot surfaces and related 3D and 2D surfaces emphasize the result's physical nature. To monitor the problem's physical activity, exact and complete solutions are necessary. The results demonstrate the potential applicability of additional nonlinear physical models from mathematical physics and under-investigation in real-world settings. In order to solve fractional differential equations, it may prove helpful to use these methods in various situations.

View PDF Chat

Dynamic study of Clannish Random Walker’s parabolic equation via extended direct algebraic method

2023-12-14

Naeem Ullah , Hamood Ur Rehman , Muhammad Imran Asjad +2 more

The Hille Yosida Theorem for Conformable Fractional Semi-Groups of Operators

2021-05-01

Sh. Al-Sharif , Mohammed Al Horani , Roshdi Khalil

In this paper, we extend the classical Hille Yosida Theorem to fractional semi-groups of operators on Banach spaces. In this paper, we extend the classical Hille Yosida Theorem to fractional semi-groups of operators on Banach spaces.

Generalized fractional Ostrowski type integral inequalities for logarithmically h-convex function

2021-03-18

Sabir Hussain , Faiza Azhar , Muhammad Amer Latif

Propagation of the ultra-short femtosecond pulses and the rogue wave in an optical fiber

2020-05-25

Maha S. M. Shehata , Hadi Rezazadeh , Emad H. M. Zahran +1 more

Modelo de operador fraccional para describir la dinámica de lossupercondensadores

2020-04-01

Diana Sofía Mendoza Contreras , Javier Solano , Rodrigo Correa

Este artículo propone un nuevo circuito equivalente para modelar supercondensadores. El circuito propuesto es unarreglo de circuitos RC serie descritos por ecuaciones diferenciales fraccionarias conformables. Se implementa unalgoritmo de identificación … Este artículo propone un nuevo circuito equivalente para modelar supercondensadores. El circuito propuesto es unarreglo de circuitos RC serie descritos por ecuaciones diferenciales fraccionarias conformables. Se implementa unalgoritmo de identificación de parámetros delcircuito equivalente, que utiliza como entrada datos experimentales. Losresultados de validación obtenidos muestran que un circuito equivalente que emplea el operador conformable puedeser utilizado para modelar el comportamiento real del supercondensador

View PDF Chat

Exact solutions of space–time fractional KdV–MKdV equation and Konopelchenko–Dubrovsky equation

2020-12-05

Bo Tang , Jiajia Tao , Shijun Chen +3 more

Abstract In the present study, we deal with the space–time fractional KdV–MKdV equation and the space–time fractional Konopelchenko–Dubrovsky equation in the sense of the conformable fractional derivative. By means of … Abstract In the present study, we deal with the space–time fractional KdV–MKdV equation and the space–time fractional Konopelchenko–Dubrovsky equation in the sense of the conformable fractional derivative. By means of the extend \left(\tfrac{G^{\prime} }{G}\right) -expansion method, many exact solutions are obtained, which include hyperbolic function solutions, trigonometric function solutions and rational solutions. The results show that the extend \left(\tfrac{G^{\prime} }{G}\right) -expansion method is an efficient technique for solving nonlinear fractional partial equations. We also provide some graphical representations to demonstrate the physical features of the obtained solutions.

View PDF Chat

Conformable fractional order controller design and optimization for sensorless control of induction motor

