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 boundary influx condition from knowledge of position moving front and initial state. derived logarithmic stability estimate that shows inversion may be severely ill-posed. The proof is based on integral equations unique continuation holomorphic functions. also proposed direct algorithm with regularization term to solve nonlinear problem. Several numerical tests using noisy data are provided relative errors.