7. Numerical results and verification

This section is devoted to describe and confirm the results achieved in the preceding sections. Also, we present the numerical results using the value of the Copper material at \(T_{0}=293\ K\) as [27]
\(\par \begin{matrix}\lambda=7.76\times 10^{10}\text{kg\ }m^{-1}s^{-2},\ \ \ \mu=3.86\times 10^{10}\text{kg\ }m^{-1}s^{-2},\ \ \ \rho=8954\ kg\ m^{-3},\\ K=386\ W\ m^{-1}K^{-1},\ \ \ C_{e}=3.381\ J\ kg\ K^{-1},\ \ \ \varepsilon=0.0168.\\ \end{matrix}\)
The obtained results are accessible in Tables (1-4) and graphically in Figs. 1-9 at several values of the distance \(x(0\leq x\leq 4\)) and distinct values of the two fractional parameters\(\ \ \alpha,\ \ \beta\)at \(t=0.15\), when the dual phase-lags \(\tau_{\theta}=0.03\) and\(\tau_{q}=0.05\) in the present external body forces\(F_{x}=e^{-\text{ωx}}\). The Mathematica programming Language is used within our numerical calculations. Comparisons are accomplished with the expected results predicted by all the models mentioned. Furthermore, the numerical calculations are prepared for three directions.