Analytical solutions to hyperbolic heat conductive models using Green's function method

Abstract

In this work, the existing theoretical heat conductive models such as: Cattaneo-Vernotte model, simplified thermomass model, and single-phase-lag two-step model are summarized, and then a general model of hyperbolic heat conduction (HHC) is presented with boundary conditions prescribed as: (1) temperature at the boundary; (2) heat flux (not the temperature gradient) at the boundary. The convective boundary condition is not considered because it is impossible to produce a fluid motion at such time scale, e.g. picoseconds. In the context of HHC, the reciprocity relation of Green's function is systematically proven, based on which Green's function solution equation of the general model is completely derived. Meanwhile, Green's functions are deduced under several different sets of boundary conditions. Thus, solution to HHC based on Green's function is obtained, and it consists of three parts, i.e. boundary conditions, initial conditions and heat generations. The priority of the present solution is that the time-dependent boundary condition and heat generation may be dealt with, and that it is easy to extend the solution to multi-dimensional case. The accuracy of the solution is verified through numerical examples, and Laplace transform method is adopted to avoid the dispersions around the front of temperature wave for jump-type boundary conditions, thus the solution is further perfected.

Identification of the thermo-physical properties of a stratified tissue. Adiabatic hypodermic wall

“…where n is the outward unit normal to the boundary ∂Ω, maintains as the zero-Neumann boundary condition −k∂ n T = 0, but a non-zero flux prescription does not maintain its usual form in terms of the normal derivative of T , [28]. On the other hand the invariance holds for a Robin boundary condition that arises from a generalized Newton's law of the form, [14],…”

“…Similarly, introducing (33) in (28), we obtain: 2∆x l+1 A l+1 u l+1 P l ,j+1 x + B l+1 u l+1 P l ,j+1 − 2A l+1 u l+1 P l +1,j+1 = −2∆x l+1 A l+1 u l+1 P l ,j x + C l+1 u l+1 P l ,j + 2A l+1 u l+1 P l +1,j + 2v l+1 P l ,j + ∆t 2…”

Reconstruction of the thermal properties in a wave-type model of bio-heat transfer

“…whereq c denotes the computed generalized flux and σ 2 f is the variance of the noise in the corresponding dimensional generalized flux measurement found using the MATLAB command var. In the case of noisy measurement,q is replaced byq ǫ (defined in (36)) in (42) and (43).…”

“…In the next subsection, we numerically simulate the generalized heat flux (35) of the dimensional model given by equations (7), (8) and (11), and the dimensionless boundary temperatures of the dimensionless model given by equations (13)-(15), to be used as additional measurements for a comparison between the presented inversion techniques discussed in Sections 4.1 and 4.2, respectively, and for the reconstruction of all possible parameters of the dimensionless model given by equations (13)-(15), respectively. It is worth pointing out that, according to equation (35), in the hyperbolic model of bio-heat transfer that is considered in this paper, the temperature gradient is not the heat flux q (0, t ) but the generalized heat flux q ˜( t ):= q (0, t )+ τ ∂ t q (0, t ) (Yu, 2018).…”

“…that is considered in this paper, the temperature gradient is not the heat flux q(0, t) but the generalized heat flux q t ð Þ :¼ q 0; t ð Þþ t @ t q 0; t ð Þ (Yu, 2018).…”