A heat polynomial method for inverse cylindrical one-phase Stefan problems

Abstract

In this paper, solutions of one-phase inverse Stefan problems are studied. The approach presented in the paper is an application of the heat polynomials method (HPM) for solving one- and two-dimensional inverse Stefan problems, where the boundary data is reconstructed on a fixed boundary. We present numerical results illustrating an application of the heat polynomials method for several benchmark examples. We study the effects of accuracy and measurement error for different degree of heat polynomials. Due to ill-conditioning of the matrix generated by HPM, optimization techniques are used to obtain regularized solution. Therefore, the sensitivity of the method to the data disturbance is discussed. Theoretical properties of the proposed method, as well as numerical experiments, demonstrate that to reach accurate results it is quite sufficient to consider only a few of the polynomials. The heat flux for two-dimensional inverse Stefan problem is reconstructed and coefficients of a solution function are found approximately.