## 3 Neural Network-Based Q-Analogies of the Numerical Scheme
(**b**) The neural network’s transition statistics curve. (**c**) The neural network’s mean square error curve. (**d**) The neural network’s regression curve. (**e**) The neural network’s fitness curve. ### Engineering Application 2: Blood Rheology Model

Blood is a non-Newtonian fluid modeled as a \"Casson Fluid\". According to the Casson fluid model, simple fluids such as water and blood will flow through a tube in such a way that the center core of the fluids will travel as a plug with little distortion and a velocity gradient will occur near the wall [52; 54]. We used the following non-linear polynomial equation to describe the plug flow of Casson fluids:

\[G=1-\frac{16}{7}\sqrt{x}+\frac{4}{3}x-\frac{1}{21}x^{4}, \tag{54}\]

where \(G\) represents the reduction in flow rate. Using \(G=0.40\) in (54), we have:

\[f_{4}(x)=\frac{1}{441}x^{8}-\frac{8}{63}x^{5}-0.05714285714x^{4}+\frac{16}{9}x ^{2}-3.624489796x+0.36. \tag{55}\]

The exact roots of (55) are:

\[\zeta_{1} = 0.1046986515,\zeta_{2}=3.822389235,\zeta_{3}=1.553919850+0.94041 49899i,\] \[\zeta_{4} = -1.238769105+3.408523568i,\zeta_{5}=-2.278694688+1.987476450i\] \[\zeta_{6} = -2.278694688-1.987476450i,\zeta_{7}=-1.238769105-3.408523568,\] \[\zeta_{8} = 1.553919850-0.9404149899i.\]

To analyze convergence, we use the randomly generated starting guesses \(X_{2}^{[0]}=\left[X_{21}^{[0]},X_{22}^{[0]},X_{23}^{[0]},\ldots\right]\) shown in Appendix B, Table A2. In Tables 12-22, we examine the numerical results of the proposed parallel numerical scheme for this engineering applications at different values of \(\mathrm{q}\). In Table 12, the error and estimated solution are given up to two decimal places whereas the approximated root is computed up to 64 digit floating point arithmetic using \(\Lambda_{i}^{[*]}=\left| \begin{matrix}[\varphi+1]\\ \dot{x}_{i}-\dot{x}_{i}^{[\varphi]}\end{matrix} \right|\) as stopping criterion. Tables 13-15 report on the maximum error, number of iterations, and elapsed CPU time for the Q-MM\({}^{\sigma_{1}}\) method using different values of \(\mathrm{q}\). On the other hand, Tables 16-19 displays the numerical results of Q-MM\({}^{\sigma_{2}}\) in terms of approximated value, residual error, maximum error, number of iterations, and elapsed CPU time for finding all roots of Equation (55) using different values of \(\mathrm{q}\). The rate of convergence of Q-MM\({}^{\sigma_{1}}\)-Q-MM\({}^{\sigma_{2}}\) increases significantly as the \(\mathrm{q}\) values increase from 0 \begin{table}
\begin{tabular}{l c c c c c c c c c c} \hline \hline
**(Q-MM\({}^{\alpha_{2}}\);q]** & **tte\({}_{1}^{(3)}\)** & **tte\({}_{2}^{(5)}\)** & **tte\({}_{3}^{(3)}\)** & **tte\({}_{4}^{(5)}\)** & **tte\({}_{5}^{(3)}\)** & **tte\({}_{6}^{(5)}\)** & **tte\({}_{7}^{(3)}\)** & **tte\({}_{8}^{(5)}\)** \\ \hline \([\)Q-MM\({}^{\alpha_{2}}\);1.1\(]\) & \(24\) & \(24\) & \(24\) & \(24\) & \(24\) & \(24\) & \(24\) & \(24\) & \(24\) \\ Q-MM\({}^{\alpha_{2}}\);1.0\(]\) & \(23\) & \(23\) & \(23\) & \(23\) & \(23\) & \(23\) & \(23\) & \(23\) & \(23\) \\ Q-MM\({}^{\alpha_{2}}\);0.9\(]\) & \(30\) & \(30\) & \(30\) & \(30\) & \(30\) & \(30\) & \(30\) & \(30\) & \(30\) \\ Q-MM\({}^{\alpha_{2}}\);0.8\(]\) & \(31\) & \(31\) & \(31\) & \(31\) & \(31\) & \(31\) & \(31\) & \(31\) \\ Q-MM\({}^{\alpha_{2}}\);0.7\(]\) & \(36\) & \(36\) & \(36\) & \(36\) & \(36\) & \(36\) & \(36\) & \(36\) \\ Q-MM\({}^{\alpha_{2}}\);0.6\(]\) & \(39\) & \(39\) & \(39\) & \(39\) & \(39\) & \(39\) & \(39\) & \(39\) & \(39\) \\ Q-MM\({}^{\alpha_{2}}\);0.5\(]\) & \(46\) & \(46\) & \(46\) & \(46\) & \(46\) & \(46\) & \(46\) & \(46\) \\ Q-MM\({}^{\alpha_{2}}\);0.4\(]\) & \(47\) & \(47\) & \(47\) & \(47\) & \(47\) & \(47\) & \(47\) & \(47\) \\ Q-MM\({}^{\alpha_{2}}\);0.3\(]\) & \(50\) & \(50\) & \(50\) & \(50\) & \(50\) & \(50\) & \(50\) & \(50\) \\ \hline \hline \end{tabular}
\end{table} 
Table 18: Number of iterations for the Q-MM\({}^{\alpha_{2}}\) method using \(X_{2}^{[0]}\).