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 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 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 No local L1 solutions for semilinear fractional heat equations

2017 ◽  
Vol 20 (6) ◽  
Author(s):  
Kexue Li
Keyword(s):  
Cauchy Problem ◽  
Heat Equation ◽  
Heat Equations ◽  
Fractional Heat Equation ◽  
The Cauchy Problem

AbstractWe study the Cauchy problem for the semilinear fractional heat equation

scholarly journals Fourier Analysis with Generalized Integration

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1199
Author(s):  
Juan H. Arredondo ◽  
Manuel Bernal ◽  
María Guadalupe Morales
Keyword(s):  
Fourier Transform ◽  
Fourier Analysis ◽  
Integral Theory ◽  
Dense Subspace ◽  
Fourier Cosine ◽  
Sine Transform ◽  
Integrable Functions ◽  
Fourier Sine Transform ◽  
Fourier Cosine Transform ◽  
The Fourier Transform

We generalize the classic Fourier transform operator F p by using the Henstock–Kurzweil integral theory. It is shown that the operator equals the H K -Fourier transform on a dense subspace of L p , 1 < p ≤ 2 . In particular, a theoretical scope of this representation is raised to approximate the Fourier transform of functions on the mentioned subspace numerically. Besides, we show the differentiability of the Fourier transform function F p ( f ) under more general conditions than in Lebesgue’s theory. Additionally, continuity of the Fourier Sine transform operator into the space of Henstock-Kurzweil integrable functions is proved, which is similar in spirit to the already known result for the Fourier Cosine transform operator. Because our results establish a representation of the Fourier transform with more properties than in Lebesgue’s theory, these results might contribute to development of better algorithms of numerical integration, which are very important in applications.

A new difference scheme for time fractional heat equations based on the Crank-Nicholson method

2013 ◽  
Vol 16 (4) ◽  
Author(s):  
Ibrahim Karatay ◽  
Nurdane Kale ◽  
Serife Bayramoglu
Keyword(s):  
Diffusion Equation ◽  
Heat Equation ◽  
Difference Scheme ◽  
Numerical Solution ◽  
Caputo Derivative ◽  
Numerical Experiments ◽  
Time Derivative ◽  
Heat Equations ◽  
First Order ◽  
Fractional Heat Equation

AbstractIn this paper, we consider the numerical solution of a time-fractional heat equation, which is obtained from the standard diffusion equation by replacing the first-order time derivative with the Caputo derivative of order α, where 0 < α < 1. The main purpose of this work is to extend the idea on the Crank-Nicholson method to the time-fractional heat equations. By the method of the Fourier analysis, we prove that the proposed method is stable and the numerical solution converges to the exact one with the order O(τ 2-α + h 2), conditionally. Numerical experiments are carried out to support the theoretical claims.

FOURIER SINE AND COSINE TRANSFORMS ON BOEHMIAN SPACES

2013 ◽  
Vol 06 (01) ◽  
pp. 1350005 ◽  
Author(s):  
R. Roopkumar ◽  
E. R. Negrin ◽  
C. Ganesan
Keyword(s):  
Fourier Transform ◽  
Cosine Transform ◽  
Fourier Cosine ◽  
Sine Transform ◽  
Fourier Sine ◽  
Fourier Sine Transform ◽  
Fourier Cosine Transform ◽  
One To One ◽  
The Fourier Transform

We construct suitable Boehmian spaces which are properly larger than [Formula: see text] and we extend the Fourier sine transform and the Fourier cosine transform more than one way. We prove that the extended Fourier sine and cosine transforms have expected properties like linear, continuous, one-to-one and onto from one Boehmian space onto another Boehmian space. We also establish that the well known connection among the Fourier transform, Fourier sine transform and Fourier cosine transform in the context of Boehmians. Finally, we compare the relations among the different Boehmian spaces discussed in this paper.

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 .

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