## 4 Numerical Experiments
The RPS approximate solution of IVP (15) and (16) can be written as follows:  
\[\omega(t)=\sum_{k=0}^{\infty}a_{k}\frac{t^{k\alpha}}{\Gamma(k\alpha+1)},\]  
with the assumptions \(\omega(t)=x_{1}(t)\), \(D_{0^{+}}^{\alpha}x_{1}(t)=x_{2}(t)\), \(D_{0^{+}}^{\alpha}x_{2}(t)=x_{3}(t)\), \(D_{0^{+}}^{\alpha}x_{3}(t)=x_{4}(t)\) and \(D_{0^{+}}^{\alpha}x_{4}(t)+x_{4}(t)+x_{1}(t)=t+1\) subject to \(x_{1}(0)=1,x_{2}(0)=0,x_{3}(0)=1\) and \(x_{4}(0)=0\), where \(0<\alpha\leq 1/2\). According to the RPS method, the 5 th approximate solution of FBTEs (14) and (15) is \(\omega_{5}(t)=1+\frac{t^{2\alpha}}{\Gamma(2\alpha+1)}\). If we keep the repeating of the RPS process, the unknown coefficients \(a_{k}\) for \(k\geq 6\) will be vanished. In particular, the RPS approximate solution at \(\alpha=1/2\) is \(\omega(t)=1+t\), which is fully compatible with the exact solution investigated earlier in [29]. To show the accuracy of the RPS algorithm, numerical results of the 5 th approximate solutions are given in Table 2 for inputs \(t\) between 0 and 1 with a step size of 0.1. Table 2 displays the comparison between the results obtained by the RPS algorithm with those obtained by the application \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \(t\) & \(\omega(t)\) & RPS & RKA & VIM & PMA & PSA & GAM \\ \hline
0.0 & 0.00 & 0.00 & 0.00 & 0.0000 & 0.00000 & 0.027300 & 0.06730 \\
0.1 & 0.01 & 0.01 & 0.01 & 0.01005 & 0.00929 & 0.012392 & 0.08925 \\
0.2 & 0.04 & 0.04 & 0.04 & 0.04063 & 0.03892 & 0.049646 & 0.12802 \\
0.3 & 0.09 & 0.09 & 0.09 & 0.09266 & 0.08873 & 0.111143 & 0.18375 \\
0.4 & 0.16 & 0.16 & 0.16 & 0.16748 & 0.15867 & 0.196074 & 0.25676 \\
0.5 & 0.25 & 0.25 & 0.25 & 0.26679 & 0.24871 & 0.303374 & 0.34756 \\
0.6 & 0.36 & 0.36 & 0.36 & 0.39277 & 0.35883 & 0.431620 & 0.45684 \\
0.7 & 0.49 & 0.49 & 0.49 & 0.54806 & 0.48903 & 0.578880 & 0.55848 \\
0.8 & 0.64 & 0.64 & 0.64 & 0.73588 & 0.63928 & 0.742531 & 0.73455 \\
0.9 & 0.81 & 0.81 & 0.81 & 0.96007 & 0.80958 & 0.919015 & 0.90530 \\
1.0 & 1.00 & 1.00 & 1.00 & 1.22519 & 0.99993 & 1.103535 & 1.09922 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Numerical comparison of the approximate solutions of Example 1. Figure 1: Plots of exact and residual power series (RPS) approximate solutions at different values of \(\alpha\): Pink-Dashed \(\alpha=0.3\); Green-Dashed \(\alpha=0.35\); Gray-Dashed \(\alpha=0.4\); Orange-Dashed \(\alpha=0.45\); Red-Dashed \(\alpha=0.5\); and Blue is exact. of reproducing kernel algorithm (RKA) [33], genetic algorithm method (GAM), Pattern search technique (PST), and GAM hybrid with PST (GA-PS) [35]. The results of this table illustrate that the result obtained by our scheme is in good agreement with the state-of-the-art numerical solvers. **Example 3**. : _We consider the following homogeneous FBTE [36]:_  
\[\omega^{\prime\prime}\left(t\right)+\beta\sqrt{\pi}D_{0^{+}}^{3/2}\omega(t)+ \omega(t)=0,\ t\geq 0, \tag{17}\]  
_with the initial conditions_  
\[\omega(0)=1,\omega^{\prime}(0)=0, \tag{18}\]  
_where \(\beta\) is a parameter._  
This fractional model was developed to design a highly accurate microelectromechanical instrument for measuring the viscosity of liquids encountered during oil drilling. The exact solution reduced to \(\omega(t)=\cos(t)\) as soon as \(\beta\to 0\). The RPS solution of IVP (17) and (18) can be written as follows:  
\[\omega(t)=\sum_{k=0}^{\infty}a_{k}\frac{t^{k\alpha}}{\Gamma(k\alpha+1)},\ 0< \alpha\leq 1/2,\]  
with the assumptions \(\omega(t)=x_{1}(t)\), \(D_{0^{+}}^{\alpha}x_{1}(t)=x_{2}(t)\), \(D_{0^{+}}^{\alpha}x_{2}(t)=x_{3}(t)\), \(D_{0^{+}}^{\alpha}x_{3}(t)=x_{4}(t)\) and \(D_{0^{+}}^{\alpha}x_{4}(t)+\beta\sqrt{\pi}x_{4}(t)+x_{1}(t)=0\) subject to \(x_{1}(0)=1\) and \(x_{2}(0)=x_{3}(0)=x_{4}(0)=0\), where \(0<\alpha\leq 1/2\). Thus, according to the RPS algorithm, the 5 th RPS approximate solution of FBTEs (17) and (18) is \(\omega_{5}(t)=1-\frac{t^{4\alpha}}{\Gamma(4\alpha+1)}+\beta\sqrt{\pi}\frac{t^ {5\alpha}}{\Gamma(5\alpha+1)}\).