On the one-dimensional two-phase inverse Stefan problems

Abstract

New formulations of the inverse nonstationary Stefan problems are considered:
(a) for $x ∈ [0,1]$ (the inverse problem IP_1;
(b) for $x ∈ [0, β(t)]$ with a degenerate initial condition (the inverse problem IP_{β}).
Necessary conditions for the existence and uniqueness of a solution to these problems are formulated. On the first phase $\{x ∈ [0, y(t)]\}$, the solution of the inverse problem is found in the form of a series; on the second phase $\{x ∈ [y(t), 1]$ or $x ∈ [y(t), β (t)]\}$, it is found as a sum of heat double-layer potentials. By representing the inverse problem in the form of two connected boundary-value problems for the heat conduction equation in the domains with moving boundaries, it can be reduced to the integral Volterra equations of the second kind. An exact solution of the problem IPβ is found for the self similar motion of the boundariesx=y(t) andx=β(t).