Identification of a boundary influx condition in a one-phase Stefan problem

We consider a one-dimensional one-phase inverse Stefan problem for the heat equation. It consists in recovering a boundary influx condition from the knowledge of the position of the moving front and the initial state. We derived a logarithmic stability estimate that shows that the inversion may be severely ill-posed. The proof is based on integral equations and unique continuation of holomorphic functions. We also proposed a direct algorithm with a regularization term to solve the nonlinear inverse problem. Several numerical tests using noisy data are provided with relative errors.

Automated summary

This study examines a one-dimensional inverse Stefan problem for the heat equation, showing through a logarithmic stability estimate that the inversion may be severely ill-posed. A direct algorithm with a regularization term is proposed to solve the nonlinear inverse problem. Numerical tests using noisy data are conducted, providing relative errors.