A new regularization method for solving a time-fractional inverse diffusion problem

In this paper, we consider an inverse problem for a time-fractional diffusion equation in a one-dimensional semi-infinite domain. The temperature and heat flux are sought from a measured temperature history at a fixed location inside the body. We show that such problem is severely ill-posed and further apply a new regularization method to solve it based on the solution given by the Fourier method. Convergence estimates are presented under the a priori bound assumptions for the exact solution. Finally, numerical examples are given to show that the proposed numerical method is effective.     

An iteration regularization for a time-fractional inverse diffusion problem

In the present paper we consider a time-fractional inverse diffusion problem, where data is given at x = 1 and the solution is required in the interval 0 < x < 1. This problem is typically ill-posed: the solution (if it exists) does not depend continuously on the data. We give a new iteration regularization method to deal with this problem, and error estimates are obtained for a priori and a posteriori parameter choice rules, respectively. Furthermore, numerical implement shows the proposed method works effectively.     

Fourier regularization for a final value time-fractional diffusion problem

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.     

Spectral regularization method for a Cauchy problem of the time fractional advection-dispersion equation

In this paper, a Cauchy problem for the time fractional advection-dispersion equation (TFADE) is investigated. Such a problem is obtained from the classical advection-dispersion equation by replacing the first-order time derivative by the Caputo fractional derivative of order . We show that the Cauchy problem of TFADE is severely ill-posed and further apply a spectral regularization method to solve it based on the solution given by the Fourier method. The convergence estimate is obtained under a priori bound assumptions for the exact solution. Numerical examples are given to show the effectiveness of the proposed numerical method.     

Tikhonov regularization method for a backward problem for the inhomogeneous time-fractional diffusion equation

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.     

Tikhonov regularization method for a backward problem for the time-fractional diffusion equation

This paper is devoted to solve a backward problem for a time-fractional diffusion equation with variable coefficients in a general bounded domain by the Tikhonov regularization method. Based on the eigenfunction expansion of the solution, the backward problem for searching the initial data is changed to solve a Fredholm integral equation of the first kind. The conditional stability for the backward problem is obtained. We use the Tikhonov regularization method to deal with the integral equation and obtain the series expression of solution. Furthermore, the convergence rates for the Tikhonov regularized solution can be proved by using an a priori regularization parameter choice rule and an a posteriori regularization parameter choice rule. Two numerical examples in one-dimensional and two-dimensional cases respectively are investigated. Numerical results show that the proposed method is effective and stable.     

Uniqueness for an inverse source problem of determining a space dependent source in a time-fractional diffusion equation

We study uniqueness of a solution for an inverse source problem arising in linear time-fractional diffusion equations with time dependent coefficients. New uniqueness results are formulated in Theorem 3.1. We also show optimality of the conditions under which uniqueness holds by explicitly constructing counterexamples, that is by constructing more than one solution in the case when the conditions for uniqueness are violated.     

Uniqueness for inverse problems of determining orders of multi-term time-fractional derivatives of diffusion equation

This article proves the uniqueness for two kinds of inverse problems of identifying fractional orders in diffusion equations with multiple time-fractional derivatives by pointwise observation. By means of eigenfunction expansion and Laplace transform, we reduce the uniqueness for our inverse problems to the uniqueness of expansions of some special function and complete the proof.     

A new fractional derivative for solving time fractional diffusion wave equation

The time fractional diffusion wave equation, which can be used to describe wave diffusion process in this article, was studied. First of all, the diffusion wave equation can be extended to a generalized form in the sense of the regularized version of the $k$-Hilfer–Prabhakar ( $k$-H-P) fractional operator involving the $k$-Mittag- function. Then, the analytical solution can be obtained for this considered equation by using the Laplace transform method and the Fourier transform method. As a result, a novel and general solution have been found. The unconventional solution may show new result and phenomenon to wave diffusion process. Thereby, this research provides a window for discovering new diffusion mechanisms.     

An inverse problem for space‐fractional backward diffusion problem

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.     

Regularization of a terminal value problem for time fractional diffusion equation

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.     

Maximum principle for the generalized time-fractional diffusion equation

In the paper, a maximum principle for the generalized time-fractional diffusion equation over an open bounded domain is formulated and proved. The proof of the maximum principle is based on an extremum principle for the Caputo-Dzherbashyan fractional derivative that is given in the paper, too. The maximum principle is then applied to show that the initial-boundary-value problem for the generalized time-fractional diffusion equation possesses at most one classical solution and this solution continuously depends on the initial and boundary conditions.     

The revised generalized Tikhonov method for the backward time-fractional diffusion equation

In this paper, we solve the backward problem for a time-fractional diffusion equation with variable coefficients in a bounded domain by using the revised generalized Tikhonov regularization method. Convergence estimates under an a-priori and a-posteriori regularization parameter choice rules are given. Numerical example shows that the proposed method is effective and stable.     

Identification of an inverse source problem for time‐fractional diffusion equation with random noise

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.     

Identification of a time-dependent source term for a time fractional diffusion problem

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.     

多层介质中逆热传导问题的傅里叶正则化方法

逆热传导问题是数学物理反问题中的热点和前沿课题之一,在钢铁生产等领域中具有重要的应用背景.讨论一个多层介质中的逆热传导问题,它是一个极度不适定问题.通过傅里叶截断方法构造正则化近似解,并给出相应的稳定性估计.     

An inverse source problem for a two dimensional time fractional diffusion equation with nonlocal boundary conditions

We consider the inverse source problem for a time fractional diffusion equation. The unknown source term is independent of the time variable, and the problem is considered in two dimensions. A biorthogonal system of functions consisting of two Riesz bases of the space L²[(0,1) × (0,1)], obtained from eigenfunctions and associated functions of the spectral problem and its adjoint problem, is used to represent the solution of the inverse problem. Using the properties of the biorthogonal system of functions, we show the existence and uniqueness of the solution of the inverse problem and its continuous dependence on the data. Copyright © 2012 John Wiley & Sons, Ltd.     

A modified quasi-boundary value method for an inverse source problem of the time-fractional diffusion equation

In this paper, we consider an inverse source problem for a time-fractional diffusion equation with variable coefficients in a general bounded domain. That is to determine a space-dependent source term in the time-fractional diffusion equation from a noisy final data. Based on a series expression of the solution, we can transform the original inverse problem into a first kind integral equation. The uniqueness and a conditional stability for the space-dependent source term can be obtained. Further, we propose a modified quasi-boundary value regularization method to deal with the inverse source problem and obtain two kinds of convergence rates by using an a priori and an a posteriori regularization parameter choice rule, respectively. Numerical examples in one-dimensional and two-dimensional cases are provided to show the effectiveness of the proposed method.     

A regularization method for solving the Cauchy problem for the Helmholtz equation

In this paper, we investigate a Cauchy problem associated with Helmholtz-type equation in an infinite “strip”. This problem is well known to be severely ill-posed. The optimal error bound for the problem with only nonhomogeneous Neumann data is deduced, which is independent of the selected regularization methods. A framework of a modified Tikhonov regularization in conjunction with the Morozov’s discrepancy principle is proposed, it may be useful to the other linear ill-posed problems and helpful for the other regularization methods. Some sharp error estimates between the exact solutions and their regularization approximation are given. Numerical tests are also provided to show that the modified Tikhonov method works well.     

A backward problem for the time‐fractional diffusion equation

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.