The meshfree strong form methods for solving one dimensional inverse Cauchy-Stefan problem

TLDR: The proposed method makes appropriate shape functions which possess the important Delta function property to satisfy the essential conditions automatically and provides the space-time approximations for the heat temperature derived by expanding the required approximate solutions using collocation scheme based on radial point interpolation method (RPIM).

Abstract: In this paper, we extend the application of meshfree node based schemes for solving one-dimensional inverse Cauchy-Stefan problem. The aim is devoted to recover the initial and boundary conditions from some Cauchy data lying on the admissible curve s(t) as the extra overspecifications. To keep matters simple, the problem has been considered in one dimensional, however the physical domain of the problem is supposed as an irregular bounded domain in $$\mathbb {R}^2$$R2. The methods provide the space-time approximations for the heat temperature derived by expanding the required approximate solutions using collocation scheme based on radial point interpolation method (RPIM). The proposed method makes appropriate shape functions which possess the important Delta function property to satisfy the essential conditions automatically. In addition, to conquer the ill-posedness of the problem, particular optimization technique has been applied for solving the system of equations $$Ax=b$$Ax=b in which A is a nonsymmetric stiffness matrix. As the consequences, reliable approximate solutions are obtained which continuously depend on input data.