## 3 Numerical Method
### Second Level of Truncation (Local Non-Similarity)  
Following [PERSON] and co-workers [22; 23], all the terms are retained in Equations (15)-(17) without any approximation and new auxiliary functions \(F(\xi,\eta),G(\xi,\eta)\) and \(\Theta(\xi,\eta)\) are assumed which are defined by:  
\[\begin{array}{l}F=\frac{\partial f}{\partial\xi},G=\frac{\partial g}{ \partial\xi},\Theta=\frac{\partial\theta}{\partial\xi}\end{array}\]  
Using these functions, Equations (15)-(17) can be re-written as:  
\[\begin{array}{l}\frac{\rho_{f}}{\rho_{nf}}\Big{(}\frac{\mu_{nf}}{\mu_{f}}+ \Delta\Big{)}f^{\prime\prime\prime}+ff^{\prime\prime}-f^{\prime 2}+\frac{\sin\xi \cos\xi}{\xi}+\lambda\frac{\sin\xi}{\xi}\frac{\rho_{f}}{\rho_{nf}}\frac{(\rho \theta)_{nf}}{(\rho\beta)_{f}}\theta\\ +\frac{\rho_{f}}{\rho_{nf}}\Big{(}\frac{\mu_{nf}}{\mu_{f}}+\Delta\Big{)}\xi \Big{(}\frac{\sin\xi}{\xi}-f^{\prime}\Big{)}+\frac{\rho_{f}}{\rho_{nf}} \Delta g^{\prime}-\xi(f^{\prime}F^{\prime}-f^{\prime\prime}F)=0\end{array} \tag{25}\]  
\[\frac{\rho_{f}}{\rho_{nf}}\bigg{(}\frac{\mu_{nf}}{\mu_{f}}+\frac{\Delta}{2} \bigg{)}g^{\prime\prime}+fg^{\prime}-f^{\prime}g-\frac{\rho_{nf}}{\rho_{nf}} \Delta B(2g+f^{\prime\prime})-\xi(f^{\prime}G-Fg^{\prime})=0 \tag{26}\]  
\[\frac{1}{\text{Pr}}\frac{\alpha_{nf}}{\alpha_{f}}\Theta^{\prime\prime}+f \theta^{\prime}-\xi(f^{\prime}\Theta-\theta^{\prime}F)=0 \tag{27}\]  
With the same boundary conditions (18).  
Differentiating Equations (25)-(27) and boundary conditions (18) with respect to \(\xi\) and neglecting \(\xi\) derivatives again, we get:\[\left.\begin{array}{l}\frac{\rho_{f}}{\rho_{nf}}\left(\frac{\mu_{nf}}{\mu_{f}}+ \Delta\right)F^{\prime\prime\prime}+Ff^{\prime\prime}+fF^{\prime\prime}-2f^{ \prime}F^{\prime}+\frac{\cos^{2}\xi-\sin^{2}\xi}{\xi}-\frac{\cos\xi-\sin\xi}{ \xi^{2}}+\\ \lambda\frac{\cos\xi}{\xi}\frac{\rho_{f}}{\rho_{nf}}\frac{(\rho\beta)_{nf}}{( \rho\beta)_{f}}\theta-\lambda\frac{\sin\xi}{\xi^{2}}\frac{\rho_{f}}{\rho_{nf} }\frac{(\rho\beta)_{nf}}{(\rho\beta)_{f}}\theta+\lambda\frac{\sin\xi}{\xi} \frac{\rho_{f}}{\rho_{nf}}\frac{(\rho\beta)_{nf}}{(\rho\beta)_{f}}\Theta+\\ \frac{\rho_{f}}{\rho_{nf}}\left(\frac{\mu_{nf}}{\mu_{f}}+\Delta\right)\xi\left( \frac{\cos\xi}{\xi}-\frac{\sin\xi}{\xi^{2}}-F^{\prime}\right)+\frac{\rho_{f}}{ \rho_{nf}}\Delta G^{\prime}-\xi\left(F^{\prime 2}-F^{\prime\prime}F\right)=0\end{array}\right\} \tag{28}\]  
\[\left.\begin{array}{l}\frac{\rho_{f}}{\rho_{nf}}\left(\frac{\mu_{nf}}{\mu_{f} }+\frac{\Delta}{2}\right)G^{\prime\prime}+2 Fg^{\prime}+fG^{\prime}-F^{\prime}g -2f^{\prime}G-\frac{\rho_{nf}}{\rho_{nf}}\Delta B(2G+F^{\prime\prime})-\xi \left(F^{\prime}G-FG^{\prime}\right)=0\end{array}\right. \tag{29}\]  
\[\frac{1}{\Pr}\frac{\alpha_{nf}}{\alpha_{f}}\Theta^{\prime\prime}+2F\theta^{ \prime}+f\Theta^{\prime}-f^{\prime}\Theta-\xi\left(F^{\prime}\Theta-\Theta^{ \prime}F\right)=0 \tag{30}\]  
With additional boundary conditions.  
\[\left.\begin{array}{l}F=F^{\prime}=0,G=0,\frac{k_{nf}}{k_{f}}\Theta^{\prime} (0)=Bi[\Theta(0)]\text{ at }\eta=0\\ F^{\prime}=\frac{\cos\xi\sin\xi}{\xi^{2}},G=0,\Theta\to 0\text{ as }\eta\to\infty\end{array}\right\} \tag{31}\]  
The system of nonlinear ordinary differential Equations (28)-(30) with boundary conditions (31) can be solved numerically using the Runge-Kutta-Fehlberg method of seventh order. In this method, twelve evaluations are found to be enough for each step. For higher accuracy, this method provides the most efficient results. The step size \(\Delta\eta=0.001\) and a convergence criterion of \(10^{-6}\) were used in the numerical computations. The asymptotic boundary conditions, given by Equations (31), were replaced by using a value of 12 for the similarity variable \(\eta_{\text{max}}\) as follows.  
\[F^{\prime}(\xi,12)=0,\Theta(\xi,12)=0,G(\xi,12)=0 \tag{32}\]  
The selection of \(\eta_{\text{max}}=12\) guarantees that all numerical results approached the asymptotic values accurately. For further details, readers are advised to see refs [27,28].