Xiang-Tuan Xiong

Fourier regularization method for solving a Cauchy problem for the Laplace equation

In this article, we use Fourier method to solve a Cauchy problem for the Laplace equation in a strip region, the method is rather simple and convenient. Meanwhile, we overstep the Hölder continuity and provide some more sharp error estimates between the exact solution and its approximation. Numerical examples also show that the method work effectively.

Two numerical methods for solving a backward heat conduction problem

We introduce a central difference method and a quasi-reversibility method for solving a backward heat conduction problem (BHCP) numerically. For these two numerical methods, we give the stability analysis. Meanwhile, we investigate the roles of regularization parameters in these two methods. Numerical results show that our algorithm is effective.

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 modified method for determining the surface heat flux of IHCP

We consider the determination of the surface heat flux of a body from a measured temperature history at a fixed location inside the body. Mathematically, it is formulated as a problem for the one-dimensional heat equation in a quarter plane with data given along the line x = 1, and the gradient of the solution is determined for 0≤x<1. This problem is of practical interest in some engineering contexts and it is severely ill-posed, in the sense that the solution (if it exists) does not depend continuously on the data. In this article, we employ a fourth-order modified method to solve the problem. Some stability estimates are given. Numerical examples show that the modified method works very well.

Central difference schemes in time and error estimate on a non-standard inverse heat conduction problem

Inverse heat conduction problems (IHCP) are severely ill-posed in the sense that the solution (if it exists) does not depend continuously on the data. But now the results available in the literature on IHCP are mainly devoted to the standard sideways heat equation. This paper remedies this by a central difference schemes in time which itself has a regularizing effect for a non-standard IHCP which appears in some applied subjects. An error estimate is obtained and the error estimate also gives information about how to choose the step length in the time discretization. A numerical example shows that the computational effect of this method is satisfactory.

Fourier regularization method for solving the surface heat flux from interior observations

In the present paper, a Fourier regularization method for solving the surface heat flux distribution from interior observations with some estimates are given. A numerical example is also provided.

Wavelet and spectral regularization methods for a sideways parabolic equation

We consider a special sideways parabolic equation which appears in some applied subjects. The problem is ill-posed in the sense that the solution (if it exists) does not depend continuously on the data. In this paper some wavelet and spectral regularization methods for this problem are given. The error estimates are also established respectively.

ON THREE SPECTRAL REGULARIZATION METHODS FOR A BACKWARD HEAT CONDUCTION PROBLEM

We introduce three spectral regularization methods for solving a backward heat conduction problem (BHCP). For the three spectral regularization methods, we give the stability error estimates with optimal order under an a-priori and an a-posteriori regularization parameter choice rule. Numerical results show that our theoretical results are effective.

Error estimates of a difference approximation method for a backward heat conduction problem.

We introduce a central difference method for a backward heat conduction problem (BHCP). Error estimates for this method are provided together with a selection rule for the regularization parameter (the space step length). A numerical experiment is presented in order to illustrate the role of the regularization parameter.

Optimal Tikhonov approximation for a sideways parabolic equation

We consider an inverse heat conduction problem with convection term which appears in some applied subjects. This problem is ill posed in the sense that the solution (if it exists) does not depend continuously on the data. A generalized Tikhonov regularization method for this problem is given, which realizes the best possible accuracy.