Browsing by Author "Long, Le Dinh"

Now showing 1 - 9 of 9

Citation Count: Long, Le Dinh...et al. (2021). "An inverse source problem for pseudo-parabolic equation with Caputo derivative", Journal of Applied Mathematics and Computing.
An inverse source problem for pseudo-parabolic equation with Caputo derivative
(2021) Long, Le Dinh; Luc, Nguyen Hoang; Tatar, Salih; Baleanu, Dumitru; Can, Nguyen Huu; 56389
In this paper, we consider an inverse source problem for a fractional pseudo-parabolic equation. We show that the problem is severely ill-posed (in the sense of Hadamard) and the Tikhonov regularization method is proposed to solve the problem. In addition, we present numerical examples to illustrate applicability and accuracy of the proposed method to some extent.
Citation Count: Phuong, Nguyen Duc;...et.al. (2022). "Fractional evolution equation with Cauchy data in spaces", Boundary Value Problems, No.100, pp.1-22.
Fractional evolution equation with Cauchy data in spaces
(2022) Phuong, Nguyen Duc; Baleanu, Dumitru; Agarwal, Ravi P.; Long, Le Dinh; 56389
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 L2 and Hs . However, there have not been any papers dealing with this problem with observed data in Lp with p = 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 Lp. To our knowledge, Lp evaluations for the inverse problem are very limited. This work generalizes some recent results on this problem
Citation Count: Luc, Nguyen Hoang...et al. (2021). "Identifying the initial condition for space-fractional sobolev equation", Journal of Applied Analysis and Computation, Vol. 11, No. 5, pp. 2402-2422.
Identifying the initial condition for space-fractional sobolev equation
(2021) Luc, Nguyen Hoang; Long, Le Dinh; Hang, Le Thi Diem; Baleanu, Dumitru; Can, Nguyen Hu; 56389
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. © 2021, Wilmington Scientific Publisher. All rights reserved.
Citation Count: Luc, Nguyen Hoang...et al. (2021). "Identifying the source function for time fractional diffusion with non-local in time conditions", Computational and Applied Mathematics, Vol. 40, No. 5.
Identifying the source function for time fractional diffusion with non-local in time conditions
(2021) Luc, Nguyen Hoang; Baleanu, Dumitru; Agarwal, Ravi P.; Long, Le Dinh; 56389
The diffusion equation has many applications in fields such as physics, environment, and fluid mechanics. In this paper, we consider the problem of identifying an unknown source for a time-fractional diffusion equation in a general bounded domain from the nonlocal integral condition. The problem is non-well-posed in the sense of Hadamard, i.e, if the problem has only one solution, the solution will not depend continuously on the input data. To get a stable solution and approximation, we need to offer the regularization methods. The first contribution to the paper is to provide a regularized solution using the modified fractional Landweber method. Two choices are proposed including a priori and a posteriori parameter choice rules, to estimate the convergence rate of the regularized methods. The second new contribution is to use truncation to give an estimate of Lp for the convergence rate. © 2021, SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional.
Citation Count: Can, Nguyen Huu...et al. (2020). "Inverse source problem for time fractional diffusion equation with Mittag-Leffler kernel", Advances in Difference Equations, Vol. 2020, No.1.
Inverse source problem for time fractional diffusion equation with Mittag-Leffler kernel
(2020) Can, Nguyen Huu; Luc, Nguyen Hoang; Baleanu, Dumitru; Zhou, Yong; Long, Le Dinh; 56389
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 Count: Luc, Nguyen Hoang...et al. (2020). "Reconstructing the right-hand side of a fractional subdiffusion equation from the final data", Journal of Inequalities and Applications, Vol. 2020, No. 1.
Reconstructing the right-hand side of a fractional subdiffusion equation from the final data
(2020) Luc, Nguyen Hoang; Baleanu, Dumitru; Long, Le Dinh; Can, Nguyen-Huu; 56389
In this study, we study an inverse source problem for the time-fractional diffusion equation, where the final data t=Tare given. We show that our problem is ill-posed in the sense of Hadamard. Applying a truncation method, we give the regularized solution. Finally, convergence estimates under a priori and a posteriori parameter choice rules are proved.
Citation Count: Huynh, Le Nhat...et al. (2021). "Recovering the space source term for the fractional-diffusion equation with Caputo–Fabrizio derivative", Journal of Inequalities and Applications, Vol. 2021, No. 1.
Recovering the space source term for the fractional-diffusion equation with Caputo–Fabrizio derivative
(2021) Huynh, Le Nhat; Luc, Nguyen Hoang; Baleanu, Dumitru; Long, Le Dinh; 56389
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 Count: Triet, Nguyen Anh...et al. (2020). "Regularization of a terminal value problem for time fractional diffusion equation", Mathematical Methods in the Applied Sciences, Vol. 43, No. 6, pp. 3850-3878.
Regularization of a terminal value problem for time fractional diffusion equation
(2020) Triet, Nguyen Anh; Au, Vo Van; Long, Le Dinh; Baleanu, Dumitru; Tuan, Nguyen Huy; 56389
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 Count: Phuong, Nguyen Duc;...et.al. (2022). "Regularization of the Inverse Problem for Time Fractional Pseudo-parabolic Equation with Non-local in Time Conditions", Acta Mathematica Sinica, English Series, Vol.38, No.12, pp.2199-2219.
Regularization of the Inverse Problem for Time Fractional Pseudo-parabolic Equation with Non-local in Time Conditions
(2022) Phuong, Nguyen Duc; Long, Le Dinh; Nguyen, Anh Tuan; Baleanu, Dumitru; 56389
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.