The boundary element method applied to moving boundary problems

Abstract

In this paper we consider free boundary problems. For such problems the free boundary is not known before the problem is solved and must be determined as part of the solution. For the problems considered in this paper, the solution is time dependent, so the free boundary is also a moving boundary. We consider a heat conduction problem associated with ice melting which involves one spatial dimension and a problem associated with diffusion which involves two spatial dimensions. For both the one dimensional problem and the two dimensional problem we introduce the fundamental solution or Green's function for our problem and apply a Green's identity to obtain the solution of our problem in terms of a boundary integral which involves an unknown function on the free boundary. We call this integral expression for our solution (FBE). By taking the limit of (FBE) to the free boundary, we obtain a nonlinear system of integral equations for the location of the free boundary and the unknown function. We call this system (BIS). We use the boundary element method to obtain a solution of the system (BIS). We then substitute this solution of (BIS) into the equation (FBE) to obtain the solution of our original problem. Results for both the one dimensional problem and a two dimensional problem are presented.