## 3 Hybrid Technique Implementation
### LRBFCM
\begin{split}& c\rho\bigg{(}\frac{\mathbf{T}^{f}(\mathbf{x})-\mathbf{T}^{f-1}( \mathbf{x})}{\Delta\tau}+\tau_{q}\frac{\mathbf{T}^{f}(\mathbf{x})-2\mathbf{T}^{f-1}(\mathbf{x})+ \mathbf{T}^{f-2}(\mathbf{x})}{(\Delta\tau)^{2}}\bigg{)}\\ &\quad=\tilde{K}\
abla^{2}\mathbf{T}^{f}(\mathbf{x})+\frac{\tilde{K}\tau_{ \rm T}}{\Delta\tau}\Big{[}\
abla^{2}\mathbf{T}^{f}(\mathbf{x})-\
abla^{2}\mathbf{T}^{f-1} (\mathbf{x})\Big{]}+W_{b}C_{b}\Big{[}\tilde{T}_{b}-\mathbf{T}^{f}(\mathbf{x})\Big{]}+Q_{m} \\ &\quad-W_{b}C_{b}\tau_{q}\frac{\mathbf{T}^{f}(\mathbf{x})-\mathbf{T}^{f-1}(\bm {x})}{\Delta\tau}\end{split}  \tag{15}\]  
Equation (15) may be expressed as  
\[\
abla^{2}\mathbf{T}^{f}(\mathbf{x})-B\mathbf{T}^{f}(\mathbf{x})+C\
abla^{2}\mathbf{T}^{f-1}(\mathbf{x })+D\mathbf{T}^{f-1}(\mathbf{x})+E\mathbf{T}^{f-2}(\mathbf{x})+F=0 \tag{16}\]  
in which  
\[E=-\frac{c\rho\tau_{q}}{\tilde{K}\Delta\tau(\Delta\tau+\tau_{T})},\ F=\frac{ \Delta\tau\big{(}W_{b}C_{b}\dot{T}_{b}+Q_{m}\big{)}}{\tilde{K}(\Delta\tau+\tau_ {T})}\]Equations (13) and (14) may be expressed as  
\[\mathbf{T}^{f}(\mathbf{x})=\mathbf{T}_{b}^{f}(\mathbf{x})\ \ \text{for}\ \mathbf{x}\in\Gamma_{1} \tag{17}\]  
\[\mathbf{Z}^{f}(\mathbf{x})=w_{b}^{f}(\mathbf{x})=-\tilde{K}\frac{\partial\mathbf{T}^{f}(\mathbf{x}) }{\partial\mathbf{n}}\ \ \text{for}\ \mathbf{x}\in\Gamma_{2} \tag{18}\]  
where  
\[w_{b}^{f}(\mathbf{x})=\frac{\Delta\tau}{\Delta\tau+\tau_{T}}\bigg{(}q_{b}^{f}(\mathbf{ x})+\tau_{q}\left.\frac{\partial q_{b}(\mathbf{x},\tau)}{\partial\tau}\right|_{t=t ^{f}}\bigg{)}-\tilde{K}\frac{\tau_{T}}{\Delta\tau+\tau_{T}}\frac{\partial\bm {T}^{f-1}(\mathbf{x})}{\partial n}\]  
By using the same technique used by [PERSON] [16], the following boundary integral equation can be obtained:  
\[B(\xi)\mathbf{U}^{(1)}(\xi)+\frac{1}{\tilde{K}}\int_{\Gamma}\mathbf{T}^{*}(\tilde{\xi },\mathbf{x})\mathbf{w}^{(1)}(\mathbf{x})d\Gamma=\frac{1}{\tilde{K}}\int_{\Gamma}\mathbf{q}^{* }(\tilde{\xi},\mathbf{x})\mathbf{U}^{(1)}(\mathbf{x})d\Gamma+\int_{\Omega}R\Big{[}\mathbf{T}_{ k-1}^{f}(\mathbf{x})\Big{]}\mathbf{T}^{*}(\tilde{\xi},\mathbf{x})d\Omega \tag{19}\]  
The fundamental solutions are  
\[\mathbf{T}^{*}(\tilde{\xi},\mathbf{x})=\frac{1}{4\pi\tau}\exp\left(-r\sqrt{B}\right) \tag{20}\]  
\[\mathbf{q}^{*}(\tilde{\xi},\mathbf{x})=\frac{\tilde{K}d}{4\pi\tau^{2}}\exp\left(-r \sqrt{B}\right)\left(\frac{1}{\tau}+\sqrt{B}\right),\ \ \ \ \ \ \ \ d=\sum_{\varepsilon=1}^{3}(\mathbf{x}_{\varepsilon}-\tilde{\xi}_{ \varepsilon})\cos\alpha_{\varepsilon} \tag{21}\]  
After the discretization process, Equation (19) is approximated as follows [16]:  
\[\sum_{j=1}^{N}G_{ij}W^{[1]}\big{(}\mathbf{x}_{j}\big{)}=\sum_{j=1}^{N}H_{ij}\mathbf{U} ^{[1]}\big{(}\mathbf{x}_{j}\big{)}+\sum_{l=1}^{L}P_{il}R\Big{[}\mathbf{T}_{k-1}^{f}( \mathbf{x}_{l})\Big{]} \tag{22}\]  
in which  
\[G_{ij}=\frac{1}{\tilde{K}}\int_{\Gamma_{j}}\mathbf{T}^{*}(\tilde{\xi}_{i},\mathbf{x}) d\Gamma_{j}\]  
\[H_{ij}= \begin{cases}\int_{\Gamma_{j}}\mathbf{q}^{*}(\tilde{\xi}_{i},\mathbf{x})d\Gamma_ {j},&i\
eq j\\ -0.5,&i=j\end{cases} \]  
\[P_{il}=\int_{\Gamma_{j}}\mathbf{T}^{*}(\tilde{\xi}_{i},\mathbf{x})d\Omega_{j}\]  
From (22), we obtain the boundary unknowns \(\mathbf{W}^{(1)}\) and \(\mathbf{U}^{(1)}\). Then, we can calculate \(\mathbf{U}^{(1)}(\xi_{i})\) using  
\[\mathbf{U}^{(1)}(\tilde{\xi}_{i})=\sum_{j=1}^{N}H_{ij}\mathbf{U}^{(1)}\big{(}\mathbf{x}_{ j}\big{)}-\sum_{j=1}^{N}G_{ij}\mathbf{W}^{(1)}\big{(}\mathbf{x}_{j}\big{)}+\sum_{j=1}^{N}P _{il}R\Big{[}\mathbf{T}_{k-1}^{f}(\mathbf{x}_{l})\Big{]} \tag{23}\]