Simultaneous determination of time-dependent coefficients in the heat equation

Abstract

(M.S. Hussein), amt5ld@maths.leeds.ac.uk (D. Lesnic), ivanchov@franko.lviv.ua (M.I. Ivanchov). AbstractIn this paper, the determination of time-dependent leading and lower-order thermal coefficients is investigated. We consider the inverse and ill-posed nonlinear problems of simultaneous identification of a couple of these coefficients in the one-dimensional heat equation from Cauchy boundary data. Unique solvability theorems of these inverse problems are supplied and, in one new case where they were not previously provided, are rigorously proved. However, since the problems are still ill-posed the solution needs to be regularized. Therefore, in order to obtain a stable solution, a regularized nonlinear least-squares objective function is minimized in order to retrieve the unknown coefficients. The stability of numerical results is investigated for several test examples with respect to different noise levels and for various regularization parameters. This study will be significant to researchers working on computational and mathematical methods for solving inverse coefficient identification problems with applications in heat transfer and porous media.

Simultaneous determination of time-dependent coefficients and heat source

“…Moreover, multiple time-dependent coefficient identifications have also been considered theoretically in the past, see e.g. [3,11], and recently been solved numerically by the authors, [8]. In these studies, the unknowns were mainly coefficients multiplying the temperature and its partial derivatives, but more recent theoretical studies, [14,15], allow for one of the time-dependent unknown to be in the free term heat source as well.…”

Reconstruction of time-dependent coefficients from heat moments

“…Finally, we mention that a comparison between Figures 8 and 9, and Figures 22 and 24 of [6], respectively, shows that the IP1 based on measuring the heat moments (5) is less stable than when measuring the heat fluxes (4). This is to be expected since supplying the bounded normal derivatives (4) contains stronger information than prescribing the integral average heat moments (5).…”

“…Prior to this study, the simultaneous identification of the coefficients a(t) and b(t) in the problem (1)-(3) with the additional flux data (4) has been considered in [6]. In this paper, we consider the simultaneous reconstruction of the same time-dependent coefficients, as well as of a(t) and c(t), but from the heat moments (5) instead of the heat fluxes (4).…”

“…In this example, we consider the IP1 given by equations (2), (3), (5) and (6) with unknown coefficients a(t) and b(t) and we solve this inverse problem with the following input data:…”

“…In a recent paper [6] by the authors we have investigated the inverse problems of simultaneous numerical reconstruction of time-dependent thermal conductivity and convection coefficients in a one-dimensional parabolic equation from Cauchy boundary data measurements represented by the boundary temperature and heat flux. In this paper, we investigate the reconstruction of the same coefficients, as well as of the absorption coefficient, using the measurement of the heat moments instead of the heat flux.…”