Zhi Qian

A modified method for a backward heat conduction problem

We consider a backward heat conduction problem in a strip, where data is given at the final time t = T(T > 0) and a solution for 0 <= t < T is sought. The problem is ill-posed in the sense that the solution(if it exists) does not depend continuously on the data. In order to numerically solve the problem, we study a modification of the equation, where a third-order mixed derivative term is added. Error estimates for this problem are given, which show that the modified problem is stable and its solution is an approximation of the backward heat conduction problem. Some numerical tests illustrate that the proposed method is feasible and effective. (c) 2006 Elsevier Inc. All rights reserved.

Regularization strategies for a two-dimensional inverse heat conduction problem

We consider a two-dimensional inverse heat conduction problem for a slab. This is a severely ill-posed problem. Two regularization strategies, one based on the modification of the equation, the other based on the truncation of high frequency components, are proposed to solve the problem in the presence of noisy data. Error estimates show that the regularized solution is dependent continuously on the data and is an approximation of the exact solution of the two-dimensional inverse heat conduction problem. The relation of these two and other regularization strategies is also discussed.

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.

A simple regularization method for stable analytic continuation

The problems of analytic continuation are frequently encountered in many practical applications. These problems are well known to be severely ill-posed and therefore several regularization methods have been suggested for solving them. In this paper we consider the problem of analytic continuation of the analytic function f(z) = f(x + iy) on a strip domain , where the data are given only on the line y = 0. We use a very simple and convenient method—the Fourier regularization method to solve this problem. Some sharp error estimates between the exact solution and its approximation are given and numerical examples show the method works 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.

Numerical approximation of solution of nonhomogeneous backward heat conduction problem in bounded region

In this paper we consider a numerical approximation of solution of nonhomogeneous backward heat conduction problem (BHCP) in bounded region based on Tikhonov regularization method. Error estimate at t=0 for this method is provided. According to the error estimate, a selection of regularization parameter is given. Meanwhile, a numerical implementation is described and the numerical results show that our algorithm is effective.

A modified Tikhonov regularization method for a spherically symmetric three-dimensional inverse heat conduction problem

This paper deals with a spherically symmetric three-dimensional inverse heat conduction problem of determining the internal surface temperature distribution of a hollow sphere from the measured data at a fixed location inside it. This is an ill-posed problem, i.e., the solution (if it exists) does not depend continuously on the data. A modified Tikhonov regularization method is given and an order optimal stability estimate is obtained.

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 high order numerical derivatives

In this paper we propose a new regularization method for computing high order numerical derivatives from one dimensional noisy data. The convergence estimate under an appropriate choice of the regularization parameter is obtained. Some interesting numerical tests show that the proposed method is effective and stable.

Optimal modified method for a fractional-diffusion inverse heat conduction problem

We consider the determination of the boundary temperature from one measured transient data temperature at some interior point of a one-dimensional semi-infinite conductor. Mathematically, it can be formulated as a fractional-diffusion inverse heat conduction problem where data are given at x = l and we want to determine a solution for 0 < x < l. This problem arises in several contexts and has important applications in science and engineering. The difficulty of the problem is its severe ill-posedness, i.e. the solution (if it exists) does not depend continuously on the data. In this article, we consider an optimal modified method from the frequency domain and obtain a Hölder-type convergence estimate with the coefficient c = 1, which is optimal. The method can be implemented numerically using discrete Fourier transforms. Three kinds of examples illustrate the behaviour of the proposed method.