A meshless method for an inverse two-phase one-dimensional nonlinear Stefan problem

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. © 2014 IMACS.

Identification of a time-dependent bio-heat blood perfusion coefficient

In the paper the identification of the time-dependent blood perfusion coefficient is formulated as an inverse problem. The bio-heat conduction problem is transformed into the classical heat conduction problem. Then the transformed inverse problem is solved using the method of fundamental solutions together with the Tikhonov regularization. Some numerical results are presented in order to demonstrate the accuracy and the stability of the proposed meshless numerical algorithm.