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We extend the meshless method of fundamental solutions proposed in B.T. Johansson, D. Lesnic, and T. Reeve, A method of fundamental solutions for the one-dimensional inverse Stefan problem, Appl. Math. Model. 35 (2011), pp. 4367–4378 for the one-dimensional one-phase inverse Stefan problem to the two-phase change case. The implementation and analysis are more complicated and meaningful since one needs to handle composite layered materials. Furthermore, the inverse problem is ill-posed since small errors in the input data lead to large deviations in the solution. Therefore, the inverse problem is intractable to classical methods of linear inversion, and regularization is employed in our study. Numerical results obtained when the input data is either exact or contaminated with random noise are compared with the analytical solution, where available, or with the numerical results obtained by other methods, D.D. Ang, A. Pham Ngoc Dinh, and D.N. Tranh, Regularization of an inverse two-phase Stefan problem, Nonlinear Anal. 34 (1998), pp. 719–731, otherwise.
Johansson et al. (Tue,) studied this question.