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.
Bibliographical noteCreative commons attribution 3.0 Unported.
Funding: EPSRC; Isaac Newton Institute for the Visiting Fellowships within the programme on “Inverse Problems” (November-December 2011).
- heat conduction
- two-phase change
- method of fundamental solutions
- inverse Stefan problem