共查询到20条相似文献,搜索用时 46 毫秒
1.
In this paper, an inverse problem for space‐fractional backward diffusion equation, which is highly ill‐posed, is considered. This 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]. We show that such a problem is severely ill‐posed, and further present a simplified Tikhonov regularization method to deal with this problem. Convergence estimate is presented under a priori choice of regularization parameter. Numerical experiments are given to illustrate the accuracy and efficiency of the proposed method. Copyright © 2013 John Wiley & Sons, Ltd. 相似文献
2.
A Riesz–Feller space‐fractional backward diffusion problem with a time‐dependent coefficient: regularization and error estimates 下载免费PDF全文
Nguyen Huy Tuan Dang Duc Trong Dinh Nguyen Duy Hai Duong Dang Xuan Thanh 《Mathematical Methods in the Applied Sciences》2017,40(11):4040-4064
In this paper, we consider a Riesz–Feller space‐fractional backward diffusion problem with a time‐dependent coefficient We show that this problem is ill‐posed; therefore, we propose a convolution regularization method to solve it. New error estimates for the regularized solution are given under a priori and a posteriori parameter choice rules, respectively. Copyright © 2016 John Wiley & Sons, Ltd. 相似文献
3.
Mohammad F. Al‐Jamal 《Mathematical Methods in the Applied Sciences》2017,40(7):2466-2474
In this paper, we are concerned with the backward problem of reconstructing the initial condition of a time‐fractional diffusion equation from interior measurements. We establish uniqueness results and provide stability analysis. Our method is based on the eigenfunction expansion of the forward solution and the Tikhonov regularization to tackle the ill‐posedness issue of the underlying inverse problem. Some numerical examples are included to illustrate the effectiveness of the proposed approach. Copyright © 2016 John Wiley & Sons, Ltd. 相似文献
4.
Identification of an inverse source problem for time‐fractional diffusion equation with random noise
Tran Ngoc Thach Tuan Nguyen Huy Pham Thi Minh Tam Mach Nguyet Minh Nguyen Huu Can 《Mathematical Methods in the Applied Sciences》2019,42(1):204-218
In this paper, we consider an inverse source problem for a time fractional diffusion equation. In general, this problem is ill posed, therefore we shall construct a regularized solution using the filter regularization method in the random noise case. We will provide appropriate conditions to guarantee the convergence of the approximate solution to the exact solution. Then, we provide examples of filters in order to obtain error estimates for their approximate solutions. Finally, we present a numerical example to show efficiency of the method. 相似文献
5.
The aim of this work is to solve the backward problem for a time‐fractional diffusion equation with variable coefficients in a general bounded domain. The problem is ill‐posed in L 2 norm sense. An iteration scheme is proposed to obtain a regularized solution. Two kinds of convergence rates are obtained using an a priori regularization parameter choice rule and an a posteriori regularization parameter choice rule. Numerical examples in one‐dimensional and two‐dimensional cases are provided to show the effectiveness of the proposed methods. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 2029–2041, 2014 相似文献
6.
Halyna Lopushanska Andriy Lopushansky 《Mathematical Methods in the Applied Sciences》2019,42(9):3327-3340
We find the conditions for the unique solvability of the inverse problem for a time‐fractional diffusion equation with Schwarz‐type distributions in the right‐hand sides. This problem is to find a generalized solution of the Cauchy problem and an unknown space‐dependent part of an equation's right‐hand side under a time‐integral overdetermination condition. 相似文献
7.
