5. The initial and boundary conditions:

We suppose that the medium initially is at rest so that initial conditions of the problem has the form:
\(\par \begin{matrix}\left.\ u\left(x,t\right)\right|_{t=0}=\left.\ \frac{\partial u\left(x,t\right)}{\partial t}\right|_{t=0}=0\ ,\ \ \ x>0,\\ \ \left.\ \theta\left(x,t\right)\right|_{t=0}=\left.\ \frac{\partial\theta\left(x,t\right)}{\partial t}\right|_{t=0}=0\ ,\ \ \ x>0,\\ \end{matrix}\text{\ \ \ \ }\) (17)
Also, the boundary conditions are taken the form
\(\text{\ \ }\par \begin{matrix}\left.\ \theta\left(x,t\right)\right|_{x=0}=G\left(t\right)=\theta_{0}e^{-\left(\frac{t}{t_{0}}\right)}\\ \left.\ u\left(x,t\right)\right|_{x=0}=0\\ \end{matrix}\text{\ \ \ \ \ \ \ \ \ \ \ \ }\) (18)
where \(t_{0}\) and \(\theta_{0}\) are constants, and the regularity boundary conditions are
\(\operatorname{}\left\{u\left(x,t\right),\ \ \theta\left(x,t\right),\ \sigma_{\text{xx}}\left(x,t\right)\right\}=0\ \ \ \ \ \ \ \ \ \)(19)