In this paper a simple and convenient new regularization method for solving backward heat equationFourier regularization method is given. Meanwhile, some quite sharp error estimates between the approximate solution and exact solution are provided. A numerical example also shows that the method works effectively.

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.

A Cauchy problem for the Laplace equation in a rectangle is considered. Cauchy data are given for y = 0, and boundary data are for x = 0 and x = π . The solution for 0 < y 1 is sought. We propose two different regularization methods on the ill-posed problem based on separation of variables. Both methods are applied to formulate regularized solutions which are stably convergent to the exact one with explicit error estimates.

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 = {z = x + iy ∈ C|x ∈ R, |y| y 0 }, 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.