AI 中文总结
该研究求解带传热边界条件的麦克斯韦-卡塔内奥-韦尔诺特非傅里叶热传导方程，构建双正交系统得到精确解析解，分析谱结构并与数值解对比，为非自伴双曲热传导提供解析基准。
AI 中文摘要
针对最著名的非傅里叶热传导模型——麦克斯韦-卡塔内奥-韦尔诺特方程，在存在传热边界条件的情况下对其进行了解析求解。与对应的傅里叶问题不同，该边界条件使得 underlying 微分算子成为非自伴的。通过合适的标量积，建立了伴随特征值问题，得到的左、右本征函数构成双正交系统，可从任意平方可积初始条件确定展开系数，从而得到方便的无穷级数解析解。针对模型参数的不同取值（包括近傅里叶和高度双曲区域）以及两种具有实际重要性的初始条件（平衡态和闪光脉冲引发），对该解析解进行了呈现和深入研究。详细分析了谱结构，包括实、虚和复共轭特征值根的出现、其渐近分布，以及它们对无量纲弛豫时间和毕渥数的依赖关系。研究发现其与对应的有限差分数值解吻合良好，该比较还阐明了谱截断（即吉布斯现象）和数值耗散在传播热波前沿附近的影响。通过数值方法证明了本征函数集的完备性，还解析推导了闪光脉冲引发的初始条件。研究结果为非自伴双曲热传导提供了解析基准，揭示了边界传热和弛豫时间如何影响类傅里叶扩散与热波传播之间的转变，并为通过实验和实际应用中的传热来发现和研究非傅里叶热传导开辟了可能性。
英文摘要
The most well-known beyond-Fourier heat conduction model, the Maxwell--Cattaneo--Vernotte equation is solved analytically in the presence of heat transfer boundary condition. In contrast to the corresponding Fourier problem, this boundary condition renders the underlying differential operator non-selfadjoint. With a suitable scalar product, the adjoint eigenvalue problem is established. The resulting left and right eigenfunctions constitute a biorthogonal system, allowing the expansion coefficients to be determined from arbitrary square-integrable initial conditions. This enables a convenient infinite-sum analytical solution, which is presented and thoroughly investigated for various values of the model parameters (including near-Fourier and highly hyperbolic regimes) and for two practically important initial conditions (equilibrium and flash pulse initiated). The spectral structure is analyzed in detail, including the occurrence of real, imaginary, and complex-conjugate eigenvalue roots, their asymptotic distribution, and their dependence on the dimensionless relaxation time and Biot number. We find good agreement with corresponding finite-difference numerical solutions. The comparison also illustrates the effects of spectral truncation (i.e., the Gibbs phenomenon) and numerical dissipation near propagating thermal-wave fronts. Completeness of the eigenfunction set is numerically demonstrated. The initial condition induced by the flash pulse is derived analytically. The results provide an analytical benchmark for non-selfadjoint hyperbolic heat conduction, unveil how boundary heat transfer and relaxation time influence the transition between Fourier-like diffusion and thermal-wave propagation, and open the possibility to find and investigate beyond-Fourier heat conduction via heat transfer in experiments and practical applications.
Comments35 pages, 19 figures
