4. An application to the constructed model

In this section, we apply our modified model for an isotropic homogeneous, thermoelastic the half-space \(x\geq 0\) under an external body force and subjected to exponential varying heat. Also, we supposed that the state of the medium depends only on \(x\), \(t\) and that the displacement vector\(\ \overrightarrow{u}=(u\left(x,t\right),0,0)\). In this case, the constitutive equation will have the following form
\(\sigma_{\text{xx}}=(\lambda+2\mu)\frac{\partial u}{\partial x}-\text{γθ}\)(9)
Also, the equation of motion in the present external force \(F_{x}\) in the one dimensional case has the form
\(\rho\frac{\partial^{2}u}{\partial t^{2}}=\left(\lambda+2\mu\right)\frac{\partial^{2}u}{\partial x^{2}}-\gamma\frac{\partial\theta}{\partial x}+F_{x}\ \)(10)
According to Eq. (7), the modified equation of heat conduction with fractional derivatives and phase lags with \(Q=0\) can be written as
\(\left(1+\tau_{q}^{\alpha}\frac{\partial^{\alpha}}{\partial t^{\alpha}}\right)\left[\rho C_{e}\frac{\partial\theta}{\partial t}+\gamma T_{0}\frac{\partial^{2}u}{\partial t\partial x}\right]=K\left(1+\tau_{\theta}^{\beta}\frac{\partial^{\beta}}{\partial t^{\beta}}\right)\frac{\partial^{2}\theta}{\partial x^{2}}\)(11)
We introduce the non-dimensional variables as
\(\par \begin{matrix}\left\{x^{{}^{\prime}},u^{{}^{\prime}}\ \right\}=c_{1}\eta\left\{x,u\right\},\ \ \left\{t^{\prime},\ \tau_{q}^{{}^{\prime}},\tau_{\theta}^{{}^{\prime}}\right\}=c_{1}^{2}\eta\left\{t,\ \tau_{q\ },\tau_{\theta\ }\right\}\ ,\ \ \eta=\frac{\rho C_{e}}{K},\\ {\text{\ \ }\theta^{\prime}=\frac{\text{γθ}}{\lambda+2\mu},\ \ \sigma^{\prime}}_{\text{xx}}=\frac{\sigma_{\text{xx}}}{\lambda+2\mu},\ \ \ c_{1}=\sqrt{\frac{\lambda+2\mu}{\rho}\ },\ \ \ F_{x}^{{}^{\prime}}=\frac{F_{x}}{\rho c_{1}^{3}\eta}\text{\ .}\\ \end{matrix}\text{\ \ \ }\) (12)
Using Eq. (12), the non-dimensional form of Eqs. (9)-(11) becomes
\(\frac{\partial^{2}u}{\partial t^{2}}=\frac{\partial^{2}u}{\partial x^{2}}-\frac{\partial\theta}{\partial x}+F_{x}\)(13)
\(\left(1+\tau_{q}^{\alpha}\frac{\partial^{\alpha}}{\partial t^{\alpha}}\right)\left[\frac{\partial\theta}{\partial t}+\varepsilon\frac{\partial^{2}u}{\partial t\partial x}\right]=\left(1+\tau_{\theta}^{\beta}\frac{\partial^{\beta}}{\partial t^{\beta}}\right)\frac{\partial^{2}\theta}{\partial x^{2}}\)(14)
\(\sigma_{\text{xx}}=\frac{\partial u}{\partial x}-\theta\) (15)
where
\(\text{\ \ \ }\varepsilon=\frac{\gamma^{2}T_{0}}{\rho C_{e}\left(\lambda+2\mu\right)}\)(16)
For convenience and clarity, we have dropped the primes in the above equations. Also, we take\(F_{x}=f\left(x\right)=e^{-\text{ωx}}\), where \(\omega\) is a constant parameter (decaying parameter).