Zhousheng Ruan 《Applicable analysis》2017,96(10):1638-1655
In this paper, we study an inverse problem of identifying a time-dependent term of an unknown source for a time fractional diffusion equation using nonlocal measurement data. Firstly, we establish the conditional stability for this inverse problem. Then two regularization methods are proposed to for reconstructing the time-dependent source term from noisy measurements. The first method is an integral equation method which formulates the inverse source problem into an integral equation of the second kind; and a prior convergence rate of regularized solutions is derived with a suitable choice strategy of regularization parameters. The second method is a standard Tikhonov regularization method and formulates the inverse source problem as a minimizing problem of the Tikhonov functional. Based on the superposition principle and the technique of finite-element interpolation, a numerical scheme is proposed to implement the second regularization method. One- and two-dimensional examples are carried out to verify efficiency and stability of the second regularization method. 相似文献
8.
Fractional (nonlocal) diffusion equations replace the integer-order derivatives in space and time by their fractional-order analogs and they are used to model anomalous diffusion, especially in physics. In this paper, we study a backward problem for an inhomogeneous time-fractional diffusion equation with variable coefficients in a general bounded domain. Such a backward problem is of practically great importance because we often do not know the initial density of substance, but we can observe the density at a positive moment. The backward problem is ill-posed and we propose a regularizing scheme by using Tikhonov regularization method. We also prove the convergence rate for the regularized solution by using an a priori regularization parameter choice rule. Numerical examples illustrate applicability and high accuracy of the proposed method. 相似文献
9.
Ming Yang 《Applicable analysis》2013,92(7):1508-1526
The evolution process of fractional order describes some phenomenon of anomalous diffusion and transport dynamics in complex system. The equation containing time-fractional derivative provides a suitable mathematical model for describing such a process. The backward problem for this system, which means to recover the initial state for some slow diffusion process from its present status, is very hard to solve due to the nonlocal property of fractional derivative and the irreversibility of time. For this ill-posed problem, we construct a regularizing solution using the Fourier transform method. Both the a-priori choice strategy and the a-posteriori choice strategy for the regularizing parameter are given, with the convergence analysis on the regularizing solution. Numerical implementations are presented to show the validity of the proposed scheme. 相似文献
10.
A modified regularized algorithm for a semilinear space‐fractional backward diffusion problem 下载免费PDF全文
Xiaoying Jiang Dinghua Xu Qifeng Zhang 《Mathematical Methods in the Applied Sciences》2017,40(16):5996-6006
In this paper, we investigate a backward problem for a space‐fractional partial differential equation. The main purpose is to propose a modified regularization method for the inverse problem. The existence and the uniqueness for the modified regularized solution are proved. To derive the gradient of the optimization functional, the variational adjoint method is introduced, and hence, the unknown initial value is reconstructed. Finally, numerical examples are provided to show the effectiveness of the proposed algorithm. Copyright © 2017 John Wiley & Sons, Ltd. 相似文献
11.
In this paper, we study the forward and the backward in time problems for a class of nonlinear diffusion equations with respect to the pseudo‐differential operator. Herein, we investigate the stability of the solution of the forward problem in relationship with parameters of the pseudo‐differential operator and initial data. Besides, as known, the backward in time problem is instability. Hence, we give a method to regularize the solution of the backward problem in the case of the parameters are perturbed. 相似文献
12.
Nguyen Anh Triet Vo Van Au Le Dinh Long Dumitru Baleanu Nguyen Huy Tuan 《Mathematical Methods in the Applied Sciences》2020,43(6):3850-3878
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. 相似文献
13.
Yun Zhang Ting Wei Yuan‐Xiang Zhang 《Numerical Methods for Partial Differential Equations》2021,37(1):24-43
This study is devoted to recovering two initial values for a time‐fractional diffusion‐wave equation from boundary Cauchy data. We provide the uniqueness result for recovering two initial values simultaneously by the method of Laplace transformation and analytic continuation. And then we use a nonstationary iterative Tikhonov regularization method to solve the inverse problem and propose a finite dimensional approximation algorithm to find good approximations to the initial values. Numerical examples in one‐ and two‐dimensional cases are provided to show the effectiveness of the proposed method. 相似文献
14.
