AI 中文总结
该研究将Fokas统一变换方法扩展到半无限区间的双曲-抛物问题，构造了四分之一平面上阻尼波动方程的闭式解，揭示了保证光滑解存在的相容性条件，其积分表示可用于数值基准及多领域建模研究。
AI 中文摘要
我们严格构造并后验验证了受迫麦克斯韦-卡塔内奥-弗尔诺特方程（也被广泛称为阻尼波动方程、双曲热方程以及有损传输线上的电报方程）在时空四分之一平面上的新闭式解，该方程带有经典函数空间中的一般初始和边界数据。为此，我们首次将Fokas的现代复分析统一变换方法（最初为椭圆型偏微分方程和具有多项式色散关系的发展方程开发）扩展用于分析半无限区间上的双曲-抛物问题。重要的是，我们随后建立了相关定理，涉及新解析公式的正则性、边界和渐近性质，以及所研究边值问题的适定性。值得注意的是，所考虑问题的性质和一般性，结合定义域的半无界性，引发了重大的分析挑战，需要在适当解释解公式的振荡积分项以及证明所提结果时进行精细处理。在此过程中，关键是揭示了原点处初始、边界和强迫数据之间的某些相容性条件，这些条件保证了整个感兴趣域上光滑解的存在性。我们的显式积分表示可直接用于数值基准测试、探索与连续介质力学、数学物理、生物学和自然科学建模的联系，以及研究非线性对应问题的适定性。
英文摘要
We rigorously construct and verify a posteriori new closed-form solutions for the forced Maxwell-Cattaneo-Vernotte equation (also broadly known as the damped wave equation, hyperbolic heat, and telegrapher's equation on lossy transmission lines) posed on the spatiotemporal quarter-plane with general initial and boundary data in classical function spaces. For this purpose, the modern complex-analytic unified transform method of Fokas (originally developed for elliptic PDE and evolution equations with polynomial dispersion relations) is here, for the first time, extended for analysis of hyperbolic-parabolic problems on the semi-infinite interval. Importantly, we then establish theorems which pertain to regularity, boundary and asymptotic properties of the new analytical formulae as well as to well-posedness of the addressed boundary-value problems. Notably, the nature and generality of problems considered, combined with the semi-unboundedness of the domain, induce substantial analytic challenges which demand delicate treatment, both in appropriately interpreting oscillatory integral terms of the solution formulae and in proving the proposed results. In this process, crucially, certain compatibility conditions, between initial, boundary and forcing data at the origin, are revealed, which guarantee the existence of a smooth solution across the whole domain of interest. Our explicit integral representations are of direct utility for numercal benchmarking purposes, for exploring connections with modelling in continuum mechanics, mathematical physics, biology and the natural sciences, and for the investigation of well-posedness for nonlinear counterparts too.
