Auxiliary functions in the study of Stefan-like problems with variable thermal properties

Abstract

We address the existence and uniqueness of the so-called modified error function that arises in the study of phase-change problems with specific heat and thermal conductivity given by linear functions of the material temperature. This function is defined from a differential problem that depends on two parameters which are closely related with the slopes of the specific heat and the thermal conductivity. We identify conditions on these parameters which allow us to prove the existence of the modified error function. In addition, we show its uniqueness in the space of nonnegative bounded analytic functions for parameters that can be negative and different from each other. This extends known results from the literature and enlarges the class of associated phasechange problems for which exact similarity solutions can be obtained. In addition, we provide some properties of the modified error function considered here.

On the existence, uniqueness, and new analytic approximate solution of the modified error function in two‐phase Stefan problems

“…In this note, a result for the existence and uniqueness of the solution to Pr. ( 1) is proved without any conditions on the parameters 𝛾 and 𝛿, which considerably extends the result in Ceretani et al 4 It is often very difficult, if not impossible, to find explicit solutions of such problems. However, the Adomian decomposition method (ADM) is the most important tool for finding solutions to this problem.…”

“…Also, explicit approximations solutions were provided in Bougoffa. 3 Recently, Ceretani et al 4 proved the existence and uniqueness of the solution of Pr. (1) under the following severely restrictive condition on 𝛿 and 𝛾 (Theorem 2.1 4 ):…”

“…Recently, Ceretani et al 4 proved the existence and uniqueness of the solution of Pr. () under the following severely restrictive condition on δ and γ (Theorem 2.1 4 ): which is also rather quite complicated.…”

An investigation of spherical micro/nanoparticle melting using asymptotic matchings in a weak formulation

“…It is found that the prescribed Gibbs-Thomson condition at the solid-molten interface leads to mathematical complexity of solving the Stefan problems analytically (Carslaw and Jaeger, 1959;Crank, 1984) so that asymptotic techniques, for example using the large Stefan number (Davis and Hill, 1982;Herrero and Velázquez, 1997; Kucera and Hill, 1986;McCue et al, 2008McCue et al, , 2009 and numerical methods (Crank, 1984;Meyer, 1973;Voller and Cross, 1981), are sought. The existence and uniqueness of certain functions related to phase-change problems are also studied (Ceretani et al, 2020). Moreover, the Gibbs-Thomson condition imposed on the moving boundary poses difficulty on using the Baiocchi transform so that one can not employ the enthalpy method (Meyer, 1973;Voller and Cross, 1981) to tackle the governing equations numerically (McCue et al, 2009).…”

Explicit solution for non-classical one-phase Stefan problem with variable thermal coefficients and two different heat source terms

“…where δ and p are given non-negative constants, k 0 = k (θ f ) and c 0 = c (θ f ) are the reference thermal conductivity and the specific heat coefficients, respectively. Some other models involving temperature-dependent thermal conductivity can also be found in [3,4,11,13,23,24,26,27,36,38,40,41,42,43,44,45].…”