In this paper, we consider an inverse problem related to a fractional diffusion equation. The model problem is governed by a nonlinear partial differential equation involving the fractional spectral Laplacian. This study is focused on the reconstruction of an unknown source term from a partial internal measured data. The considered ill‐posed inverse problem is formulated as a minimization one. The existence, uniqueness, and stability of the solution are discussed. Some theoretical results are established. The numerical reconstruction of the unknown source term is investigated using an iterative process. The proposed method involves a denoising procedure at each iteration step and provides a sequence of source term approximations converging in norm to the actual solution of the minimization problem. Some numerical results are presented to show the efficiency and the accuracy of the proposed approach. 相似文献
15.
In this paper, a time‐fractional diffusion equation with singular source term is considered. The Caputo fractional derivative with order 0<α ?1 is applied to the temporal variable. Under specific initial and boundary conditions, we find that the time‐fractional diffusion equation presents quenching solution that is not globally well‐defined as time goes to infinity. The quenching time is estimated by using the eigenfunction of linear fractional diffusion equation. Moreover, by implementing a finite difference scheme, we give some numerical simulations to demonstrate the theoretical analysis. Copyright © 2017 John Wiley & Sons, Ltd. 相似文献
16.
Nguyen Huy Tuan Nguyen Hoang Tuan Dumitru Baleanu Tran Ngoc Thach 《Mathematical Methods in the Applied Sciences》2020,43(3):1292-1312
In the present paper, we study the initial inverse problem (backward problem) for a two-dimensional fractional differential equation with Riemann-Liouville derivative. Our model is considered in the random noise of the given data. We show that our problem is not well-posed in the sense of Hadamard. A truncated method is used to construct an approximate function for the solution (called the regularized solution). Furthermore, the error estimate of the regularized solution in L2 and Hτ norms is considered and illustrated by numerical example. 相似文献
17.
Nguyen Huy Tuan Vo Au Le Nhat Huynh Yong Zhou 《Mathematical Methods in the Applied Sciences》2020,43(8):5450-5463
In this paper, we consider a backward problem for an inhomogeneous time-fractional wave equation in a general bounded domain. Such a backward problem is of practically great importance because we often do not know the initial density of substance, but we can observe the density at a positive moment. The existence and regularity for the backward problem are investigated. The backward problem is ill-posed, and we propose a regularizing scheme by using a modified regularization method. We also prove the convergence rate for the regularized solution by using some a priori regularization parameter choice rule. 相似文献
18.
Nguyen Huy Tuan Le Nhat Huynh Dumitru Baleanu Nguyen Huu Can 《Mathematical Methods in the Applied Sciences》2020,43(6):2858-2882
In this paper, we consider an inverse problem of recovering the initial value for a generalization of time-fractional diffusion equation, where the time derivative is replaced by a regularized hyper-Bessel operator. First, we investigate the existence and regularity of our terminal value problem. Then we show that the backward problem is ill-posed, and we propose a regularizing scheme using a fractional Tikhonov regularization method. We also present error estimates between the regularized solution and the exact solution using two parameter choice rules. 相似文献
19.
《Mathematical Methods in the Applied Sciences》2018,41(8):2987-2999
The purpose of the study is to analyze the time‐fractional reaction‐diffusion equation with nonlocal boundary condition. The proposed model is used to predict the invasion of tumor and its growth. Further, we establish the existence and uniqueness of a weak solution of the proposed model using the Faedo‐Galerkin method and compactness arguments. 相似文献
20.
Danh Hua Quoc Nam Vo Van Au Nguyen Huy Tuan Donal O'Regan 《Mathematical Methods in the Applied Sciences》2019,42(18):6672-6685
In this paper, we consider the nonlinear biharmonic equation. The problem is ill‐posed in the sense of Hadamard. To obtain a stable numerical solution, we consider a regularization method. We show rigourously, with error estimates provided, that the corresponding regularized solutions converge to the true solution strongly in uniformly with respect to the space coordinate under some a priori assumptions on the solution. 相似文献