Browsing by Author "Le Dinh Long"

Now showing 1 - 7 of 7

Citation - WoS: 2
Citation - Scopus: 2
Fractional Evolution Equation With Cauchy Data in L^p Spaces
(Springer, 2022) Baleanu, Dumitru; Agarwal, Ravi P.; Le Dinh Long; Nguyen Duc Phuong; Long, Le Dinh; Phuong, Nguyen Duc
In this paper, we consider the Cauchy problem for fractional evolution equations with the Caputo derivative. This problem is not well posed in the sense of Hadamard. There have been many results on this problem when data is noisy in L-2 and H-s,H- However, there have not been any papers dealing with this problem with observed data in L-p with p not equal 2. We study three cases of source functions: homogeneous case, inhomogeneous case, and nonlinear case. For all of them, we use a truncation method to give an approximate solution to the problem. Under different assumptions on the smoothness of the exact solution, we get error estimates between the regularized solution and the exact solution in L-p. To our knowledge, L-p evaluations for the inverse problem are very limited. This work generalizes some recent results on this problem.
Citation - WoS: 7
Citation - Scopus: 7
Identifying the Initial Condition for Space-Fractional Sobolev Equation
(Wilmington Scientific Publisher, Llc, 2021) Le Dinh Long; Le Thi Diem Hang; Baleanu, Dumitru; Nguyen Huu Can; Nguyen Hoang Luc; Long, Le Dinh; Luc, Nguyen Hoang; Hang, Le Thi Diem; Can, Nguyen Huu
In this work, a final value problem for a fractional pseudo-parabolic equation is considered. Firstly, we present the regularity of solution. Secondly, we show that this problem is ill-posed in Hadamard's sense. After that we use the quasi-boundary value regularization method to solve this problem. To show that the proposed theoretical results are appropriate, we present an illustrative numerical example.
Citation - WoS: 20
Citation - Scopus: 24
Inverse Source Problem for Time Fractional Diffusion Equation With Mittag-Leffler Kernel
(Springer, 2020) Nguyen Hoang Luc; Baleanu, Dumitru; Zhou, Yong; Le Dinh Long; Nguyen Huu Can; Long, Le Dinh; Can, Nguyen Huu; Luc, Nguyen Hoang
In this work, we study the problem to identify an unknown source term for the Atangana-Baleanu fractional derivative. In general, the problem is severely ill-posed in the sense of Hadamard. We have applied the generalized Tikhonov method to regularize the instable solution of the problem. In the theoretical result, we show the error estimate between the regularized and exact solutions with a priori parameter choice rules. We present a numerical example to illustrate the theoretical result. According to this example, we show that the proposed regularization method is converged.
Citation - WoS: 4
Citation - Scopus: 4
Recovering the Source Term for Parabolic Equation With Nonlocal Integral Condition
(Wiley, 2021) Baleanu, Dumitru; Tran Thanh Phong; Le Dinh Long; Nguyen Duc Phuong; Thanh Phong, Tran; Duc Phuong, Nguyen; Dinh Long, Le; Long, Le Dinh; Phuong, Nguyen Duc; Phong, Tran Thanh
The main purpose of this article is to present a Tikhonov method to construct the source function f(x) of the parabolic diffusion equation. This problem is well known to be severely ill-posed. Therefore, regularization is required. The error estimates between the sought solution and the regularized solution are obtained under an a priori parameter choice rule and an a posteriori parameter choice rule, respectively. One numerical test illustrates that the proposed method is feasible and effective.
Citation - WoS: 6
Citation - Scopus: 8
Recovering the Space Source Term for the Fractional-Diffusion Equation With Caputo-Fabrizio Derivative
(Springer, 2021) Nguyen Hoang Luc; Baleanu, Dumitru; Le Dinh Long; Le Nhat Huynh; Long, Le Dinh; Huynh, Le Nhat; Luc, Nguyen Hoang
This article is devoted to the study of the source function for the Caputo-Fabrizio time fractional diffusion equation. This new definition of the fractional derivative has no singularity. In other words, the new derivative has a smooth kernel. Here, we investigate the existence of the source term. Through an example, we show that this problem is ill-posed (in the sense of Hadamard), and the fractional Landweber method and the modified quasi-boundary value method are used to deal with this inverse problem and the regularized solution is also obtained. The convergence estimates are addressed for the regularized solution to the exact solution by using an a priori regularization parameter choice rule and an a posteriori parameter choice rule. In addition, we give a numerical example to illustrate the proposed method.
Citation - WoS: 20
Regularization of a Terminal Value Problem for Time Fractional Diffusion Equation
(Wiley, 2020) Vo Van Au; Le Dinh Long; Baleanu, Dumitru; Nguyen Huy Tuan; Nguyen Anh Triet; Au, Vo Van; Long, Le Dinh; Tuan, Nguyen Huy; Triet, Nguyen Anh
In this article, we study an inverse problem with inhomogeneous source to determine an initial data from the time fractional diffusion equation. In general, this problem is ill-posed in the sense of Hadamard, so the quasi-boundary value method is proposed to solve the problem. In the theoretical results, we propose a priori and a posteriori parameter choice rules and analyze them. Finally, two numerical results in the one-dimensional and two-dimensional case show the evidence of the used regularization method.
Citation - WoS: 10
Citation - Scopus: 12
Regularization of the Inverse Problem for Time Fractional Pseudo-Parabolic Equation With Non-Local in Time Conditions
(Springer Heidelberg, 2022) Le Dinh Long; Anh Tuan Nguyen; Baleanu, Dumitru; Nguyen Duc Phuong; Long, Le Dinh; Phuong, Nguyen Duc; Nguyen, Anh Tuan
This paper is devoted to identifying an unknown source for a time-fractional diffusion equation in a general bounded domain. First, we prove the problem is non-well posed and the stability of the source function. Second, by using the Modified Fractional Landweber method, we present regularization solutions and show the convergence rate between regularization solutions and sought solution are given under a priori and a posteriori choice rules of the regularization parameter, respectively. Finally, we present an illustrative numerical example to test the results of our theory.