两相源与反应系数的斯蒂芬型问题
Two-phase source and reaction coefficient Stefan type problems
AI总结：
研究抛物型热方程逆两相斯蒂芬型问题，通过变换和傅里叶谱展开推导未知系数重构公式，分两种表述识别源和反应系数，利用移动边界的斯蒂芬条件确定系数，建立解的性质，示例验证方法适用性及系数在噪声下的稳定性。
AI中文摘要：
本文研究具有未知时间相关源和反应系数的抛物型热方程的逆两相斯蒂芬型问题。通过适当变换将原始自由边界问题转化为固定空间域上的等价方程，利用傅里叶谱展开推导未知系数的重构公式。在第一种表述中，从积分、逐点和非局部附加数据识别源系数，得到沃尔泰拉积分方程。在第二种表述中，应用指数变换恢复反应系数。具有两个移动边界的模型的主要优点是每个边界都提供自己的斯蒂芬条件，足以确定两个未知的时间相关系数而无需额外的超定条件。在适当假设下建立了弱解和强解的存在性、唯一性、有界性和正则性。示例证实了重构过程的适用性，并表明系数在噪声数据下保持稳定。所提出的方法为识别两相相变过程中的时间相关热参数提供了严格的基础。
英文摘要：
This paper investigates inverse two-phase Stefan-type problems for parabolic heat equations with unknown time-dependent source and reaction coefficients. Suitable transformations reduce the original free-boundary problems to equivalent equations on fixed spatial domains, and Fourier spectral expansions are used to derive reconstruction formulas for the unknown coefficients. In the first formulation, the source coefficients are identified from integral, pointwise, and nonlocal additional data, leading to Volterra integral equations. In the second formulation, an exponential transformation is applied to recover the reaction coefficients. A principal advantage of the model with two moving boundaries is that each boundary provides its own Stefan condition which sufficient to determine the two unknown time-dependent coefficients without additional overdetermination conditions. Under appropriate assumptions the existence, uniqueness, boundedness and regularity of weak and strong solutions are established. Illustrative examples confirm the applicability of the reconstruction procedure and show that the coefficients remain stable under the noisy data. The proposed approach provides a rigorous base for identifying time-dependent thermal parameters in two-phase phase-change processes.
