## 5 Conclusions  
A new numerical scheme was proposed for solving ODEs that achieves fourth-order accuracy. A stability and consistency analysis of the proposed scheme for linear ODEs was also conducted. In addition to proposing a numerical scheme, we put forward a mathematical model for the Darcy-Forchheimer flow of non-Newtonian micropolar fluid over a sheet, considering the effects of homogeneous-heterogeneous reactions. Since the exact solution of the reduced ODEs obtained from the set of equations governing the flow problem was unknown, the numerical solution obtained by the Matlab solver \(bvp4c\) was considered for comparison in place of the exact solution. The absolute error between the Euler/proposed scheme and the Matlab solver \(bvp4c\) was depicted graphically, showing that the proposed scheme is more accurate than the existing forward Euler scheme. The following conclusions can be drawn:  
* The angular velocity displayed dual behavior as the coupling and microrotation parameters increased. * The velocity profile also showed dual behavior as the coupling constant parameter increased. * The concentration profiles of species \(A\) and \(B\) displayed dual behavior as the values of the Schmidt number and the homogeneous and heterogeneous reaction parameters increased. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{\(\mathbf{P_{r}}\)} & \multirow{2}{*}{\(\mathbf{\mathrm{Wang}}\)[41]} & \multirow{2}{*}{\(\mathbf{\mathrm{Khan}}\) and \(\mathbf{\mathrm{Pop}}\)[42]} & \multirow{2}{*}{\(\mathbf{\mathrm{Mabood}}\)} & \multirow{2}{*}{\(\mathbf{\mathrm{A}}\)} & \multirow{2}{*}{\(\mathbf{\mathrm{Mabood}}\)} & \multirow{2}{*}{\(\mathbf{\mathrm{A}}\)} & \multirow{2}{*}{\(\mathbf{\mathrm{Proposed}}\)} \\  & & & & & & & & \\ \hline
0.7 & 0.4539 & 0.4539 & 0.453918 & 0.469378 & 0.587843 \\
2 & 0.9114 & 0.9113 & 0.911358 & 0.902463 & 1.014149 \\
7 & 1.8954 & 1.8954 & 1.895403 & 1.859681 & 1.973166 \\
20 & 3.3539 & 3.3539 & 3.353904 & 3.244681 & 3.421444 \\
70 & 6.4622 & 6.4621 & 6.462199 & 6.069466 & 6.458617 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of results obtained by the proposed scheme and by past research for calculation of \(-\theta^{\prime}(0)\) using \(\lambda=F_{r}=K_{b}=R_{d}=Q=0\). \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{\(\mathbf{G}\)} & \multirow{2}{*}{\(\mathbf{\mathrm{K}_{b}}\)} & \multirow{2}{*}{\(\mathbf{\mathrm{F}_{r}}\)} & \multirow{2}{*}{\(\mathbf{\mathrm{Q}}\)} & \multirow{2}{*}{\(\mathbf{\mathrm{B}}\)} & \multicolumn{2}{c}{\(-\mathbf{f}\)\((0)\)} & \multicolumn{2}{c}{\(-\mathbf{\theta}^{\prime}\)\((0)\)} \\ \cline{5-10}  & & & & & & \(\mathbf{\mathrm{\mathbf{bcp4c}}}\) & & \(\mathbf{\mathrm{\mathbf{Proposed}}\)} & \(\mathbf{bvp4c}\) & **Proposed** \\ \hline
0.5 & 0.5 & 0.1 & 0.1 & 1 & 3 & 1.031707 & 1.031706 & 0.414644 & 0.414644 \\
1 & 0.9 & 0.5 & & & & 1.044032 & 1.044032 & 0.416028 & 0.416028 \\
0.9 & 0.5 & & & & & 1.042270 & 1.042270 & 0.415837 & 0.415837 \\  & 0.9 & & & & & 1.011661 & 1.011660 & 0.413135 & 0.413135 \\  & 0.7 & 0.1 & & & & 1.027045 & 1.027045 & 0.414596 & 0.414596 \\  & 0.2 & & & & 1.056069 & 1.056069 & 0.412853 & 0.412852 \\  & 0.1 & 0.1 & & & 1.027045 & 1.027045 & 0.414596 & 0.414596 \\  & & 0.2 & & & 1.027045 & 1.027045 & 0.357609 & 0.357608 \\  & & 0.3 & 1 & & 1.027045 & 1.027045 & 0.259314 & 0.259313 \\  & & & 10 & & 1.027045 & 1.027045 & 0.338257 & 0.338256 \\  & & & 3 & 1 & 1.027045 & 1.027045 & 0.114308 & 0.114308 \\  & & & 2 & 1.027045 & 1.027045 & 0.169927 & 0.169926 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: List of numerical values for \(-f^{\prime}(0)\) using the proposed scheme and Matlab solver \(bvp4c\) when \(\lambda=0.1\) and \(R_{d}=0.5\). * The proposed scheme achieved a larger stability region than the existing Euler scheme. * The proposed scheme provided higher accuracy and a lower absolute error compared with the existing numerical scheme.