SUMMARYIn this paper, we consider a finite difference approximation to an inverse problem of determining a spacewise-dependent heat source in the parabolic heat equation using the usual conditions of the direct problem and information from one supplementary temperature measurement at a given instant of time.Since the resulting matrix equation is ill-conditioned, a regularized solution is obtained by employing the truncated singular value decomposition to solve the matrix equation arising from the finite difference method, with the optimal regularization parameter determined by the generalized cross-validation criterion. The effectiveness of the proposed numerical scheme is illustrated by several continuous and discontinuous numerical examples.

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.

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