Determination of time-dependent coefficients for a weakly degenerate heat equation

In this paper, we consider solving numerically for the first time inverse problems of determining the time-dependent thermal diffusivity coefficient for a weakly degenerate heat equation, which vanishes at the initial moment of time, and/or the convection coefficient along with the temperature for a one-dimensional parabolic equation, from some additional information about the process (the so-called over-determination conditions). Although uniquely solvable these inverse problems are still ill-posed since small changes in the input data can result in enormous changes in the output solution. The finite difference method with the Crank-Nicolson scheme combined with the nonlinear Tikhonov regularization are employed. The resulting minimization problem is computationally solved using the MATLAB toolbox routine lsqnonlin. For both exact and noisy input data, accurate and stable numerical results are obtained

Determination of a Time-Dependent Free Boundary in a Two-Dimensional Parabolic Problem

The retrieval of the timewise-dependent intensity of a free boundary and the temperature in a two-dimensional parabolic problem is, for the first time, numerically solved. The measurement, which is sufficient to provide a unique solution, consists of the mass/energy of the thermal system. A stability theorem is proved based on the Green function theory and Volterra’s integral equations of the second kind. The resulting nonlinear minimization is numerically solved using the lsqnonlin MATLAB optimization routine. The results illustrate the reliability, in terms of accuracy and stability, of the time-dependent free surface reconstruction

Time-Dependent Reaction Coefficient Identification Problems with a Free Boundary

The determination of time-dependent reaction coefficients in free boundary heat transfer problems is numerically investigated. The additional data which provides a unique solution is given by the Stefan boundary condition and the heat moments. The finite difference method with the Crank-Nicolson scheme combined with a regularized nonlinear optimization is employed. The resulting nonlinear system of equations is solved numerically using the MATLAB toolbox routine lsqnonlin for minimizing the Tikhonov regularization functional. A discussion of the choice of regularization parameters is provided. Numerical results are presented and discussed

An inverse problem of finding the time-dependent thermal conductivity from boundary data

We consider the inverse problem of determining the time-dependent thermal conductivity and the transient temperature satisfying the heat equation with initial data, Dirichlet boundary conditions, and the heat flux as overdetermination condition. This formulation ensures that the inverse problem has a unique solution. However, the problem is still ill-posed since small errors in the input data cause large errors in the output solution. The finite difference method is employed as a direct solver for the inverse problem. The inverse problem is recast as a nonlinear least-squares minimization subject to physical positivity bound on the unknown thermal conductivity. Numerically, this is effectively solved using the lsqnonlin routine from the MATLAB toolbox. We investigate the accuracy and stability of results on a few test numerical examples

Reconstruction of the timewise conductivity using a linear combination of heat flux measurements

The reconstruction of the timewise conductivity in the heat equation from an observation consisting of a linear combination of heat flux measurement data is considered. This inverse formulation results in a local uniquely solvable problem. The two-dimensional inverse problem is discretized using an alternating direction explicit method. The resulting constrained optimization problem is minimized iteratively by employing a MATLAB toolbox subroutine

Determination of time-dependent coefficients and multiple free boundaries

A difficult inverse problem consisting of determining the time-dependent coefficients and multiple free boundaries, together with the temperature in the heat equation with Stefan condition and several-orders heat moment measurements is, for the first time, numerically solved. The time-dependent missing information matches up quantitatively with the time-dependent additional information that is supplied. Although the inverse problem has a unique local solution, this problem is still ill-posed since small errors in input data cause large errors in the output solution. For the numerical realization, the finite difference method with the Crank-Nicolson scheme combined with the Tikhonov regularization are employed in order to obtain an accurate and stable numerical solution. The resulting nonlinear minimization problem is computationally solved using the MATLAB toolbox routine lsqnonlin. A couple of numerical examples are presented and discussed to verify the accuracy and stability of the approximate solutions

Reconstruction of an orthotropic thermal conductivity from nonlocal heat flux measurements

Raw materials are anisotropic and heterogeneous in nature, and recovering their conductivity is of utmost importance to the oil, aerospace and medical industries concerned with the identification of soils, reinforced fibre composites and organs. Due to the ill-posedness of the anisotropic inverse conductivity problem certain simplifications are required to make the model tracktable. Herein, we consider such a model reduction in which the conductivity tensor is orthotropic with the main diagonal components independent of one space variable. Then, the conductivity components can be taken outside the divergence operator and the inverse problem requires reconstructing one or two components of the orthotropic conductivity tensor of a two-dimensional rectangular conductor using initial and Dirichlet boundary conditions, as well as non-local heat flux over-specifications on two adjacent sides of the boundary. We prove the unique solvability of this inverse coefficient problem. Afterwards, numerical results indicate that accurate and stable solutions are obtained