Fibonacci collocation spectral method and error analysis for heat conduction waves in an n-dimensional domain

This paper investigates the coupled thermoelastic interactions within an \(n\)-dimensional rectangular parallelepiped domain under time-dependent boundary conditions, formulating a hyperbolic system based on the Cattaneo-Vernotte principle to account for finite-speed thermal wave propagation. The mixed boundary value problem, incorporating non-homogeneous Dirichlet conditions and Cauchy initial data for displacement and temperature fields, is solved analytically via the Generalized Fourier Principle, yielding a unified solution expressed as an \(n\)-dimensional eigenfunction expansion. To validate the analytical findings and address complex configurations, a Fibonacci Collocation Spectral Method (FCSM) evaluated at Chebyshev–Gauss–Lobatto nodes is developed. Rigorous error analysis in \(L^2\) and \(L^\infty\) norms confirms spectral convergence under appropriate regularity assumptions. Numerical experiments in one, two, and three dimensions demonstrate exponential error decay from \(\mathcal{O}(10^{-3})\) to \(\mathcal{O}(10^{-14})\) with moderate polynomial degrees, establishing a robust theoretical and computational framework for analyzing wave-like thermoelastic behavior in high-precision engineering and advanced materials applications.