The work in this paper concerns the study of different approximations for one-dimensional one-phase Stefan-like problems with a space-dependent latent heat. It is considered two different problems, which differ from each other in their boundary condition imposed at the fixed face: Dirichlet and Robin conditions. The approximate solutions are obtained by applying the heat balance integral method (HBIM), the modified HBIM and the refined integral method (RIM). Taking advantage of the exact analytical solutions, we compare and test the accuracy of the approximate solutions. The analysis is carried out using the dimensionless generalised Stefan number (Ste) and Biot number (Bi). It is also studied the case when Bi goes to infinity in the problem with a convective condition, recovering the approximate solutions when a temperature condition is imposed at the fixed face. Some numerical simulations are provided in order to assert which of the approximate integral methods turns out to be optimal. Moreover, we pose an approximate technique based on minimising the least-squares error, obtaining also approximate solutions for the classical Stefan problem.

中文翻译：

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具有空间相关潜热的单相 Stefan 类问题的近似解

本文的工作涉及研究具有空间相关潜热的一维单相 Stefan 类问题的不同近似。它被认为是两个不同的问题，它们在固定面上施加的边界条件彼此不同：Dirichlet 和 Robin 条件。采用热平衡积分法(HBIM)、改进的HBIM和精积分法(RIM)得到了近似解。利用精确解析解，我们比较和测试近似解的准确性。使用无量纲广义 Stefan 数 (Ste) 和 Biot 数 (Bi) 进行分析。还研究了在对流条件下 Bi 趋于无穷大的情况，当在固定面施加温度条件时恢复近似解。提供了一些数值模拟以断言哪种近似积分方法是最佳的。此外，我们提出了一种基于最小二乘误差的近似技术，也获得了经典 Stefan 问题的近似解。