## 3 Results and Discussions  
In this study, we adopt the local non-similarity method [36; 37; 38; 39; 40] to solve Equations (13)-(15). Considering equating up to the second level of truncation gives almost-accurate results compared with the solutions from other methods, such as the finite-difference method. To do this, we introduce the following new function \(G=\partial f/\partial\xi\), \(H=\partial\theta/\partial\xi\), and \(I=\partial\phi/\partial\xi\) into Equations (13)-(15), to obtain  
\[ \begin{array}{l}\frac{1}{f}f^{\prime\prime}+\theta+N\phi-\frac{1}{5e^{2} \Pr}\Big{[}(2n+3)(f^{\prime})^{2}-(n+4)ff^{\prime\prime}\Big{]}-\xi f^{\prime }-\xi^{5/4}\frac{\gamma^{1/2}}{\Pr}(f^{\prime})^{2}\\ =\frac{2(1-n)}{5e^{2}\Pr}\xi(f^{\prime}G^{\prime}-f^{\prime\prime}G)\\ \end{array}  \tag{20}\]  
\[\theta^{\prime\prime}+\frac{n+4}{5}f\theta^{\prime}+\frac{-1-4n}{5}f^{\prime }\theta=\frac{2(1-n)}{5}\xi\big{(}f^{\prime}H-\theta^{\prime}G\big{)} \tag{21}\]  
\[ \begin{array}{l}\frac{1}{5}\xi\theta^{\prime\prime}+\frac{n+4}{5\Pr}f\phi^{ \prime}+\kappa\frac{1}{Nt+\theta}\theta^{\prime}\phi^{\prime}+\kappa\frac{1} {Nt+\theta}\theta^{\prime\prime}\phi-\kappa\Big{(}\frac{1}{Nt+\theta}\Big{)} ^{2}(\theta^{\prime})^{2}\phi\\ =\frac{2(1-n)}{5\Pr}\xi(f^{\prime}I-\phi^{\prime}G)\\ \end{array}  \tag{22}\]  
\[ \begin{array}{l}\frac{1}{\epsilon}G^{\prime\prime}+H+NI-\frac{2(2n+3)}{5e^{2 }\Pr}f^{\prime}G^{\prime}+\frac{(n+4)}{5e^{2}\Pr}(f^{\prime\prime}G+fG^{\prime \prime})-(f^{\prime}+\xi G^{\prime})\\ -\frac{1/2}{\Pr}\Big{[}\frac{5}{4}\xi^{1/4}(f^{\prime})^{2}+2\xi^{5/4}f^{ \prime}G^{\prime}\Big{]}=\frac{2(1-n)}{5e^{2}\Pr}(f^{\prime}G^{\prime}-f^{ \prime\prime}G)\\ \end{array}  \tag{23}\]  
\[H^{\prime\prime}+\frac{n+4}{5}\big{(}\theta^{\prime}G+fH^{\prime}\big{)}+\frac {-1-4n}{5}\big{(}\theta G^{\prime}+f^{\prime}H\big{)}=\frac{2(1-n)}{5}\big{(}f ^{\prime}H-\theta^{\prime}G\big{)} \tag{24}\]\[ \begin{array}{l}\frac{1}{Sc}I^{\prime\prime}+\frac{n+4}{5\Gamma r}(\phi^{\prime}G +fl^{\prime})+\kappa\left[\frac{\phi^{\prime}H^{\prime}}{(Nt+\theta)}+\frac{ \theta^{\prime}I^{\prime}}{(Nt+\theta)}-\frac{\theta^{\prime}\phi^{\prime}H}{( Nt+\theta)^{2}}\right]+\kappa\left[\frac{\phi H^{\prime\prime}}{(Nt+\theta)}\right. \\ \left.+\frac{\theta^{\prime\prime}I}{(Nt+\theta)}-\frac{\theta^{\prime\prime} \phi H}{(Nt+\theta)^{2}}\right]-\kappa\left[\frac{2\theta^{\prime}\phi H^{ \prime}}{(Nt+\theta)^{2}}+\frac{(\theta^{\prime})^{2}I}{(Nt+\theta)^{2}}- \frac{2(\theta^{\prime})^{2}\phi H}{(Nt+\theta)^{3}}\right]\\ =\frac{2(1-m)}{5\Gamma r}(f^{\prime}I-\phi^{\prime}G)\end{array}  \tag{25}\]  
which constitute a system of ordinary differential equations. The associated boundary conditions are  
\[ \begin{array}{l}f(\xi,0)=f_{w}\quad f^{\prime}(\xi,0)=0\quad\theta^{\prime}( \xi,0)=-1\quad\phi(\xi,0)=1\\ f^{\prime}(\xi,\infty)=0\quad\theta(\xi,\infty)=0\quad\phi(\xi,\infty)=0\\ G(\xi,0)=0\quad G^{\prime}(\xi,0)=0\quad H^{\prime}(\xi,0)=0\quad I(\xi,0)=0\\ G^{\prime}(\xi,\infty)=0\quad H(\xi,\infty)=0\quad I(\xi,\infty)=0\end{array}  \tag{26}\]  
The solutions are obtained by integrating numerically using the fourth-order Runge-Kutta method with the shooting technique. The grid mesh selected \(\Delta\eta=0.01\) is used to obtain the numerical solution with the boundary condition as \(\eta\to\infty\) approximated by \(\eta_{\text{max}}=6.0\), which is sufficiently large to approach the relevant stream velocity. To confirm the numerical skills, we compared the local Nusselt and Sherwood numbers with a previous study from [PERSON] and [PERSON] [12]. Table 1 presents the comparison data for various values of buoyancy ratio \(N\) and Lewis number \(Le\); the results are in good agreement.