scholarly journals Fourier Truncation Regularization Method for a Time-Fractional Backward Diffusion Problem with a Nonlinear Source

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 865 ◽  
Author(s):  
Fan Yang ◽  
Ping Fan ◽  
Xiao-Xiao Li ◽  
Xin-Yi Ma
Keyword(s):  
Regularization Method ◽  
Regularization Parameter ◽  
A Priori ◽  
Fractional Diffusion ◽  
Diffusion Problem ◽  
Nonlinear Source ◽  
Convergence Error ◽  
Strip Domain ◽  
Ill Posed ◽  
Regularization Parameter Selection

In present paper, we deal with a backward diffusion problem for a time-fractional diffusion problem with a nonlinear source in a strip domain. We all know this nonlinear problem is severely ill-posed, i.e., the solution does not depend continuously on the measurable data. Therefore, we use the Fourier truncation regularization method to solve this problem. Under an a priori hypothesis and an a priori regularization parameter selection rule, we obtain the convergence error estimates between the regular solution and the exact solution at 0 ≤ x < 1 .

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.

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.

scholarly journals A Regularization Method to Solve a Cauchy Problem for the Two-Dimensional Modified Helmholtz Equation

Mathematics ◽  
2019 ◽  
Vol 7 (4) ◽  
pp. 360 ◽  
Author(s):  
Shangqin He ◽  
Xiufang Feng
Keyword(s):  
Helmholtz Equation ◽  
Regularization Method ◽  
Numerical Approximation ◽  
Regularization Parameter ◽  
Two Dimensional ◽  
Approximation Solution ◽  
Modified Helmholtz Equation ◽  
Strip Domain ◽  
Ill Posed ◽  
De La Vallée Poussin

In this paper, the ill-posed problem of the two-dimensional modified Helmholtz equation is investigated in a strip domain. For obtaining a stable numerical approximation solution, a mollification regularization method with the de la Vallée Poussin kernel is proposed. An error estimate between the exact solution and approximation solution is given under suitable choices of the regularization parameter. Two numerical experiments show that our procedure is effective and stable with respect to perturbations in the data.

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 A Stable Approach for Numerical Differentiation by Local Regularization Method with its Regularization Parameter Selection Strategies

2020 ◽  
pp. 27-35
Author(s):  
Huilin Xu ◽  
Xiaoyan Xiang ◽  
Yanling He
Keyword(s):  
Regularization Method ◽  
Regularization Parameter ◽  
A Priori ◽  
Numerical Differentiation ◽  
Selection Strategy ◽  
Numerical Comparison ◽  
Local Regularization ◽  
Selection Strategies ◽  
The Given ◽  
Regularization Parameter Selection

The local regularization method for solving the first-order numerical differentiation problem is considered in this paper. The a-priori and a-posteriori selection strategy of the regularization parameter is introduced, and the convergence rate of local regularization solution under some assumption of the exact derivative is also given. Numerical comparison experiments show that the local regularization method can reflect sharp variations and oscillations of the exact derivative while suppress the noise of the given data effectively.

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 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.

scholarly journals A modified Tikhonov regularization method based on Hermite expansion for solving the Cauchy problem of the Laplace equation

2020 ◽  
Vol 18 (1) ◽  
pp. 1685-1697
Author(s):  
Zhenyu Zhao ◽  
Lei You ◽  
Zehong Meng
Keyword(s):  
Cauchy Problem ◽  
Laplace Equation ◽  
Tikhonov Regularization ◽  
Regularization Method ◽  
Regularization Parameter ◽  
Solution Process ◽  
Tikhonov Regularization Method ◽  
Hermite Expansion ◽  
The Cauchy Problem ◽  
Ill Posed

Abstract In this paper, a Cauchy problem for the Laplace equation is considered. We develop a modified Tikhonov regularization method based on Hermite expansion to deal with the ill posed-ness of the problem. The regularization parameter is determined by a discrepancy principle. For various smoothness conditions, the solution process of the method is uniform and the convergence rate can be obtained self-adaptively. Numerical tests are also carried out to verify the effectiveness of the method.

scholarly journals Regularization of backward time-fractional parabolic equations by Sobolev-type equations

2020 ◽  
Vol 28 (5) ◽  
pp. 659-676
Author(s):  
Dinh Nho Hào ◽  
Nguyen Van Duc ◽  
Nguyen Van Thang ◽  
Nguyen Trung Thành
Keyword(s):  
Error Estimates ◽  
Parabolic Equations ◽  
Regularization Parameter ◽  
A Priori ◽  
Sobolev Type ◽  
A Posteriori ◽  
Parameter Choice ◽  
Numerical Tests ◽  
Ill Posed ◽  
Parameter Choice Rules

AbstractThe problem of determining the initial condition from noisy final observations in time-fractional parabolic equations is considered. This problem is well known to be ill-posed, and it is regularized by backward Sobolev-type equations. Error estimates of Hölder type are obtained with a priori and a posteriori regularization parameter choice rules. The proposed regularization method results in a stable noniterative numerical scheme. The theoretical error estimates are confirmed by numerical tests for one- and two-dimensional equations.

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