2022-02-11

Erdem İlten

Purpose In recent years, use of sensorless control methods for electrical motor-based variable speed drive systems has been increasing rapidly to compensate the increasing costs in industrial systems. Also, use … Purpose In recent years, use of sensorless control methods for electrical motor-based variable speed drive systems has been increasing rapidly to compensate the increasing costs in industrial systems. Also, use of induction motors is popular for a long time to decrease the cost of these industrial systems. This study aims to design an effective controller to improve the sensorless speed control performance of induction motor. To achieve this, a conformable fractional order proportional integral (CFOPI) controller is designed. Design/methodology/approach The system is modeled based on small signal analysis by using the input–output data, experimentally. To do this, system identification toolbox of Matlab is used. The proposed controller is established on conformable fractional integral approach proposed by Khalil et al. (2014). CFOPI controller coefficients are optimized using particle swarm optimization method on the created small signal-based simulation model of the system to minimize the integral time absolute error. To prove the success of the proposed method, a traditional fractional order proportional integral (TFOPI) controller is tested under the same experimental system with the CFOPI controller. Findings TFOPI and CFOPI controllers are tested with the optimum parameters. Reference and actual speed trends are obtained for both methods. In induction motor start-up test, settling-times are measured as 8.73 and 8.44 s and steady-state oscillations are 2.66% and 0% (almost) for TFOPI and CFOPI controllers, respectively. In variable referenced speed tracking test, CFOPI performs well at all speed levels, while TFOPI fails to reach the reference speed at most speed levels. Practical implications Proposed CFOPI control method can be easily implemented in industrial systems, thanks to its simple algorithm. digital signal peripheral interface controller (dsPIC) based driver circuit with designed CFOPI controller used in this study can be applied directly to industrial systems such as elevators, conveyors, cranes and drills. Moreover, it can improve the performance of induction motor-based variable speed drive systems. Originality/value The proposed method provides robust performance for induction motor used in control systems. Additionally, it does this by using less complex algorithm written on the processors according to the traditional fractional order controllers.

Ostrowski type inequalities involving conformable integrals via preinvex functions

2020-05-01

Yousaf Khurshid , Muhammad Adil Khan , Yu–Ming Chu

In this research article, we use preinvex functions to develop Ostrowski type inequalities for conformable integrals. First, we aim for an identity linked with the Ostrowski inequality. We obtain several … In this research article, we use preinvex functions to develop Ostrowski type inequalities for conformable integrals. First, we aim for an identity linked with the Ostrowski inequality. We obtain several results of Ostrowski type inequalities for conformal integrals by using some well-known inequalities, preinvexity, and the derived identity.

View PDF Chat

Special Fractional Curve Pairs with Fractional Calculus

2022-04-28

Aykut Has , Beyhan Yılmaz

In this study, the effect of fractional derivatives, whose application area is increasing day by day, on curve pairs is investigated. As it is known, there are not many studies … In this study, the effect of fractional derivatives, whose application area is increasing day by day, on curve pairs is investigated. As it is known, there are not many studies on a geometric interpretation of fractional calculus. When examining the effect of fractional analysis on a curve, the Conformable fractional derivative that fits the algebraic structure of differential geometry derivative is used. This effect is examined with the help of examples consistent with the theory and visualized for different values of the Conformable fractional derivative. The difference of this study from others is the use of Conformable fractional derivatives and integrals in calculations. Fractional calculus has applications in many fields such as physics, engineering, mathematical biology, fluid mechanics,signal processing, etc. Fractional derivatives and integrals have become an extremely important and new mathematical method in solving various problems in many sciences.

View PDF Chat

Opial-type inequalities for conformable fractional integrals

2019-11-20

Mehmet Zeki Sarıkaya , Hüseyin Budak

Abstract In this paper, we establish the Opial-type inequalities for a conformable fractional integral and give some results in special cases of α. The results presented here would provide generalizations … Abstract In this paper, we establish the Opial-type inequalities for a conformable fractional integral and give some results in special cases of α. The results presented here would provide generalizations of those given in earlier works.

Inverse problems for a conformable fractional Sturm–Liouville operator

2020-06-07

İbrahi̇m Adalar , A. Sinan Ozkan

In this paper, a Sturm-Liouville boundary value problem equiped with conformable fractional derivates is considered. We give some uniqueness theorems for the solutions of inverse problems according to the Weyl … In this paper, a Sturm-Liouville boundary value problem equiped with conformable fractional derivates is considered. We give some uniqueness theorems for the solutions of inverse problems according to the Weyl function, two given spectra and classical spectral data. We also study on half-inverse problem and prove a Hochstadt and Lieberman-type theorem.

