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.

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

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 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 Finite Element Solutions for the Space Fractional Diffusion Equation with a Nonlinear Source Term

2012 ◽  
Vol 2012 ◽  
pp. 1-25 ◽  
Author(s):  
Y. J. Choi ◽  
S. K. Chung
Keyword(s):  
Finite Element ◽  
Diffusion Equation ◽  
Fractional Derivative ◽  
Source Term ◽  
Approximate Solutions ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Nonlinear Source ◽  
Backward Euler ◽  
Space Fractional Diffusion Equation

We consider finite element Galerkin solutions for the space fractional diffusion equation with a nonlinear source term. Existence, stability, and order of convergence of approximate solutions for the backward Euler fully discrete scheme have been discussed as well as for the semidiscrete scheme. The analytical convergent orders are obtained asO(k+hγ˜), whereγ˜is a constant depending on the order of fractional derivative. Numerical computations are presented, which confirm the theoretical results when the equation has a linear source term. When the equation has a nonlinear source term, numerical results show that the diffusivity depends on the order of fractional derivative as we expect.

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 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.

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