Determination of one or two unknown thermal coefficients of a semi-infinite material through a two-phase stefan problem

Abstract

This paper consists of two parts: 1. (1) We consider the two-phase (fusion) Stefan problem for a semi-infinite material with one unknown thermal coefficient. We have an overspecified condition of type k 2 ∂T 2 ∂x (0, t) = −ho/√t with ho > 0 on the fixed face x = 0 of the phase-change material. We obtain: (i) if ρ is unknown, then the corresponding free boundary problem always has a unique solution of the Neumann type; (ii) if one of the remaining five coefficients is unknown, then the corresponding free boundary problem has a unique solution of the Neumann type iff a complementary condition is verified. 2. (2) We consider the inverse two-phase (fusion) Stefan problem for a semi-infinite material with an overspecified condition k 2 ∂T 2 ∂x (0, t) = −ho/√t with ho > 0 on the fixed face and with two unknown thermal coefficients. We obtain: (i) if ( p, k 2) are unknown, the corresponding moving boundary problem always has a unique solution of the Neumann type; (ii) if ( l, k 1), ( l, c 1) or (, k 1, c 1) are unknown, the corresponding moving boundary problem has infinite solutions whenever the complementary conditions are verified; (iii) in the remaining eleven cases, the corresponding moving boundary problem has a unique solution of the Neumann type iff complementary conditions are verified. Moreover, in both parts, we obtain formulas for the unknown thermal coefficients.