## 2 Problem Description and Mathematical Modelling
### Integral Form of Surplus Temperature  
Substituting \((t-\tau)\) for \(t\), separated variable method has been utilized to work out the one-dimensional Green's function in \(x\) direction formed as [23]  
\[G\left(x,t\big{|}x^{\prime},\tau\right)=\frac{\alpha}{k}\frac{1}{\sqrt{4\pi \alpha(t-\tau)}}\exp\left[-\frac{\left(x-x^{\prime}\right)^{2}}{4\alpha(t- \tau)}\right] \tag{7}\]  
[PERSON]'s functions in the other two directions have the similar forms so that corresponding Green's function of the multidimensional heat transfer problem in an infinite medium can be expressed as the product form of three one-dimensional problem solutions. It is worth noting that the coefficient \((\alpha/k)\) can only appear once, no matter whether the heat transfer problem is one or multidimensional [22]. So, the [PERSON]'s function of above 3D heat transfer problem can be expressed as  
\[G\left(x,y,z,t\big{|}x^{\prime},y,z,\tau\right)=\frac{\alpha}{k}\frac{1}{\left[ 4\pi\alpha(t-\tau)\right]^{3/2}}\exp[-\frac{\left[x-S(\tau)\right]^{2}+y^{2}+ z^{2}}{4\alpha(t-\tau)}] \tag{8}\]  
In our study, the heating power is a constant unchangeable with time so that operator \(g_{\rm{P}}(t)=Q\). For a reciprocating motion heat source, \(x^{\prime}=S(\tau)\) is introduced in calculation. Then, the integral form of surplus temperature can be calculated as  
\[\theta=\frac{Q}{\left(4\pi\alpha\right)^{3/2}}\frac{\alpha}{k}\left(\sum_{N=0 }^{K-1}\int_{NT}^{(N+\frac{1}{2})T}+\sum_{N=0}^{K-1}\int_{(N+\frac{1}{2})T}^{( N+1)T}+\int_{KT}^{t}\right)\frac{1}{\left(t-\tau\right)^{3/2}}\exp\left[-\frac{ \left[x-S(\tau)\right]^{2}+y^{2}+z^{2}}{4\alpha(t-\tau)}\right]{\rm{d}}\tau \tag{9}\]  
where \(Q\) is the rate of heat transfer from a point source [24], W. Locating point heat source at \(x=x^{\prime}\), the temperature distribution produced by point heat source can be represented as \(T(x)=(Q/\rho c)\times\delta(x-x^{\prime})\). This formula constructs a bridge between temperature and heat conversion. The formula can be solved in three sections, introducing  
\[\theta_{1}^{N}=\frac{Q/(\rho c)}{\left(4\pi\alpha\right)^{3/2}}\int_{NT}^{(N+ \frac{1}{2})T}\frac{1}{\left(t-\tau\right)^{3/2}}\exp\left[-\frac{\left[x-S( \tau)\right]^{2}+y^{2}+z^{2}}{4\alpha(t-\tau)}\right]{\rm{d}}\tau \tag{10}\]  
Bringing in the displacement calculation formula of the first half cycle, the position variable \((x-x^{\prime})\) over time can be noted: \(x-U(\tau-NT)=x+UNT-Ut+U(t-\tau)\)Substituting variables as \(X_{N}=x+U\!INT-Ut\); \({R_{N}}^{2}={X_{N}}^{2}+y^{2}+z^{2}\) and \(\varphi={R_{N}}^{2}/[4\alpha(t-\tau)]\), then Equation (10) can be rewritten in the following form  
\[ \begin{array}{ll}\theta_{1}^{N}=&\frac{Q}{4\pi\kappa^{3/2}{R_{N}}}\exp(- \frac{X_{N}U}{2\alpha})\\ &\times\int_{{R_{N}}^{2}/[4\alpha(t-(N+1/2)T]]}^{\frac{\alpha}{2}}\phi^{-\frac{ 1}{2}}\exp(-\phi-\frac{U^{2}{R_{N}}^{2}}{16\alpha^{2}}\phi^{-1})\mathrm{d} \phi\end{array}  \tag{11}\]  
The expression can be denoted by the generalized incomplete Gamma functions [12] in the form as  
\[ \begin{array}{ll}\theta_{1}^{N}=&\frac{Q}{4\pi\kappa^{3/2}{R_{N}}}\exp(- \frac{X_{N}U}{2\alpha})\\ &\times\Big{[}\Gamma\Big{(}\frac{1}{2},\frac{{R_{N}}^{2}}{4\alpha(t-NT)};\frac {U^{2}{R_{N}}^{2}}{16\alpha^{2}}\Big{)}-\Gamma\Big{(}\frac{1}{2},\frac{{R_{N} }^{2}}{4\alpha[t-(N+1/2)T]};\frac{U^{2}{R_{N}}^{2}}{16\alpha^{2}}\Big{)}\Big{]} \end{array}  \tag{12}\]  
The generalized incomplete Gamma function is proved useful in the analytical study of heat conduction problems in an infinite medium [16], of which the general form can be expressed as  
\[\Gamma(\alpha,t;b)=\int_{t}^{\infty}x^{\alpha-1}\exp(-x-bx^{-1})\mathrm{d}x \tag{13}\]  
Similarly, substituting variables as \(Y_{N}=x+Ut-U(N+1)T\); \({S_{N}}^{2}={Y_{N}}^{2}+y^{2}+z^{2}\) and \(\eta={R_{N}}^{2}/[4\alpha(t-\tau)]\). The second half cycle part can be expressed as  
\[