## 3 Hybrid Technique Implementation  
### LRBFCM  
_GBEM Implementation for the Temperature Field_  
3.1.1. LRBFCM Implementation for the Time-Fractional-Order Bioheat Equation without Dual-Phase Lag  
According to the finite difference scheme of [PERSON] [16], the temperature time derivative can be expressed as  
\[\dot{T}(x,\,\tau)=\frac{T^{f+1}(x)-T^{f}(x)}{\Delta\tau}+O(\Delta\tau),\,\,\, \,\tau^{f}=f\Delta\tau,\,f=0,\,1,\,2,\,\ldots,\,F,\,\,\dot{T}(x,\,\tau)\in \left[\tau^{f},\,\tau^{f+1}\right]\]  
where \(D_{\tau}^{a}\) is the Caputo derivative of order \(a\), which can be written as [9]  
\[D_{\tau}^{a}T(x,\,\tau)=\frac{1}{\Gamma(1-a)}\int\limits_{0}^{\tau}\frac{ \partial T(r,\,s)}{\partial s}\frac{ds}{(\tau-s)^{a}},\,0<a<1 \tag{6}\]  
Based on the Caputo derivative (6), the following formula can be presented:  
\[D_{\tau}^{a}T^{f+1}+D_{\tau}^{a}T^{f}\approx\sum\limits_{j=0}^{k}W_{a,j}\Big{(} T^{f+1-j}(x)-T^{f-j}(x)\Big{)},(f=1,\,2,\,\ldots,\,F) \tag{7}\]  
in which  
\[W_{a,0}=\frac{(\Delta\tau)^{-a}}{\Gamma(2-a)} \tag{8}\]  
\[W_{a,j}=W_{a,0}\Big{(}(j+1)^{1-a}-(j-1)^{1-a}\Big{)},\,j=1,\,2,\,\ldots,\,F \tag{9}\]  
Figure 1: Schematic flowchart representation of the considered problem. According to Equation (7), we can write the bioheat transfer as Equation (5):  
\[ \begin{split} W_{a,0}& T^{f+1}(\mathbf{x})-\tilde{K}(\mathbf{x })T^{f+1}_{,II}(\mathbf{x})-\tilde{K}_{,I}(\mathbf{x})T^{f+1}_{,I}(\mathbf{x})\\ &=W_{a,0}T^{f}(\mathbf{x})-\tilde{K}(\mathbf{x})T^{f}_{,II}(\mathbf{x})- \tilde{K}_{,I}(\mathbf{x})T^{f}_{,J}(\mathbf{x})-\sum\limits_{j=1}^{f}W_{a,j}\Big{(}T ^{f+1-j}(\mathbf{x})-T^{f-j}(\mathbf{x})\Big{)}\\ &+Q^{f+1}_{m}(\mathbf{x},\tau)+Q^{f}_{m}(\mathbf{x},\tau),f=0,1,2,\dots,F \end{split}  \tag{10}\]  
[PERSON] et al. (2017) used the LBBFCM to solve (10) as follows:  
\[ \begin{split}&\Big{(}W_{a,0}E-\Lambda\dot{\Phi}_{II}\dot{\Phi}^{ -1}-\Lambda_{I}\dot{\Phi}_{I}\dot{\Phi}^{-1}\Big{)}\tilde{T}^{f+1}\\ &=\Big{(}W_{a,0}E+\Lambda\dot{\Phi}_{II}\dot{\Phi}^{-1}+\Lambda_{ I}\dot{\Phi}_{I}\dot{\Phi}^{-1}\Big{)}\tilde{T}^{f}-\sum\limits_{j=1}^{f}W_{a,j} \Big{(}T^{f+1-j}(x)-T^{f-j}(x)\Big{)}+Q^{f+1}_{m}(\mathbf{x},\ \tau)+Q^{f}_{m}(\mathbf{x},\ \tau),\ f\\ &=0,\ 1,\ 2,\ \dots,\ F\end{split} \]  
1.2 GBEM Implementation for the Dual-Phase-Lag Bioheat Equation without a Fractional-Order Derivative  
The dual-phase-lag bioheat equation in the absence of a fractional derivative may be written as ([PERSON] et al., 2017)  
\[c\rho\bigg{[}\frac{\partial\mathbf{T}}{\partial\tau}+\tau_{q}\frac{\partial^{2}\bm {T}}{\partial\tau^{2}}\bigg{]}=\tilde{K}\
abla^{2}\mathbf{T}+\tilde{K}\tau_{\rm T} \frac{\partial}{\partial\tau}\Big{(}\
abla^{2}\mathbf{T}\Big{)}+W_{b}C_{b}\big{(} \tilde{T}_{b}-\mathbf{T}\big{)}+Q_{m}-W_{b}C_{b}\tau_{q}\frac{\partial\mathbf{T}}{ \partial\tau} \tag{11}\]  
where \(\tau_{q}\), \(\tau_{T}\), \(W_{b}\), \(C_{b}\), and \(\tilde{T}_{b}\) are the heat-flux phase lag, temperature-gradient phase lag, perfusion rate, specific heat, and arterial temperature, respectively. In the proposed model, we considered the following conditions:  
\[\mathbf{T}(\mathbf{x},0)=T_{0},\quad\frac{\partial\mathbf{T}(\mathbf{x},\tau)}{\partial\tau} \bigg{|}_{\tau=0}=0 \tag{12}\]  
\[\mathbf{T}(\mathbf{x},\tau)=T_{b}(\mathbf{x},\tau)\ \text{for}\ \mathbf{x}\in\Gamma_{1} \tag{13}\]  
\[\mathbf{q}_{b}(\mathbf{x},\tau)+\tau_{q}\frac{\partial\mathbf{q}_{b}(\mathbf{x},\tau)}{\partial \tau}=-\tilde{K}\bigg{[}\frac{\partial\mathbf{T}(\mathbf{x},\tau)}{\partial\mathbf{n}}+ \tau_{\rm T}\frac{\partial}{\partial\tau}\bigg{(}\frac{\partial T(\mathbf{x},\tau) }{\partial\mathbf{n}}\bigg{)}\bigg{]}\ \text{for}\ \mathbf{x}\in\Gamma_{2} \tag{14}\]  
where \(T^{0}(\mathbf{x})=T^{1}(\mathbf{x})=T_{0}\) and \(\tau^{f-1}\to\tau^{f}\) (\(f\geq 2\)). Thus, Equation (11) may be approximated as follows ([PERSON] et al., 2017):  
\[