An inverse problem for a stationary heat transfer process is studied for a totally isolated bar on its lateral surface, made up of two consecutive sections of different, isotropic and homogeneous materials, perfectly assembly, where one of the materials, that is unreachable and unknown, has to be identified. The length of the bar is assumed to be much greater that the diameter so that a 1D heat transfer process is considered. A constant temperature is assumed at the end of the unknown part of the rod while the other end is let free for convection. We propose a procedure to identify the unknown material of the bar based on a noisy flow measurement at the opposite end. Necessary and sufficient conditions are derived together with a bound for the estimation error. Moreover, elasticity analysis is performed to study the influence of the data in the conductivity estimation and numerical examples are included to illustrate the proposed ideas and show the estimation performance.

In this work, a bar fully insulated on its lateral surface. composed by two different unknown materials is considered. For the analytical solution, it is assumed a perfectly assembly solid-solid interface, so no heat loss due to friction is present. This is an ideal scenario, so this loss and possible measurement errors are included by simulating noisy data for the estimation of the thermal conductivity of the unknown materials. A stationary heat transfer process along the bar is considered where a Dirichlet condition is imposed at the left that represents a source of constant temperature. At the other end of the bar, a Robin condition that models heat dissipation by convection, is assumed. The constant thermal conductivity coefficients of both solids are determined under two different situations: a) two noisy temperature measurements are available, one at the interface and the other at the right boundary; b) a temperature measurement at the interface and a heat flow measurement at the right edge of the bar are given. The bounds for the errors in the identification of the unknown coefficients are obtained based on the data measurements, the room temperature and temperature values at the boundary and interface. Numerical examples are given to illustrate the ideas used for the parameter identification and elasticity analysis is carried out to measure the dependence of the data on the estimated parameters.

An inverse problem for a stationary heat transfer process is studied for a totally isolated bar on its lateral surface, of negligible diameter, made up of two consecutive sections of different, isotropic and homogeneous materials. At the left boundary, a Dirichlet type condition is imposed that represents a constant temperature source while a Robin type condition that models the heat dissipation by convection is considered at the right one. Many articles in the literature focus on thermal and stress analysis at the interface but no one is dedicated to the estimation of the contact point location between the two materials. In this work, it is assumed that the interface position is unknown. A technique to determine it from a unique noisy flow measurement at the right boundary is introduced. Necessary and sufficient conditions are derived in order to obtain the estimation of the interface point from a heat flux measured at the right boundary. Numerical solutions are obtained together with an expression for the estimation error. Moreover, an elasticity analysis is included to study the influence of data errors. The results show that our approach is useful for determining the location of the materials interface.

In this work we study the characteristics of the dissipation by convection of a solid circular section of a diameter d in a fluid. We assume that this section increases its temperature homogeneously over its whole surface from an initial temperature 𝒕𝒂 to the (asymptotic) temperature value 𝒕𝒎𝒂𝒙. To simulate its temperature behavior, we model the transfer of heat by conduction in an isolated one-dimensional bar of length L, where a constant temperature source F is considered at the left end, while keeping free the right end causing heat dissipation by convection. We propose a novel approach to estimate the heat transfer coefficient in the transient state. Numerical experiments are carry out for different materials. In order to measure the performance in the estimation, we conduct elasticity analysis. Also a comparison with data used in the literature is included.

In this work, we consider the problem of identifying the time independent source for full parabolic equations in Rn from noisy data. This is an ill-posed problem in the sense of Hadamard. To compensate the factor that causes the instability, a family of parametric regularization operators is introduced, where the rule to select the value of the regularization parameter is included. This rule, known as regularization parameter choice rule, depends on the data noise level and the degree of smoothness that it is assumed for the source. The proof for the stability and convergence of the regularization criteria is presented and a Hölder type bound is obtained for the estimation error. Numerical examples are included to illustrate the effectiveness of this regularization approach.