scholarly journals A Mollification Regularization Method for the Inverse Source Problem for a Time Fractional Diffusion Equation

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1048
Author(s):  
Le Dinh Long ◽  
Yong Zhou ◽  
Tran Thanh Binh ◽  
Nguyen Can
Keyword(s):  
Diffusion Equation ◽  
Regularization Method ◽  
A Priori ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Regularized Methods ◽  
A Posteriori Parameter Choice ◽  
Ill Posed ◽  
Time Fractional Diffusion Equation ◽  
Parameter Choice Rules

We consider a time-fractional diffusion equation for an inverse problem to determine an unknown source term, whereby the input data is obtained at a certain time. In general, the inverse problems are ill-posed in the sense of Hadamard. Therefore, in this study, we propose a mollification regularization method to solve this problem. In the theoretical results, the error estimate between the exact and regularized solutions is given by a priori and a posteriori parameter choice rules. Besides, the proposed regularized methods have been verified by a numerical experiment.

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.

A quasi-boundary regularization method for identifying the initial value of time-fractional diffusion equation on spherically symmetric domain

2019 ◽  
Vol 27 (5) ◽  
pp. 609-621 ◽  
Author(s):  
Fan Yang ◽  
Ni Wang ◽  
Xiao-Xiao Li ◽  
Can-Yun Huang
Keyword(s):  
Diffusion Equation ◽  
Regularization Method ◽  
Regularization Parameter ◽  
A Priori ◽  
Symmetric Domain ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Spherically Symmetric ◽  
Initial Value ◽  
Time Fractional Diffusion Equation

Abstract In this paper, an inverse problem to identify the initial value for high dimension time fractional diffusion equation on spherically symmetric domain is considered. This problem is ill-posed in the sense of Hadamard, so the quasi-boundary regularization method is proposed to solve the problem. The convergence estimates between the regularization solution and the exact solution are presented under the a priori and a posteriori regularization parameter choice rules. Numerical examples are provided to show the effectiveness and stability of the proposed method.

scholarly journals A truncation regularization method for a time fractional diffusion equation with an in-homogeneous source

2018 ◽  
Vol 20 ◽  
pp. 02007
Author(s):  
Luu Vu Cam Hoan ◽  
Ho Duy Binh ◽  
Tran Bao Ngoc
Keyword(s):  
Diffusion Equation ◽  
Regularization Method ◽  
Fourier Method ◽  
Fractional Diffusion ◽  
Diffusion Problem ◽  
Fractional Diffusion Equation ◽  
Ill Posed ◽  
Inverse Diffusion ◽  
Time Fractional Diffusion Equation

In the present paper, we consider a time-fractional inverse diffusion problem with an in-homogeneous source, where data is given at x = 1 and the solution is required in the interval 0 < x < 1. This problem is ill-posed, i.e. the solution (if it exists) does not depend continuously on the data. We propose a regularization method to solve it based on the solution given by the Fourier method.

scholarly journals Identification of Source Term for the Time-Fractional Diffusion-Wave Equation by Fractional Tikhonov Method

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 934
Author(s):  
Le Dinh Long ◽  
Nguyen Hoang Luc ◽  
Yong Zhou ◽  
and Can Nguyen
Keyword(s):  
Source Term ◽  
A Priori ◽  
Fractional Diffusion ◽  
Behavior Changes ◽  
Diffusion Wave ◽  
Tikhonov Method ◽  
A Posteriori Parameter Choice ◽  
Ill Posed ◽  
Well Posed ◽  
Parameter Choice Rules

In this article, we consider an inverse problem to determine an unknown source term in a space-time-fractional diffusion equation. The inverse problems are often ill-posed. By an example, we show that this problem is NOT well-posed in the Hadamard sense, i.e., this problem does not satisfy the last condition-the solution’s behavior changes continuously with the input data. It leads to having a regularization model for this problem. We use the Tikhonov method to solve the problem. In the theoretical results, we also propose a priori and a posteriori parameter choice rules and analyze them.

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