Ting Wei

Identifying an unknown source in time-fractional diffusion equation by a truncation method

In this paper, an inverse source problem for the time-fractional diffusion equation is investigated. We prove a conditional stability for this problem. A truncation method is presented to deal with the ill-posedness of the problem and error estimates are obtained with an a priori strategy and an a posteriori choice rule to find the regularization parameter. Numerical examples show that the proposed method is effective and stable.

High order numerical derivatives for one-dimensional scattered noisy data

Based on the smoothing spline approximation, in this paper we propose a regularization method for computing high order numerical derivatives from one-dimensional noisy data. The convergence rates under two different choices of the regularization parameter are obtained. Numerical examples show that the proposed method is effective and stable.

Some filter regularization methods for a backward heat conduction problem

In this paper, a one-dimensional backward heat conduction problem is investigated. It is well known that such problem is ill-posed. Some filter regularization methods are used to solve it. Convergence estimates under two a-priori bound assumptions for the exact solution are given based on the conditional stabilities. Finally, numerical examples are given to show that our used numerical methods are effective and stable.

Fourier truncation method for high order numerical derivatives

We consider a classical ill-posed problem-numerical differentiation with a new method. We propose Fourier truncation method to compute high order numerical derivatives. A Holder-type stability estimate is obtained. A numerical implementation is described. Numerical examples show that the proposed method is effective and stable.

A new a posteriori parameter choice strategy for the convolution regularization of the space-fractional backward diffusion problem

In this paper, we consider a backward space-fractional diffusion problem. We propose an a posteriori parameter choice rule for the regularization method given in Zheng and Wei (2010), where the authors proposed a regularization method called convolution regularization method, and gave an a priori parameter choice strategy. In this paper, we study the same problem but give a new a posteriori parameter choice based on a modified version of the discrepancy principle, and obtain a log -type error estimate under an additional source condition. Numerical results show that our method is feasible.

Numerical solution for an inverse heat source problem by an iterative method

Abstract In this paper, we consider a typical inverse heat source problem, that is, we determine two separable source terms in a heat equation from the initial and boundary data along with two additional measurements. By a simple transformation, the original problem is changed into a nonlinear problem, and then we use an iterative method to solve it. After giving an algorithm, we prove some Holder convergence rates for both the reconstructed heat source terms and the temperature distribution subject to certain bounds of the data. Numerical results show that our method is accurate and efficient.

A quasi-reversibility regularization method for an inverse heat conduction problem without initial data

In this paper, we study an inverse heat conduction problem without initial data in a bounded domain in which the Cauchy data at x=0 are given and the solution in 0

Optimal error bound and simplified Tikhonov regularization method for a backward problem for the time-fractional diffusion equation

In this paper, we consider a backward problem for a time-fractional diffusion equation. Such a problem is ill-posed. The optimal error bound for the problem under a source condition is analyzed. A simplified Tikhonov regularization method is utilized to solve the problem, and its convergence rates are analyzed under an a priori regularization parameter choice rule and an a posteriori regularization parameter choice rule, respectively. Numerical examples show that the proposed regularization method is effective and stable, and both parameter choice rules work well.

Spectral regularization method for solving a time-fractional inverse diffusion problem

Abstract In this paper, we consider an inverse problem for a time-fractional diffusion equation with 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 spectral regularization method to solve it based on the solution given by the Fourier method. Convergence estimates are presented under a priori bound assumptions for the exact solution. Finally, numerical examples are given to show that the proposed numerical method is effective.

Identification of time-dependent convection coefficient in a time-fractional diffusion equation

Abstract In the present paper, we devote our effort to solve a nonlinear inverse problem for identifying a time-dependent convection coefficient in a time-fractional diffusion equation from the measured data at an interior point for one-dimensional case. We prove the existence, uniqueness and regularity of solution for the direct problem by using the fixed point theorem. The stability of inverse convection coefficient problem is obtained based on the regularity of solution for the direct problem and some generalized Gronwall’s inequalities. We use a modified optimal perturbation regularization algorithm to solve the inverse convection coefficient problem. Two numerical examples are provided to show the effectiveness of the proposed method.