Exact analytical solution for the non-selfadjoint problem of Maxwell--Cattaneo--Vernotte heat conduction with heat-transfer boundary condition

Abstract

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.