The inverse problem of determining the temperature and the timedependent thermal diffusivity from various additional nonlocal information is investigated. These nonlocal conditions can come in the form of an internal or boundary energy, or, in the one-dimensional case, as a difference boundary temperature or heat flux so as to ensure the uniqueness of solution for the heat conduction equation with unknown thermal diffusivity coefficient. The Ritz-Galerkin method with satisfier function is employed to solve the inverse problems numerically. Numerical results are presented and discussed.

(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.

Multiple time-dependent coefficient identification thermal problems with an unknown free boundary are investigated. The difficulty in solving such inverse and ill-posed free boundary problems is amplified by the fact that several quantities of physical interest (conduction, convection/advection and reaction coefficients) have to be simultaneously identified. The additional measurements which render a unique solution are given by the heat moments of various orders together with a Stefan boundary condition on the unknown moving boundary. Existence and uniqueness theorems are provided. The nonlinear and ill-posed problems are numerically discretised using the finite-difference method and the resulting system of equations is solved numerically using the MATLAB toolbox routine lsqnonlin applied to minimizing the nonlinear Tikhonov regularization functional subject to simple physical bounds on the variables. Numerically obtained results from some typical test examples are presented and discussed in order to illustrate the efficiency of the computational methodology adopted.

517.95We consider the inverse problem of determining the time-dependent coefficient of the leading derivative in a full parabolic equation under the assumption that this coefficient is equal to zero at the initial moment of time. We establish conditions for the existence and uniqueness of a classical solution of the problem under consideration.The present work is a continuation of our study of inverse parabolic equations with degeneration [1]. Problems of this type have broad applications in the oil-extractive industry, biology, medicine, finances, etc., i.e., in the fields where it is impossible, for some obvious reasons, to exactly measure the parameters of the process under consideration and where the application of the mathematical apparatus of inverse problems is especially efficient. One of the first researchers who studied the inverse problem of determining the coefficient of the leading derivative in a parabolic equation was B. F. Jones [2]. Inverse problems with degeneration in spatial variables were considered in [3 -5] for hyperbolic and elliptic equations with unknown free term and lower coefficient. However, the problem of determining the unknown coefficient that is degenerate in time variable remains open. Problems with weak and strong degeneracies are essentially different because the reconstruction of the unknown coefficient and its behavior depend on different initial data. In the case of weak degeneration, which differs slightly from the nondegenerate case, the influence of lower terms does not change the results obtained for the heat equation in [6]. In the case of a full parabolic equation with strong degeneration, the situation is completely different. Statement of the Problem and Main ResultsIn the domain Q T = { ( x, t ) : 0 < x < h, 0 < t < T }, we consider the parabolic equationwith an unknown coefficient a ( t ) > 0, t ∈ ( 0, T ], the initial conditionthe boundary conditions

517.95We establish conditions for unique determination of an unknown source in a parabolic equation for the case of general boundary conditions and overdetermined conditions.