scholarly journals On some properties of the conformable fractional derivative

2020 ◽  
Vol 6 (2) ◽  
pp. 210-217
Author(s):  
Radouane Azennar ◽  
Driss Mentagui
Keyword(s):  
Fractional Derivative ◽  
Conformable Fractional Derivative ◽  
Intermediate Value Theorem ◽  
Definition Of ◽  
Intermediate Value

AbstractIn this paper, we prove that the intermediate value theorem remains true for the conformable fractional derivative and we prove some useful results using the definition of conformable fractional derivative given in R. Khalil, M. Al Horani, A. Yousef, M. Sababhehb [4].

scholarly journals The Limited Validity of the Conformable Euler Finite Difference Method and an Alternate Definition of the Conformable Fractional Derivative to Justify Modification of the Method

2021 ◽  
Vol 26 (4) ◽  
pp. 66
Author(s):  
Dominic Clemence-Mkhope ◽  
Belinda Clemence-Mkhope
Keyword(s):  
Finite Difference ◽  
Fractional Derivative ◽  
Initial Value Problem ◽  
Euler Method ◽  
Market Model ◽  
Alternative Methods ◽  
Conformable Fractional Derivative ◽  
Initial Value ◽  
Relaxation Equation ◽  
Definition Of

A method recently advanced as the conformable Euler method (CEM) for the finite difference discretization of fractional initial value problem Dtαyt = ft;yt, yt0 = y0, a≤t≤b, and used to describe hyperchaos in a financial market model, is shown to be valid only for α=1. The property of the conformable fractional derivative (CFD) used to show this limitation of the method is used, together with the integer definition of the derivative, to derive a modified conformable Euler method for the initial value problem considered. A method of constructing generalized derivatives from the solution of the non-integer relaxation equation is used to motivate an alternate definition of the CFD and justify alternative generalizations of the Euler method to the CFD. The conformable relaxation equation is used in numerical experiments to assess the performance of the CEM in comparison to that of the alternative methods.

scholarly journals A New Conformable Fractional Derivative and Applications

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Ahmed Kajouni ◽  
Ahmed Chafiki ◽  
Khalid Hilal ◽  
Mohamed Oukessou
Keyword(s):  
Fractional Derivative ◽  
Fractional Derivatives ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Conformable Fractional Derivative ◽  
First Order ◽  
Rolle’S Theorem ◽  
Power Rule ◽  
The Mean ◽  
Definition Of

This paper is motivated by some papers treating the fractional derivatives. We introduce a new definition of fractional derivative which obeys classical properties including linearity, product rule, quotient rule, power rule, chain rule, Rolle’s theorem, and the mean value theorem. The definition D α f t = lim h ⟶ 0 f t + h e α − 1 t − f t / h , for all t > 0 , and α ∈ 0,1 . If α = 0 , this definition coincides to the classical definition of the first order of the function f .

scholarly journals The solutions of time and space conformable fractional heat equations with conformable Fourier transform

2015 ◽  
Vol 7 (2) ◽  
pp. 130-140 ◽  
Author(s):  
Yücel Çenesiz ◽  
Ali Kurt
Keyword(s):  
Fourier Transform ◽  
Heat Equation ◽  
Fractional Derivative ◽  
Heat Equations ◽  
Conformable Fractional Derivative ◽  
Time And Space ◽  
Fourier Cosine ◽  
Fractional Heat Equation ◽  
Fourier Cosine Transform ◽  
Definition Of

Abstract In this paper our aim is to find the solutions of time and space fractional heat differential equations by using new definition of fractional derivative called conformable fractional derivative. Also based on conformable fractional derivative definition conformable Fourier Transform is defined. Fourier sine and Fourier cosine transform definitions are given and space fractional heat equation is solved by conformable Fourier transform.

scholarly journals Approximate integration through remarkable points using the Intermediate Value Theorem

2020 ◽  
Vol 25 (1) ◽  
pp. 142-149
Author(s):  
Jaime Castro Pérez ◽  
Andrés González Nucamendi ◽  
Gerardo Pioquinto Aguilar Sánchez
Keyword(s):  
Numerical Integration ◽  
Polynomial Interpolation ◽  
Gaussian Quadrature ◽  
Middle Point ◽  
Intermediate Value Theorem ◽  
Riemann Sums ◽  
Riemann Sum ◽  
Definition Of ◽  
Intermediate Value ◽  
Approximate Integration

