## 4 Numerical Experiments
In particular, for \(\alpha=1/2\) and \(\beta=0\), we have \(\omega_{5}(t)=1-\frac{1}{2}t^{2}\). Absolute errors of the 5 th approximate solution for FPTE (17) and (18) are computed for \(\alpha=1/2\), with selected nods of \(t\) with step size 0.16 and summarized in Table 3, while Table 4 shows the numerical results of the RPS algorithm and exponential integrators method (EIM) [36] for the parameter value \(\beta=0.2\) and different values of \(t\) in \([0,5]\). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(t\) & \(\omega(t)\) & RPS & RKA & PSA & GAM & GA-PS \\ \hline
0.0 & 1.00 & 1.00 & 1.00 & 0.691604 & 1.024862 & 1.016007 \\
0.1 & 1.01 & 1.01 & 1.01 & 0.623749 & 1.121206 & 1.104733 \\
0.2 & 1.04 & 1.04 & 1.04 & 0.859697 & 1.220821 & 1.199804 \\
0.3 & 1.09 & 1.09 & 1.09 & 1.122266 & 1.323041 & 1.299333 \\
0.4 & 1.16 & 1.16 & 1.16 & 1.337736 & 1.426952 & 1.401629 \\
0.5 & 1.25 & 1.25 & 1.25 & 1.501839 & 1.531330 & 1.504972 \\
0.6 & 1.36 & 1.36 & 1.60 & 1.628907 & 1.634569 & 1.607429 \\
0.7 & 1.49 & 1.49 & 1.49 & 1.734458 & 1.734591 & 1.706705 \\
0.8 & 1.64 & 1.64 & 1.64 & 1.830520 & 1.828738 & 1.799987 \\
0.9 & 1.81 & 1.81 & 1.90 & 1.925308 & 1.913640 & 1.883785 \\
1.0 & 2.00 & 1.00 & 2.00 & 2.024099 & 1.985057 & 1.953762 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Numerical comparison of the approximate solution of Example 2. **Example 4**. : _We consider the following FBTE [37; 38]:_  
\[\omega^{\prime\prime}\left(t\right)+\frac{1}{2}D_{0^{+}}^{3/2}\omega(t)+\frac{1} {2}\omega(t)=\varphi(t),\;t\geq 0, \tag{19}\]  
_with the initial conditions_  
\[\omega(0)=1,\;\omega^{\prime}(0)=0, \tag{20}\]  
_where_  
\[\varphi(t)=\left\{ \begin{array}{cc}8,\;0\leq t\leq 1,\\ 0,\;t>1.\end{array} \right.\]  
Here, the values of the assuming parameter are \(B=C=\frac{1}{2}\), and the exact solution is given by  
\[\omega(t)=\sum_{k=0}^{\infty}\frac{\left(-1\right)^{k}}{2^{k}k!}t^{2k+1}E_{1/ 2,3k/2+2}^{(k)}\left(\frac{-\sqrt{t}}{2}\right),\]  
where \(E_{\zeta,\eta}^{(k)}(t)\) is the Mittag-Leffler function of the two parameters. This fractional model is the most popular case of FBTE, which was developed to design a highly accurate microelectromechanical instrument for measuring the viscosity of fluids encountered during oil well exploration. Anyhow, the RPS solution of IVP (19) and (20) can be written as follows:  
\[\omega(t)=\sum_{k=0}^{\infty}a_{k}\frac{t^{ka}}{\Gamma(k\alpha+1)},\]  
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 \(D_{0^{+}}^{\alpha}x_{4}(t)+\frac{1}{2}x_{4}(t)+\frac{1}{2}x_{1}(t)=\varphi(t)\) 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 approximate solution of FBTEs (19) and (20) is given as \(\omega_{5}(t)=1-\left(\varphi(0)-\frac{1}{2}\right)\frac{t^{ka}}{\Gamma(4x_{1 })}+\left(\frac{1}{4}-\frac{1}{2}\varphi(0)\right)\frac{t^{ka}}{\Gamma(5\alpha +1)}\). The resulting values of the RPS algorithm and some numerical methods, including the Fermat Tau method (FTM) [38], the generalized Taylor method (GTM) [36], and the fractional Taylor method (FrTM) [37], for inputs \(t\) between \(0\) and \(1\) with a step of \(0.1\), are given in Table 5. From this table, it can be illustrated that the result obtained by our scheme is in good agreement with the state-of-the-art numerical solvers. \begin{table}
\begin{tabular}{c c c c c} \hline \(t\) & \(0\) & \(1\) & \(2\) & \(5\) \\ \hline
**RPS** & \(1\) & \(0.6100\) & \(-0.3966\) & \(-0.5372\) \\ \hline
**EIM** & \(1\) & \(0.6188\) & \(-0.1398\) & \(-0.2436\) \\ \hline \end{tabular}
\end{table} 
Table 4: Comparison of the results with \(\beta=0.2\) of Example 3.