On defects of Taylor series approximation in heat conduction models

Taylor series approximation like q(t+τ)≈q+τ∂q∂t are often used to derive, extend or interpret typical heat conduction models. Researchers may take it for granted that the single-phase-lagging (SPL) model can be considered as an extension of the Cattaneo-Vernotte (CV) model because there is such approximation relationship between them. We point out in this paper that this Taylor series approximation itself has some defects based on analyses in mathematics, physics and some examples first. Then, we show essential differences in both mathematics and physics between the CV and SPL models. It is found that their mathematical characteristics and accordance with the laws of thermodynamics are significantly different, which indicates that using this approximation to connect the two models may be defective in some cases. What's more, higher order approximation can't solve these problems and defects.