MODELING PERIODICALLY FORCED HEAT WAVES IN INHOMOGENEOUS MEDIA VIA THE GENERALIZED FOURIER PRINCIPLE

This paper investigates the 𝑛-dimensional hyperbolic-parabolic heat conduction problem in inhomogeneous media under the influence of periodically forced thermal sources. The governing dynamics are described by a generalized Cattaneo-Vernotte type equation, supplemented by third-type generalized boundary conditions and initial Cauchy conditions. For the first time, this complex boundary value problem is solved by transitioning from classical Euclidean settings to an infinite-dimensional Hilbert space framework using the Generalized Fourier Principle. Unlike traditional numerical methods that rely on spatial discretization, our approach utilizes the spectral decomposition of the inhomogeneous operator, allowing for a precise analytical representation of heat wave propagation. The internal heat source is modeled as a time-periodic trigonometric Fourier series, enabling the study of steady-state periodic responses and resonance phenomena in functionally graded materials. The results provide a rigorous mathematical foundation for understanding finite-speed heat signals in complex media, offering a benchmark for future thermal shock analysis in advanced engineering applications.