## 3 Numerical Solution Using Local Non-Similarity Method (LNSM)  
Numerous researchers have used the LNSM, which was proposed by [PERSON] and [PERSON] [40], for example, [PERSON] and [PERSON] [43] in thermal boundary layers, [PERSON] et al. [44] in liquid metal forced convection magnetic induction boundary layers, [PERSON] [45] in dissipative magnetohydrodynamic convective boundary layer flows, [PERSON] et al. [46] in cross-diffusive magnetic boundary layers in porous media and [PERSON] et al. [47] in inclined solar collector thermos-solutal convection boundary layers. It is an excellent technique for tackling non-similar boundary layer physics. Using this process, it is possible to extract two crucial properties from the resulting differential equations: the local solutions and the non-similar solutions at any streamwise location. The wall's unidentified boundary conditions included using a shooting approach and forward integration, two typical methods for computing the numerical solutions to these equations. The technique also permits some self-verification of the numerical results' accuracy. Considering the following transformations for the velocity, magnetic induction, and temperature fields, respectively, the LNSM retains all the terms in the altered equations with \(\xi-\) derivatives:  
\[\frac{\partial f}{\partial\xi}=G_{1}(\xi,\eta); \tag{16}\]  
\[\frac{\partial g}{\partial\xi}=G_{2}(\xi,\eta); \tag{17}\]  
\[\frac{\partial\theta}{\partial\xi}=G_{3}(\xi,\eta) \tag{18}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Estimated Values** \\ \hline Magnetic parameter (\(\beta\)) & \(1,2,3\) \\ \hline Upsthurst force variable (\(\xi\)) & \(0.5,2.5\) \\ \hline Velocity exponent parameter (\(p\)) & \(0,\,0.25,\,0.55,\,0.56,\,0.65,\,0.75,\,1\) \\ \hline Prandtl number (\(Pr\)) & \(0.1,\,0.7,\,0.72,\,13\) \\ \hline Constant parameter (\(\delta\)) & \(3\) \\ \hline Temperature exponent parameter (\(\lambda\)) & \(-0.75,\,-0.6,\,0.6,\,-0.4,\,0,\,0.75,\,1\) \\ \hline Radiation parameter (\(N\)) & \(0,\,1,\,2,\,3\) \\ \hline Reciprocal of magnetic Prandtl number (\(\gamma\)) & \(0.5\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Estimated values in present study. As a result, three new equations must be derived in order to identify \(G_{1}(\xi,\eta)\), \(G_{2}(\xi,\eta)\). These present three extra unknown functions. Creating the subsidiary equations by differentiating the modified equations with regard to \(\xi\). The secondary equations for \(G_{1}(\xi,\eta)\), \(G_{2}(\xi,\eta)\), \(G_{3}(\xi,\eta)\) contain the terms \(\frac{\partial G_{1}}{\partial\xi}\), \(\frac{\partial G_{2}}{\partial\xi}\), \(\frac{\partial G_{3}}{\partial\xi}\) and their \(\eta\) derivatives. The systems of equations for \(f(\xi,\eta)\), \(g(\xi,\eta)\), \(\theta(\xi,\eta)\), \(G_{1}(\xi,\eta)\), \(G_{2}(\xi,\eta)\) and \(G_{3}(\xi,\eta)\) transmuted to an ODE (Ordinary Differential Equations) system of equations with the terms ignored. This LNS (Local Non-Similarity) technique configuration is referred to as considering that approximations are obtained by omitting the words as the second degree of truncation. The level of truncation will determine how accurate the LNS results are. Now differentiating Equations (12)-(14) with respect to \(\xi\) we have:  
\[\left\{