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 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 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 Controllability of the one-dimensional fractional heat equation under positivity constraints

2020 ◽  
Vol 19 (4) ◽  
pp. 1949-1978 ◽  
Author(s):  
Umberto Biccari ◽  
◽  
Mahamadi Warma ◽  
Enrique Zuazua ◽  
◽  
...  
Keyword(s):  
Heat Equation ◽  
One Dimensional ◽  
Fractional Heat Equation ◽  
The One

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 Controllability of a one-dimensional fractional heat equation: theoretical and numerical aspects

2018 ◽  
Vol 36 (4) ◽  
pp. 1199-1235 ◽  
Author(s):  
Umberto Biccari ◽  
Víctor Hernández-Santamaría
Keyword(s):  
Heat Equation ◽  
Approximate Controllability ◽  
Fractional Laplacian ◽  
Controllability Problem ◽  
Finite Element Scheme ◽  
Element Scheme ◽  
One Dimensional ◽  
Fractional Heat Equation ◽  
Hilbert Uniqueness Method ◽  
Uniqueness Method

Abstract We analyse the controllability problem for a one-dimensional heat equation involving the fractional Laplacian $(-d_x^{\,2})^{s}$ on the interval $(-1,1)$. Using classical results and techniques, we show that, acting from an open subset $\omega \subset (-1,1)$, the problem is null-controllable for $s>1/2$ and that for $s\leqslant 1/2$ we only have approximate controllability. Moreover, we deal with the numerical computation of the control employing the penalized Hilbert Uniqueness Method and a finite element scheme for the approximation of the solution to the corresponding elliptic equation. We present several experiments confirming the expected controllability properties.

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