Abstract

We extend a meshless method of fundamental solutions recently proposed by the authors for the one-dimensional two-phase inverse linear Stefan problem, to the nonlinear case. In this latter situation the free surface is also considered unknown which is more realistic from the practical point of view. Building on the earlier work, the solution is approximated in each phase by a linear combination of fundamental solutions to the heat equation. The implementation and analysis are more complicated in the present situation since one needs to deal with a nonlinear minimization problem to identify the free surface. Furthermore, the inverse problem is ill-posed since small errors in the input measured data can cause large deviations in the desired solution. Therefore, regularization needs to be incorporated in the objective function which is minimized in order to obtain a stable solution. Numerical results are presented and discussed.

Highlights

  • The two-phase direct Stefan problem requires determining the temperature distribution and the moving free interface when the initial and boundary conditions, as well as the thermal properties of the bi-material involved, are known, see e.g. [19]

  • In contrast to the direct problem, inverse Stefan problems require determining some initial temperature and/or boundary conditions, and/or thermal properties from additional information, which may involve the partial knowledge of the free surface, the temperature measured at some points inside the medium or on the boundary, the heat flux, etc., see [10]

  • When the position of the moving interface is unknown and no temperature or heat flux boundary conditions are specified on a part of the boundary one deals with a nonlinear and ill-posed inverse Stefan problem, see [11]

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Summary

Introduction

The two-phase direct Stefan problem requires determining the temperature distribution and the moving free interface when the initial and boundary conditions, as well as the thermal properties of the bi-material involved, are known, see e.g. [19]. The two-phase direct Stefan problem requires determining the temperature distribution and the moving free interface when the initial and boundary conditions, as well as the thermal properties of the bi-material involved, are known, see e.g. In contrast to the direct problem, inverse Stefan problems require determining some initial temperature and/or boundary conditions, and/or thermal properties from additional information, which may involve the partial knowledge of the free surface, the temperature measured at some points inside the medium or on the boundary, the heat flux, etc., see [10]. When the position of the moving interface is unknown and no temperature or heat flux boundary conditions are specified on a part of the boundary one deals with a nonlinear and ill-posed inverse Stefan problem, see [11].

Mathematical formulation
The method of fundamental solutions
Numerical results and discussion
Example 1
Example 2
Example 3
Conclusions
Full Text
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