On an initial inverse problem for a diffusion equation with a conformable derivative

Binh, Tran Thanh; Luc, Nguyen Hoang; O’Regan, Donal; Can, Nguyen Huu

doi:10.1186/s13662-019-2410-z

Cited by 10 publications

(7 citation statements)

References 30 publications

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“…Recently, the authors in [36] and [37] have studied the positive mild solutions of (1.1) with Caputo fractional derivative by using the characteristics of positive operators, semigroups and the monotone iterative scheme. Binh et al [12] have considered the initial inverse problem for a diusion equation with a conformable derivative in a general bounded domain. Tuan et al [44] have studied a backward problem for a nonlinear diusion equation with a conformable derivative in the case of multidimensional and discrete data.…”

Section: Introductionmentioning

confidence: 99%

Monotone Iterative Technique for Nonlinear Periodic Time Fractional Parabolic Problems

Alaoui

Azroul

Hamou

2020

Advances in the Theory of Nonlinear Analysis and Its Application

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In this paper, existence and uniqueness of weak solutions for a linear parabolic problem with conformable derivative are proved, the existence of weak periodic solutions for conformable fractional parabolic nonlinear dierential equation is proved by using a more generalized monotone iterative method combined with the method of upper and lower solutions. We prove the convergence of monotone sequence to weak periodic minimal and maximal solutions. Moreover, the conformable version of the Lions-Magness and AubinLions lemmas are also proved.

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Section: Introductionmentioning

confidence: 99%

Monotone Iterative Technique for Nonlinear Periodic Time Fractional Parabolic Problems

Alaoui

Azroul

Hamou

2020

Advances in the Theory of Nonlinear Analysis and Its Application

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“…This novel fractional derivative is very simple and verifies all the properties of the classical deriva-tive. Actually, the conformable fractional derivative becomes the subject of many research contributions [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39].…”

Section: Introductionmentioning

confidence: 99%

Controllability of Mild Solution of Nonlocal Conformable Fractional Differential Equations

Hannabou

Bouaouid

Hilal

2022

Advances in Mathematical Physics

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In many research works Bouaouid et al. have proved the existence of mild solutions of an abstract class of nonlocal conformable fractional Cauchy problem of the form: d α x t / d t α = A x t + f t , x t , x 0 = x 0 + g x , t ∈ 0 , τ . The present paper is a continuation of these works in order to study the controllability of mild solution of the above Cauchy problem. Precisely, we shall be concerned with the controllability of mild solution of the following Cauchy problem d α x t / d t α = A x t + f t , x t + B u t , x 0 = x 0 + g x , t ∈ 0 , τ , where d α . / d t α is the vectorial conformable fractional derivative of order α ∈ 0 , 1 in a Banach space X and A is the infinitesimal generator of a semigroup T t t ≥ 0 on X . The element x 0 is a fixed vector in X and f , g are given functions. The control function u is an element of L 2 0 , τ , U with U is a Banach space and B is a bounded linear operator from U into X .

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“…For example, when the initial temperature or the final temperature for heat equation is not given immediately, but there is information regarding the temperature over a given period of time that can be described by a nonlocal initial condition. PDEs with nonlocal conditions were considered in many works, for example, see [25] for reaction-diffusion equations and [26][27][28][29][30][31][32][33][34][35] for some other PDEs. As we said before, there are not any results for considering our model (1.1) with the nonlocal final condition and the integral condition…”

Section: Introductionmentioning

confidence: 99%

On a time fractional diffusion with nonlocal in time conditions

Tuấn

Triet²,

Luc³

et al. 2021

Adv Differ Equ

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In this work, we consider a fractional diffusion equation with nonlocal integral condition. We give a form of the mild solution under the expression of Fourier series which contains some Mittag-Leffler functions. We present two new results. Firstly, we show the well-posedness and regularity for our problem. Secondly, we show the ill-posedness of our problem in the sense of Hadamard. Using the Fourier truncation method, we construct a regularized solution and present the convergence rate between the regularized and exact solutions.

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On an initial inverse problem for a diffusion equation with a conformable derivative

Cited by 10 publications

References 30 publications

Monotone Iterative Technique for Nonlinear Periodic Time Fractional Parabolic Problems

Monotone Iterative Technique for Nonlinear Periodic Time Fractional Parabolic Problems

Controllability of Mild Solution of Nonlocal Conformable Fractional Differential Equations

On a time fractional diffusion with nonlocal in time conditions

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