## 3 Solution Procedure  
For a solution of the modeled Equations (13)-(15) with boundary conditions (16) are solved with the homotopy analysis method (HAM) [44; 45; 46; 47; 48; 49; 50]. HAM is used due to its outstanding outcomes. The preliminary guesses are selected as follows:  
\[f_{0}(f)=\eta,\ g_{0}(g)=1,\theta_{0}(\theta)=1. \tag{20}\]  
The linear operators are denoted as \(L_{f}\), \(L_{g}\) and \(\text{L}_{\theta}\) are represented as  
\[L_{f}(f)=f_{\eta\eta\eta},\ L_{g}(g)=g_{\eta\eta},\ L_{\theta}\big{(}\theta)= \theta_{\eta\eta}.\] (  
obtained. Table 5 demonstrates the individual residual square errors for velocities and temperature functions. A fast convergence of the solutions is observed in Table 5. Table 4 represents the individual residual square errors for velocity \(f^{\prime}(\eta),\ g(\eta)\) and \(\theta(\eta)\). \begin{table}
\begin{tabular}{c c c c} \hline \hline \(\eta\) & **HAM Solution** & **ND-Solution** & **Absolute Error** \\ \hline
0.0 & 0.000000 & \(3.245990\times 10^{-9}\) & \(3.245990\times 10^{-9}\) \\
0.2 & 0.185094 & 0.185094 & \(1.205310\times 10^{-7}\) \\
0.4 & 0.344936 & 0.344935 & \(4.805570\times 10^{-7}\) \\
0.6 & 0.486721 & 0.486720 & \(1.053600\times 10^{-6}\) \\
0.8 & 0.617341 & 0.617339 & \(1.733100\times 10^{-6}\) \\
1.0 & 0.742857 & 0.742855 & \(2.627600\times 10^{-6}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The comparison of HAM and ND-Solve for different values of the dimensionless variable \(\eta\). \begin{table}
\begin{tabular}{c c c c} \hline \hline \(\eta\) & **HAM Solution** & **ND Solution** & **Absolute Error** \\ \hline
0.0 & 0.000000 & \(3.126830\times 10^{-9}\) & \(3.126830\times 10^{-9}\) \\ \hline
0.1 & 0.096140 & 0.096140 & \(6.091750\times 10^{-8}\) \\ \hline
0.2 & 0.185994 & 0.185094 & \(2.502260\times 10^{-7}\) \\ \hline
0.3 & 0.277724 & 0.267724 & \(5.615220\times 10^{-7}\) \\ \hline
0.4 & 0.344936 & 0.344935 & \(9.907840\times 10^{-7}\) \\ \hline
0.5 & 0.417640 & 0.417639 & \(1.529500\times 10^{-6}\) \\ \hline
0.6 & 0.486722 & 0.486720 & \(2.889860\times 10^{-6}\) \\ \hline
0.7 & 0.553026 & 0.553023 & \(2.167090\times 10^{-6}\) \\ \hline
0.8 & 0.617343 & 0.617339 & \(3.680790\times 10^{-6}\) \\ \hline
0.9 & 0.680401 & 0.680396 & \(4.519730\times 10^{-6}\) \\ \hline
1.0 & 0.742860 & 0.742855 & \(5.384880\times 10^{-6}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of HAM and ND-Solve for velocity function \(f^{\prime}(\eta)\) w.r.t. different values of the dimensionless variable \(\eta\). \begin{table}
\begin{tabular}{c c c c} \hline \hline \(\eta\) & **HAM Solution** & **ND Solution** & **Absolute Error** \\ \hline
0.0 & 1.000000 & \(1.000000\) & \(2.768567\times 10^{-8}\) \\ \hline
0.1 & 0.891532 & 0.891533 & \(7.671790\times 10^{-7}\) \\ \hline
0.2 & 0.803873 & 0.803874 & \(1.483510\times 10^{-6}\) \\ \hline
0.3 & 0.732482 & 0.732484 & \(2.160850\times 10^{-6}\) \\ \hline
0.4 & 0.674149 & 0.674152 & \(2.768330\times 10^{-6}\) \\ \hline
0.5 & 0.626630 & 0.626634 & \(3.297270\times 10^{-6}\) \\ \hline
0.6 & 0.588423 & 0.588427 & \(3.731420\times 10^{-6}\) \\ \hline
0.7 & 0.558623 & 0.558627 & \(2.062710\times 10^{-6}\) \\ \hline
0.8 & 0.536846 & 0.536850 & \(4.284390\times 10^{-6}\) \\ \hline
0.9 & 0.523221 & 0.523225 & \(4.404990\times 10^{-6}\) \\ \hline
1.0 & 0.518424 & 0.518428 & \(4.433930\times 10^{-6}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The Comparison of HAM and ND-Solve for velocity function \(g(\eta)\) w.r.t different values of the dimensionless variable \(\eta\).