The Method Of Fundamental Solutions For Detection Of A Moving Boundary In Heat Equation

In this paper,the method of fundamental solutions(MFS)is used to deter-mine a moving boundary from Cauchy data in one dimensional heat equation. This problem is severely ill-posed[1].The main process of solving this problem is follows:First,according to the main idea of the MFS,we approximate an unknown solution of the heat equation by a linear combination of fundamental solutions whose singularities are located outside the solution domain.Second,the coefficients and the unknown moving boundary will be determined by imposing the boundary conditions in an uncon-strained optimization problem,i.e.minimize an objective function.Because of the ill-posedness of this problem,i.e.any small change on the input Cauchy data on boundary can result in a dramatic change to the solution (unknown moving boundary),in order to obtain a stable solution when the Cauchy data is disturbed, we will amend the objective function for adjusting the weight of residual terms in the objective function.Then,we discrete the objective function amended into a sum of squares of nonlinear multivariate function.Finally, we will use numerical software MATLAB to solve the unconstrained optimization problem and obtain the minimizer which contains the numerical solution of the moving boundary.