Determination of a two-dimensional heat source: Uniqueness, regularization and error estimate

Abstract

Let Q be a heat conduction body and let ' = '(t) be given. We consider the problem of nding a two-dimensional heat source having the form '(t)f(x; y) in Q. The problem is ill-posed. Assuming @Q is insulated and ' 6 0, we show that the heat source is de ned uniquely by the temperature history on @Q and the temperature distribution in Q at the initial time t = 0 and at the nal time t = 1. Using the method of truncated integration and the Fourier transform, we construct regularized solutions and derive explicitly error estimate.

Determine the special term of a two-dimensional heat source

“…For a long time, it has been investigated for the heat souce which is only time-dependent [11][12][13] or only spacedependent [13][14][15]. Recently, the regularization problem for the heat source Fð; tÞ ¼ 'ðtÞ f ðÞ, where ' is a given function, was regarded for one-dimensional case [16] and two-dimensional case [17]. However, these authors needed in addition an essential datum, that is the the final condition uð; TÞ.…”

Two regularization methods for a Cauchy problem for the Laplace equation

“…Here we want to point out that the idea of truncation was applied to analyze and compute a one-dimensional (1D) IHCP by Eldén et al [8] in which they called it Fourier method. Trong et al also applied the idea of truncation to the 1D and 2D source identification problems [25,26]. Regińska et al applied the idea of truncation to a Cauchy problem for the Helmholtz equation [23].…”

Determination of the body force of a two-dimensional isotropic elastic body

“…Although we have many works on the uniqueness and the stability of inverse source problems, the literature on the regularization problem is quite scarce. Very recently, in [13,14], the authors considered the regularization problem under both lateral and final overdetermination. The ideas of using the Fourier transform and truncated integration in the two papers are used in the present paper.…”

“…He proved the uniqueness of problem with prescribed lateral and final data. In the last three decades, the problem has been considered by many authors (see [4][5][6]8,13,14]). Although we have many works on the uniqueness and the stability of inverse source problems, the literature on the regularization problem is quite scarce.…”