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

Abstract

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.

Keywords:

• Inverse problem
• stability estimates
• free boundary
• Stefan problem

2010 Mathematics Subject Classifications:

• 35R30
• 35R35

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The work of F. T. is supported in part by the grant ANR-17-CE40-0029 of the French National Research Agency ANR (project MultiOnde).