Using the Intermediate Value Theorem we demonstrate the rules of Trapeze and Simpson's. Demonstrations with this approach and its generalization to new formulas are less laborious than those resulting from methods such as polynomial interpolation or Gaussian quadrature. In addition, we extend the theory of approximate integration by finding new approximate integration formulas. The methodology we used to obtain this generalization was to use the definition of the integral defined by Riemann sums. Each Riemann sum provides an approximation of the result of an integral. With the help of the Intermediate Value Theorem and a detailed analysis of the Middle Point, Trapezoidal and Simpson Rules we note that these rules of numerical integration are Riemann sums. The results we obtain with this analysis allowed us to generalize each of the rules mentioned above and obtain new rules of approximation of integrals. Since each of the rules we obtained uses a point in the interval we have called them according to the point of the interval we take. In conclusion we can say that the method developed here allows us to give new formulas of numerical integration and generalizes those that already exist.

scholarly journals Solution of One Dimensional Conformable fractional Heat Equation

2020 ◽  
Vol 9 (5) ◽  
pp. 693-695
Keyword(s):  
Heat Equation ◽  
Fractional Derivative ◽  
Conformable Fractional Derivative ◽  
One Dimensional ◽  
Fractional Heat Equation ◽  
Definition Of

The aim of this paper is to present the solution of one dimensional conformable fractional heat equation by applying conformable fractional derivative which is considered as more convenient definition of fractional derivative.

scholarly journals Fuzzy Generalized Conformable Fractional Derivative

2020 ◽  
Vol 2020 ◽  
pp. 1-7 ◽  
Author(s):  
Atimad Harir ◽  
Said Melliani ◽  
Lalla Saadia Chadli
Keyword(s):  
Fractional Derivative ◽  
Conformable Fractional Derivative ◽  
Definition Of

We give a new definition of fuzzy fractional derivative called fuzzy conformable fractional derivative. Using this definition, we prove some results and we introduce new definition of generalized fuzzy conformable fractional derivative.

scholarly journals Improvement on Conformable Fractional Derivative and Its Applications in Fractional Differential Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Feng Gao ◽  
Chunmei Chi
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Caputo Fractional Derivative ◽  
Physical Meaning ◽  
Fractional Integral ◽  
Conformable Fractional Derivative ◽  
Conformable Derivative ◽  
Fractional Differential ◽  
Definition Of

In this paper, we made improvement on the conformable fractional derivative. Compared to the original one, the improved conformable fractional derivative can be a better replacement of the classical Riemann-Liouville and Caputo fractional derivative in terms of physical meaning. We also gave the definition of the corresponding fractional integral and illustrated the applications of the improved conformable derivative to fractional differential equations by some examples.

Hybrid Method for Solution of Fractional Order Linear Differential Equation with Variable Coefficients

2018 ◽  
Vol 19 (6) ◽  
pp. 621-626
Author(s):  
Amit Ujlayan ◽  
Ajay Dixit
Keyword(s):  
Fractional Derivative ◽  
Hybrid Method ◽  
Hybrid Methods ◽  
Variable Coefficients ◽  
Order Linear Differential Equation ◽  
Conformable Fractional Derivative ◽  
Variation Of Parameters ◽  
Fractional Differential ◽  
Linear Differential ◽  
Definition Of

AbstractIn this paper, we proposed a new analytical hybrid methods for the solution of conformable fractional differential equations (CFDE), which are based on the recently proposed conformable fractional derivative (CFD) in R. Khalil, M. Al Horani, A. Yusuf and M. Sababhed, A New definition of fractional derivative, J. Comput. Appl. 264 (2014). Moreover, we use the method of variation of parameters and reduction of order based on CFD, for the CFDE. Furthermore, to show the efficiency of the proposed analytical hybrid method, some examples are also presented.

scholarly journals Motion equations and non-Noether symmetries of Lagrangian systems with conformable fractional derivative

2021 ◽  
pp. 35-35
Author(s):  
Jing-Li Fu ◽  
Lijun Zhang ◽  
Chaudry Khalique ◽  
Ma-Li Guo
Keyword(s):  
Fractional Derivative ◽  
Fractional Derivatives ◽  
Lagrangian Systems ◽  
Conserved Quantities ◽  
Conformable Fractional Derivative ◽  
Noether Symmetries ◽  
Hamilton’S Principle ◽  
Hamilton's Principle ◽  
Motion Equations ◽  
Definition Of

In this paper, we present the fractional motion equations and fractional non-Noether symmetries of Lagrangian systems with the conformable fractional derivatives. The exchanging relationship between isochronous variation and fractional derivative, and the fractional Hamilton's principle of the holonomic conservative and non-conservative systems under the conformable fractional derivative are proposed. Then the fractional motion equations of these systems based on the Hamilton's principle are established. The fractional Euler operator, the definition of fractional non-Noether symmetries, non-Noether theorem and Hojman's conserved quantities for the Lagrangian systems are obtained with conformable fractional derivative. An example is given to illustrate the results.

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