scholarly journals Regularization for a Riesz-Feller space fractional backward diffusion problem with a time-dependent coefficient

2018 ◽  
Vol 1 (T5) ◽  
pp. 172-183
Author(s):  
Hai Nguyen Duy Dinh
Keyword(s):  
Diffusion Equation ◽  
A Priori ◽  
Fractional Diffusion ◽  
Diffusion Problem ◽  
Time Dependent ◽  
Convergence Estimate ◽  
Backward Problem ◽  
A Priori Bound ◽  
Space Fractional Diffusion Equation ◽  
Ill Posed

In the present paper, we consider a backward problem for a space-fractional diffusion equation (SFDE) with a time-dependent coefficient. Such the problem is obtained from the classical diffusion equation by replacing the second-order spatial derivative with the Riesz-Feller derivative of order α∈(0,2]. This problem is ill-posed, i.e., the solution (if it exists) does not depend continuously on the data. Therefore, we propose one new regularization solution to solve it. Then, the convergence estimate is obtained under a priori bound assumptions for exact solution.

Stepwise regularization method for a nonlinear Riesz–Feller space-fractional backward diffusion problem

2019 ◽  
Vol 27 (6) ◽  
pp. 759-775
Author(s):  
Dang Duc Trong ◽  
Dinh Nguyen Duy Hai ◽  
Nguyen Dang Minh
Keyword(s):  
Exact Solution ◽  
Initial Data ◽  
Diffusion Equation ◽  
Regularization Method ◽  
Fractional Diffusion ◽  
Diffusion Problem ◽  
Convergence Estimate ◽  
Nonlinear Source ◽  
Space Fractional Diffusion Equation ◽  
Final Data

Abstract In this paper, we consider the backward diffusion problem for a space-fractional diffusion equation (SFDE) with a nonlinear source, that is, to determine the initial data from a noisy final data. Very recently, some papers propose new modified regularization solutions to solve this problem. To get a convergence estimate, they required some strongly smooth conditions on the exact solution. In this paper, we shall release the strongly smooth conditions and introduce a stepwise regularization method to solve the backward diffusion problem. A numerical example is presented to illustrate our theoretical result.

Filter regularization method for a nonlinear Riesz-Feller space-fractional backward diffusion problem with temporally dependent thermal conductivity

2021 ◽  
Vol 24 (4) ◽  
pp. 1112-1129
Author(s):  
Dinh Nguyen Duy Hai
Keyword(s):  
Thermal Conductivity ◽  
Diffusion Equation ◽  
Diffusion Problem ◽  
Filter Function ◽  
Convergence Estimate ◽  
Backward Problem ◽  
Special Cases ◽  
Ill Posed ◽  
The Fourier Transform ◽  
Regularized Solution

Abstract This paper concerns a backward problem for a nonlinear space-fractional diffusion equation with temporally dependent thermal conductivity. Such a problem is obtained from the classical diffusion equation by replacing the second-order space derivative with a Riesz-Feller derivative of order α ∈ (0, 2), which is usually used to model the anomalous diffusion. We show that the problem is severely ill-posed. Using the Fourier transform and a filter function, we construct a regularized solution from the data given inexactly and explicitly derive the convergence estimate in the case of the local Lipschitz reaction term. Special cases of the regularized solution are also presented. These results extend some earlier works on the space-fractional backward diffusion problem.

Quasi solution of a backward space fractional diffusion equation

2019 ◽  
Vol 27 (6) ◽  
pp. 795-814 ◽  
Author(s):  
Amir Hossein Salehi Shayegan ◽  
Ali Zakeri
Keyword(s):  
Diffusion Equation ◽  
Existence And Uniqueness ◽  
Linear Equations ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
The Finite Element Method ◽  
System Of Linear Equations ◽  
Solution Approach ◽  
Space Fractional Diffusion Equation ◽  
Ill Posed

Abstract In this paper, based on a quasi solution approach, i.e., a methodology involving minimization of a least squares cost functional, we study a backward space fractional diffusion equation. To this end, we give existence and uniqueness theorems of a quasi solution in an appropriate class of admissible initial data. In addition, in order to approximate the quasi solution, the finite element method is used. Since the obtained system of linear equations is ill-posed, we apply TSVD regularization. Finally, three numerical examples are given. Numerical results reveal the efficiency and applicability of the proposed method.

scholarly journals Regularization of a sideways problem for a time-fractional diffusion equation with nonlinear source

2020 ◽  
Vol 28 (2) ◽  
pp. 211-235
Author(s):  
Tran Bao Ngoc ◽  
Nguyen Huy Tuan ◽  
Mokhtar Kirane
Keyword(s):  
Inverse Problem ◽  
Diffusion Equation ◽  
Regularization Method ◽  
A Priori ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Nonlinear Source ◽  
Numerical Example ◽  
Ill Posed ◽  
Time Fractional Diffusion Equation

AbstractIn this paper, we consider an inverse problem for a time-fractional diffusion equation with a nonlinear source. We prove that the considered problem is ill-posed, i.e., the solution does not depend continuously on the data. The problem is ill-posed in the sense of Hadamard. Under some weak a priori assumptions on the sought solution, we propose a new regularization method for stabilizing the ill-posed problem. We also provide a numerical example to illustrate our results.

scholarly journals Identifying the space source term problem for time-space-fractional diffusion equation

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Erdal Karapinar ◽  
Devendra Kumar ◽  
Rathinasamy Sakthivel ◽  
Nguyen Hoang Luc ◽  
N. H. Can
Keyword(s):  
Diffusion Equation ◽  
Source Term ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Choice Rule ◽  
Parameter Choice ◽  
Time Space ◽  
Space Fractional Diffusion Equation ◽  
Ill Posed ◽  
Regularized Solution

Abstract In this paper, we consider an inverse source problem for the time-space-fractional diffusion equation. Here, in the sense of Hadamard, we prove that the problem is severely ill-posed. By applying the quasi-reversibility regularization method, we propose by this method to solve the problem (1.1). After that, we give an error estimate between the sought solution and regularized solution under a prior parameter choice rule and a posterior parameter choice rule, respectively. Finally, we present a numerical example to find that the proposed method works well.

Inverse problem for nonlinear backward space-fractional diffusion equation

2017 ◽  
Vol 25 (4) ◽  
Author(s):  
Hai Dinh Nguyen Duy ◽  
Tuan Nguyen Huy ◽  
Long Le Dinh ◽  
Gia Quoc Thong Le
Keyword(s):  
Inverse Problem ◽  
Diffusion Equation ◽  
Fractional Diffusion ◽  
Diffusion Problem ◽  
Second Order ◽  
Fractional Diffusion Equation ◽  
Nonlinear Source ◽  
Classical Diffusion ◽  
Space Fractional Diffusion Equation

AbstractIn this paper, a backward diffusion problem for a space-fractional diffusion equation (SFDE) with nonlinear source in a strip is investigated. This problem is obtained from the classical diffusion equation by replacing the second-order space derivative with a Riesz–Feller derivative of order

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