View PDF Chat

On New Unified Bounds for a Family of Functions via Fractional<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mi>q</mml:mi></mml:math>-Calculus Theory

2020-07-16

Li Xu , Yu–Ming Chu , Saima Rashid +2 more

The present article deals with the new estimates in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mi>q</mml:mi></mml:math>-calculus and fractional<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mi>q</mml:mi></mml:math>-calculus on a time scale<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:msub><mml:mrow><mml:mi mathvariant="double-struck">T</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mfenced open="{" close="}"><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:mfenced><mml:mo>∪</mml:mo><mml:mfenced open="{" close="}"><mml:mrow><mml:mi>t</mml:mi><mml:mo>:</mml:mo><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mi>n</mml:mi><mml:mtext> </mml:mtext><mml:mtext>is</mml:mtext><mml:mtext> </mml:mtext><mml:mtext>a</mml:mtext><mml:mtext> </mml:mtext><mml:mtext>nonnegative</mml:mtext><mml:mtext> </mml:mtext><mml:mtext>integer</mml:mtext></mml:mrow></mml:mfenced><mml:mo>,</mml:mo></mml:math>where<mml:math … The present article deals with the new estimates in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mi>q</mml:mi></mml:math>-calculus and fractional<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mi>q</mml:mi></mml:math>-calculus on a time scale<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:msub><mml:mrow><mml:mi mathvariant="double-struck">T</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mfenced open="{" close="}"><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:mfenced><mml:mo>∪</mml:mo><mml:mfenced open="{" close="}"><mml:mrow><mml:mi>t</mml:mi><mml:mo>:</mml:mo><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mi>n</mml:mi><mml:mtext> </mml:mtext><mml:mtext>is</mml:mtext><mml:mtext> </mml:mtext><mml:mtext>a</mml:mtext><mml:mtext> </mml:mtext><mml:mtext>nonnegative</mml:mtext><mml:mtext> </mml:mtext><mml:mtext>integer</mml:mtext></mml:mrow></mml:mfenced><mml:mo>,</mml:mo></mml:math>where<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M5"><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>∈</mml:mo><mml:mi>ℝ</mml:mi></mml:math>and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M6"><mml:mn>0</mml:mn><mml:mo><</mml:mo><mml:mi>q</mml:mi><mml:mo><</mml:mo><mml:mn>1</mml:mn><mml:mo>.</mml:mo></mml:math>The role of fractional time scale<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M7"><mml:mi>q</mml:mi></mml:math>-calculus can be found as one of the prominent techniques to generate some variants for a class of positive functions<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M8"><mml:mi>n</mml:mi><mml:mtext> </mml:mtext><mml:mfenced open="(" close=")"><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi>ℕ</mml:mi></mml:mrow></mml:mfenced><mml:mo>.</mml:mo></mml:math>Finally, our work will provide foundation and motivation for further investigation on time-fractional<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M9"><mml:mi>q</mml:mi></mml:math>-calculus systems that have an intriguing application in quantum theory and special relativity theory.

View PDF Chat

Optical soliton solutions for time-fractional Fokas system in optical fiber by new Kudryashov approach

2023-03-20

Muhammad Amin S. Murad , Faraidun K. Hamasalh , Hajar F. Ismael

The modified extended tanh technique ruled to exploration of soliton solutions and fractional effects to the time fractional couple Drinfel'd–Sokolov–Wilson equation

2023-04-23

Md. Habibul Bashar , Hure Zannatul Mawa Mawa , Anita Biswas +3 more

The modified extended tanh technique is used to investigate the conformable time fractional Drinfel'd-Sokolov-Wilson (DSW) equation and integrate some precise and explicit solutions in this survey. The DSW equation was … The modified extended tanh technique is used to investigate the conformable time fractional Drinfel'd-Sokolov-Wilson (DSW) equation and integrate some precise and explicit solutions in this survey. The DSW equation was invented in fluid dynamics. The modified extended tanh technique executes to integrate the nonlinear DSW equation for achieve diverse solitonic and traveling wave envelops. Because of this, trigonometric, hyperbolic and rational solutions have been found with a few acceptable parameters. The dynamical behaviors of the obtained solutions in the pattern of the kink, bell, multi-wave, kinky lump, periodic lump, interaction lump, and kink wave types have been illustrated with 3D and density plots for arbitrary chose of the permitted parameters. By characterizing the particular benefits of the exemplified boundaries by the portrayal of sketches and by deciphering the actual events, we have laid out acceptable soliton plans and managed the actual significance of the acquired courses of action. New precise voyaging wave arrangements are unambiguously gained with the aid of symbolic computation using the procedures that have been announced. Therefore, the obtained outcomes expose that the projected schemes are very operative, easier and efficient on realizing natures of waves and also introducing new wave strategies to a diversity of NLEEs that occur within the engineering sector.

