A Stefan problem with nonlinear thermal conductivity

Abstract

The solution is obtained and validated by an existence and uniqueness theorem for the following nonlinear boundary value problemwhich was proposed in 1974 by Cho and Sunderland to represent a Stefan problem with a nonlinear temperature-dependent thermal conductivity on the semi-infinite line (0, ∞). The modified error function of two parameters 𝜑 𝛿,𝛾 is introduced to represent the solution of the problem above, and some properties of the function are established. This generalizes the results obtained in earlier studies.

A Nonclassical Stefan Problem with Nonlinear Thermal Parameters of General Order and Heat Source Term

“…This solution was called: "the modified error function of two parameters", because it can be viewed as a generalization to the modified error function obtained by [1,3], when γ = 0, and to the classical error function when δ = γ = 0. As shown in [5], the solution φ δ,γ shares some properties with the classical error function.…”

“…This implies that Pr. ( 4)-( 5) generalize all the problems proposed by the preceding papers [1][2][3][4][5], and the solution generalizes the error function φ δ,γ . The second goal is to provide an analytic solution in addition to lower and upper bounds to Pr.…”

“…These problems have a deep connection with heat transfer theory since they tend to model phase-change problems due to melting or liquidation processes. The scientific studies concerning these problems have significantly increased in the last two decades due to the high importance and demands of describing and analyzing many industrial and physical processes, see, for example, [3][4][5][6][7][8][9][10][11][12][13][14][15][16]. The classical Stefan problems deal with constant thermal parameters (thermal conductivity and specific heat) for the substances, but due to the recent developments in technology and science, researchers realized that models that described temperature-dependent parameters would be more realistic.…”

“…Oliver and Sunderland [2] investigated a model similar to (1) where thermal conductivity and specific heat are linear functions of temperature. No existence and uniqueness theorems were established in the preceding two articles, which has motivated subsequent researchers [3][4][5][6][7][8][9] to establish the existence and uniqueness theorems for the solutions to such problems. In particular, the authors in [3] proved the existence and uniqueness of the modified error function for small values of δ > 0.…”

“…The general case δ > −1 was investigated and established in [4]. In [5], the authors investigated problem (2), with a nonlinear thermal conductivity of the form (1 + δy + γy 2 ) n , where δ > −1 and γ > −1. Existence and uniqueness theorems for the solution were obtained, and the solution was obtained in the following form:…”