## 3 Numerical Approach: SHAM
Thus, the values of \(f_{l}(\xi_{0}),\ldots,f_{l}(\xi_{N}),\theta_{l}(\xi_{0}),\ldots,\theta_{l}( \xi_{N}),\phi_{l}(\xi_{0}),\ldots,\phi_{l}(\xi_{N})\) can be determined from  
\[F_{L}=A^{-1}G \tag{46}\]  
Equation (46) provides the initial function to determine the SHAM outcome of the flow model. Hence, to find the SHAM outcomes of (26)-(28), it requires that we define the linear operator:  
\[L_{f}(\overline{f}(\eta;q),\overline{\phi}(\eta;q),\overline{\phi}(\eta;q))= \tag{47}\]  
\[A_{12}=GtI,\ \ A_{13}=GmI,\ \ A_{21}=b_{1}+b_{3}D+b_{5}D^{2}\ \ A_{22}=(1+R)D^{2}+b_{2}D+\alpha+b_{4}\]0  
\[A_{12}=GtI,\ \ A_{13}=GmI,\ \ A_{21}=b_{1}+b_{3}D+b_{5}D^{2}\ \ A_{22}=(1+R)D^{2}+b_{2}D+\alpha+b_{4}\]1  
\[A_{12}=GtI,\ \ A_{13}=GmI,\ \ A_{21}=b_{1}+b_{3}D+b_{5}D^{2}\ \ A_{22}=(1+R)D^{2}+b_{2}D+\alpha+b_{4}\]2  
In the above equations, \(q\in[0,1]=\) functions embedded and \(\overline{f}(\eta;q),\overline{\phi}(\eta;q)\) and \(\overline{\phi}(\eta;q)\) are undetermined terms. The deformation zero equation is expressed as:  
\[A_{12}=GtI,\ \ A_{13}=GmI,\ \ A_{21}=b_{1}+b_{3}D+b_{5}D^{2}\ \ A_{22}=(1+R)D^{2}+b_{2}D+\alpha+b_{4}\]3  
\[A_{12}=GtI,\ \ A_{13}=GmI,\ \ A_{21}=b_{1}+b_{3}D+b_{5}D^{2}\ \ A_{22}=(1+R)D^{2}+b_{2}D+\alpha+b_{4}\]4  
\[A_{12}=GtI,\ \ A_{13}=GmI,\ \ A_{21}=b_{1}+b_{3}D+b_{5}D^{2}\ \ A_{22}=(1+R)D^{2}+b_{2}D+\alpha+b_{4}\]5  
In the above equations, \(\hbar_{\overline{f}},\hbar_{\overline{g}}\) and \(\hbar_{\overline{g}}=\) nonzero convergence controlling auxiliary parameters, and \(N_{hf},N_{hf},N_{hp}\), \(\overline{N_{hp}}\) = nonlinear operators defined by:  
\[A_{12}=GtI,\ \ A_{13}=GmI,\ \ A_{21}=b_{1}+b_{3}D+b_{5}D^{2}\ \ A_{22}=(1+R)D^{2}+b_{2}D+\alpha+b_{4}\]6  
\[A_{12}=GtI,\ \ A_{13}=GmI,\ \ A_{21}=b_{1}+b_{3}D+b_{5}D^{2}\ \ A_{22}=(1+R)D^{2}+b_{2}D+\alpha+b_{4}\]7  
\[A_{12}=GtI,\ \ A_{13}=GmI,\ \ A_{21}=b_{1}+b_{3}D+b_{5}D^{2}\ \ A_{22}=(1+R)D^{2}+b_{2}D+\alpha+b_{4}\]8  
\[A_{12}=GtI,\ \ A_{13}=GmI,\ \ A_{21}=b_{1}+b_{3}D+b_{5}D^{2}\ \ A_{22}=(1+R)D^{2}+b_{2}D+\alpha+b_{4}\]9  
\[A_{23}=DfPrD^{2},\ \ A_{31}=c_{1}+c_{3}D,\ \ A_{32}=ScSrD^{2},\ \ A_{33}=D^{2}+c_{2}D+c_{4}\]0  
Differentiating (50)-(52) m times \(w.r.t.\) q and taking \(q=0\) and dividing the outcome expressions by \(m!\), we obtain the \(m\)th deformation order equations:  
\[A_{23}=DfPrD^{2},\ \ A_{31}=c_{1}+c_{3}D,\ \ A_{32}=ScSrD^{2},\ \ A_{33}=D^{2}+c_{2}D+c_{4}\]1  
\[A_{23}=DfPrD^{2},\ \ A_{31}=c_{1}+c_{3}D,\ \ A_{32}=ScSrD^{2},\ \ A_{33}=D^{2}+c_{2}D+c_{4}\]2  
\[A_{23}=DfPrD^{2},\ \ A_{31}=c_{1}+c_{3}D,\ \ A_{32}=ScSrD^{2},\ \ A_{33}=D^{2}+c_{2}D+c_{4}\]3 \[Gm\overline{\phi}_{m-1}+a_{8}\overline{f}^{\prime}_{m-1}+\sum_{n =0}^{m-1}\biggl{(}-2\beta\overline{f}^{\prime}_{n}\overline{f}^{\prime\prime \prime}_{m-1-n}+\beta\overline{f}^{\prime\prime}_{n}\overline{f}^{\prime\prime }_{m-1-n}+\beta\overline{f}_{n}\overline{f}^{\prime\prime\prime}_{m-1-n}- \overline{f}_{n}\overline{f}^{\prime\prime}_{m-1-n}-\overline{f}^{\prime}_{n }\overline{f}^{\prime}_{m-1-n}-\overline{f}^{\prime}_{n}\overline{f}^{\prime} _{m-1-nFigure 3 represents the contribution of the magnetic term to the velocity, concentration, and temperature plot. From Figure 3a, it is seen that increases in the magnetic term produces a reduction in the velocity profile. This is owing to the fact the Lorentz force produced by the applied magnetism strength in the direction of the flow. In Figure 3b,c, it observed that an increase in the magnetism term increases the temperature and concentration plots. Prandtl number behavior is elucidated in \[A_{23}=DfPrD^{2},\ \ A_{31}=c_{1}+c_{3}D,\ \ A_{32}=ScSrD^{2},\ \ A_{33}=D^{2}+c_{2}D+c_{4}\]4 It is noticeable that an increase in Prandtl (Pr) reduces the velocity and temperature profiles. This is because small values of Pr lead to elevation in thermal conductivities, which causes heat to diffuse out of the heated plate faster than when the Pr number is high. It is noted that when the Prandtl number is small (meaning Pr \(<\) 1), the fluid will be conducive.