View PDF Chat

Novel analytical solutions and optical soliton structures of fractional-order perturbed Kaup–Newell model and its application

2022-10-15

Muhammad Arshad , Aly R. Seadawy , Ambreen Sarwar +1 more

The Kaup–Newell equation is used to model sub-picoseconds pulses that travel throughout optical fibers. The fractional-order perturbed Kaup–Newell model, which represents extensive waves parallel to the field of magnetic, is … The Kaup–Newell equation is used to model sub-picoseconds pulses that travel throughout optical fibers. The fractional-order perturbed Kaup–Newell model, which represents extensive waves parallel to the field of magnetic, is examined. In this paper, two analytical techniques named, improved F-expansion and generalized exp[Formula: see text]-expansion techniques, are employed and new analytical solutions in generalized forms like bright solitons, dark solitons, multi-peak solitons, peakon solitons, periodic solitons and further wave results are assembled. These soliton solutions and other waves findings have important applications in applied sciences. The configurations of some solutions are shown in the form of graphs through assigning precise values to parameters, and their dynamics are described. The illustrated novel structures of some solutions also assist engineers and scientists in better grasping the physical phenomena of this fractional model. A comparison analysis has been given to explain the originality of the current findings compared to the previously achieved results. The results of computer simulations show that the procedures described are effective, simple, and efficient.

Solitary wave solutions of Camassa–Holm and Degasperis–Procesi equations with Atangana’s conformable derivative

2023-02-28

Muhammad Shakeel , Aysha Bibi , Asim Zafar +1 more

Analysis of Kudryashov’s equation with conformable derivative via the modified Sardar sub-equation algorithm

2024-04-16

Muhammad Amin S. Murad , Waqas Ali Faridi , Mujahid Iqbal +3 more

In the present work, we utilize a new Sardar sub-equation approach, leading to the successful derivation of several exact solutions for the time-fractional Kudryashov's equation, which describes the propagation pulses … In the present work, we utilize a new Sardar sub-equation approach, leading to the successful derivation of several exact solutions for the time-fractional Kudryashov's equation, which describes the propagation pulses in optical fibers. These solutions encompass a range of categories, including singular, wave, bright, mixed dark-bright, and bell-shaped optical solutions. To effectively showcase these novel optical soliton solutions, we utilized contour plots, three-dimensional graphs, and three-dimensional surface plots. Through multiple graphical simulations, we provide a comprehensive demonstration of the dynamic behavior and physical significance of these optical solutions within the proposed model. Moreover, we investigate the magnitude of the time-fractional Kudryashov's equation by analyzing the influence of the fractional order derivative and the impact of the time parameter on the newly constructed optical solutions. Our findings highlight the versatility of the presented method, as it can readily be applied to other differential equations in various fields, such as non-linear optics and plasma physics. The proposed technique is a generalized form that incorporates various methods, including the improved Sardar sub-equation method, the modified Kudryashov method, the tanh-function extension method, and others. To the best of our knowledge, these solutions are novel and have not been reported in the literature and have potential application in nonlinear optics.

A Fractional-Order Chaotic System With Infinite Attractor Coexistence and its DSP Implementation

2020-01-01

Tianming Liu , Jiawu Yu , Huizhen Yan +1 more

In this article, the fractional calculus is introduced into a simplest memristive circuit to construct a new four-dimensional fractional-order chaotic system. Combining conformable differential definition and Adomian decomposition method (ADM) … In this article, the fractional calculus is introduced into a simplest memristive circuit to construct a new four-dimensional fractional-order chaotic system. Combining conformable differential definition and Adomian decomposition method (ADM) algorithm is used to solve the numerical solution of the system. The attractor coexistence of the fractional-order system is investigated from the attractor phase diagram, coexistence bifurcation model, coexistence Lyapunov exponent spectrum and attractor basin. In addition, the hardware circuit of the system is implemented on the DSP platform. The simulation results show that the fractional-order chaotic system exhibits rich dynamic characteristics. In particular, the initial value of the system could control the offset, amplitude and frequency of the attractor better, and increase the complexity and randomness of the chaotic sequences. The research provides theoretical basis and guidance for the applications of fractional-order chaotic system.

View PDF Chat

On the optical soliton solutions of time-fractional Biswas–Arshed equation including the beta or M-truncated derivatives

2023-01-04

Melih Çinar , Aydın Seçer , Mustafa Bayram

WKB approximation with conformable operator

2022-07-20

Mohamed Ghaleb Al-Masaeed , Eqab M. Rabei , A. Al-Jamel

In this paper, the WKB method is extended to be applicable for conformable Hamiltonian systems where the concept of conformable operator with fractional order $\alpha$ is used. The WKB approximation … In this paper, the WKB method is extended to be applicable for conformable Hamiltonian systems where the concept of conformable operator with fractional order $\alpha$ is used. The WKB approximation for the $\alpha$-wavefunction is derived when the potential is slowly varying in space. The paper is furnished with some illustrative examples to demonstrate the method. The quantities of the conformable form are found to be inexact agreement with traditional quantities when $\alpha=1$.

View PDF Chat

Application of Caputo Fractional Derivatives to the Convective Flow of Casson Fluids in a Microchannel with Thermal Radiation

2022-03-25

Marjan Mohd Daud , Lim Yeou Jiann , Rahimah Mahat +1 more

In this paper, the application of Caputo fractional derivative on unsteady boundary layer Casson fluid flow in a microchannel is studied. The partial differential equations which governed the problem are … In this paper, the application of Caputo fractional derivative on unsteady boundary layer Casson fluid flow in a microchannel is studied. The partial differential equations which governed the problem are considered with the presence of thermal radiation. The fractional partial differential equations are transformed into dimensionless governing equations using appropriate dimensionless variables. It is then solved analytically using the Laplace transform technique which transforms the equations into linear ordinary differential equations. These transformed equations are then solved using the appropriate method, and the inverse Laplace transform technique is applied to obtain the solution in form of velocity and temperature profiles. Graphical illustrations are acquired using Mathcad software and the influence of important physical parameters on velocity and temperature profiles are analyzed. Results show that thermal radiation and fractional parameter have enhanced the velocity and temperature profiles.

View PDF Chat

A novel Elzaki transform homotopy perturbation method for solving time-fractional non-linear partial differential equations

2022-11-22

Sajad Iqbal , Francisco Martínez , Mohammed K. A. Kaabar +1 more

Abstract This paper presents the solution of important types of non-linear time-fractional partial differential equations via the conformable Elzaki transform Homotopy perturbation method. We apply the proposed technique to solve … Abstract This paper presents the solution of important types of non-linear time-fractional partial differential equations via the conformable Elzaki transform Homotopy perturbation method. We apply the proposed technique to solve four types of non-linear time-fractional partial differential equations. In addition, we establish the results on the uniqueness and convergence of the solution. Finally, the numerical results for a variety of α values are briefly examined. The proposed method performs well in terms of simplicity and efficiency.

View PDF Chat

A Review of Hermite–Hadamard Inequality for α-Type Real-Valued Convex Functions

2022-04-19

Ohud Almutairi , Adem Kılıçman

Inequalities play important roles not only in mathematics but also in other fields, such as economics and engineering. Even though many results are published as Hermite–Hadamard (H-H)-type inequalities, new researchers … Inequalities play important roles not only in mathematics but also in other fields, such as economics and engineering. Even though many results are published as Hermite–Hadamard (H-H)-type inequalities, new researchers to these fields often find it difficult to understand them. Thus, some important discoverers, such as the formulations of H-H-type inequalities of α-type real-valued convex functions, along with various classes of convexity through differentiable mappings and for fractional integrals, are presented. Some well-known examples from the previous literature are used as illustrations. In the many above-mentioned inequalities, the symmetrical behavior arises spontaneously.

View PDF Chat

The Modified Exponential Function Method for Beta Time Fractional Biswas-Arshed Equation

2023-04-27

Yusuf Pandır , Tolga Aktürk , Yusuf Gürefe +1 more

In this study, the exact solutions of the Biswas-Arshed equation with the beta time derivative, which has an important role and physically means that it represents the pulse propagation in … In this study, the exact solutions of the Biswas-Arshed equation with the beta time derivative, which has an important role and physically means that it represents the pulse propagation in an optical fiber, nuclear, and particle physics, are obtained using the modified exponential function method. Exact solutions consisting of hyperbolic, trigonometric, rational trigonometric, and rational function solutions demonstrate the competence and relevance of the proposed method. In addition, the physical properties of the obtained exact solutions are shown by making graphical representations according to different parameter values. It is seen that the method used is an effective technique, since these solution functions obtained with all these cases have periodic function properties.

View PDF Chat

Consistent travelling wave characteristic of space–time fractional modified Benjamin–Bona–Mahony and the space–time fractional Duffing models

2024-01-30

Mohammad Asif Arefin , U.H.M. Zaman , M. Hafiz Uddin +1 more

Abstract Study on solitary wave phenomenon are closely related on the dynamics of the plasma and optical fiber system, which carry on broad range of wave propagation. The space–time fractional … Abstract Study on solitary wave phenomenon are closely related on the dynamics of the plasma and optical fiber system, which carry on broad range of wave propagation. The space–time fractional modified Benjamin–Bona–Mahony equation and Duffing model are important modeling equations in acoustic gravity waves, cold plasma waves, quantum plasma in mechanics, elastic media in nonlinear optics, and the damping of material waves. This study has effectively developed analytical wave solutions to the aforementioned models, which may have significant consequences for characterizing the nonlinear dynamical behavior related to the phenomenon. Conformable derivatives are used to narrate the fractional derivatives. The expanded tanh-function method is used to look into such kinds of resolutions. An ansatz for analytical traveling wave solutions of certain nonlinear evolution equations was originally a power sequence in tanh. The discovered explanations are useful, reliable, and applicable to chaotic vibrations, problems of optimal control, bifurcations to global and local, also resonances, as well as fusion and fission phenomena in solitons, scalar electrodynamics, the relation of relativistic energy–momentum, electromagnetic interactions, theory of one-particle quantum relativistic, and cold plasm. The solutions are drafted in 3D, contour, listpoint, and 2D patterns, and include multiple solitons, bell shape, kink type, single soliton, compaction solitary wave, and additional sorts of solutions. With the aid of Maple and MATHEMATICA, these solutions were verified and discovered that they were correct. The mentioned method applied for solving NLFPDEs has been designed to be practical, straightforward, rapid, and easy to use.

View PDF Chat

New k-Conformable Fractional Integral Inequalities

2020-10-12

Muhammad Uzair Awan , Muhammad Aslam Noor , Sadia Talib +2 more

Some Results on the Oscillatory Behavior of Integro-differential Equations

2021-12-22

Raziye Mert , Selami Bayeğ

In this paper, we investigate the oscillation of a class of generalized proportional fractional integro-differential equations with forcing term. We present sufficient conditions to prove some oscillation criteria in both … In this paper, we investigate the oscillation of a class of generalized proportional fractional integro-differential equations with forcing term. We present sufficient conditions to prove some oscillation criteria in both of the Riemann-Liouville and Caputo cases. Besides, we present some numerical examples for applicability of our results.

View PDF Chat

A new fractional-order chaotic system with plane equilibrium: Bifurcation analysis, Multi-stability and DSP implementation

2021-01-01

Tianming Liu , Xiaoyu Zhang , Peng Li +1 more

In this paper, based on the definition of conformable differential, a 4D chaotic system is discretized by Adomian decomposition method (ADM) and its numerical solution is obtained. The stability of … In this paper, based on the definition of conformable differential, a 4D chaotic system is discretized by Adomian decomposition method (ADM) and its numerical solution is obtained. The stability of the system at the equilibrium point is analyzed. From the coexisting phase diagram, bifurcation model

View PDF Chat

An Introduction to the Fractional Calculus and Fractional Differential Equations

1993-05-19

Kenneth S. Miller , Bertram Ross

Historical Survey The Modern Approach The Riemann-Liouville Fractional Integral The Riemann-Liouville Fractional Calculus Fractional Differential Equations Further Results Associated with Fractional Differential Equations The Weyl Fractional Calculus Some Historical Arguments. Historical Survey The Modern Approach The Riemann-Liouville Fractional Integral The Riemann-Liouville Fractional Calculus Fractional Differential Equations Further Results Associated with Fractional Differential Equations The Weyl Fractional Calculus Some Historical Arguments.