## 4 Numerical Experiments  
In this section, some illustrative examples are performed to demonstrate the efficiency and superiority of the RPS algorithm. All computations are done using Wolfram Mathematica 10.0 software package (Wolfram Research, Inc.: Champaign, IL, USA) [30]. **Example 1**. : _We consider the following FBTE [31]:_  
\[\omega^{\prime\prime}\left(t\right)+D_{0^{+}}^{3/2}\omega(t)+\omega(t)=\varphi(t) \,\ t>0, \tag{13}\]  
_with the initial conditions_  
\[\omega(0)=0,\ \omega^{\prime}(0)=0 \tag{14}\]  
_where the forcing term is_  
\[\varphi(t)=v(v-1)t^{v-2}+b\frac{\Gamma(v+1)}{\Gamma(v-1/2)}t^{v-3/2}+ct^{v},\]  
_and the exact solution is \(\omega(t)=t^{v}\)._  
This model is a special case of FBTE that arises in the modelling of the motion of a rigid plate immersed in a Newtonian fluid [32]. To apply the proposed algorithm, we have to solve the equivalent system by letting \(\omega(t)=x_{1}(t)\), \(D_{0^{+}}^{\alpha}x_{1}(t)=x_{2}(t)\), \(D_{0^{+}}^{\alpha}x_{2}(t)=x_{3}(t)\), \(D_{0^{+}}^{\alpha}x_{3}(t)=x_{4}(t)\), and \(D_{0^{+}}^{\alpha}x_{4}(t)+x_{4}(t)+x_{1}(t)=\varphi(t)\) subject to the initial conditions \(x_{i}(0)=0,\ i=1,2,3,4\), where \(0<\alpha\leq 1/2\), and the \(n\)th-truncated series is  
\[x_{i}^{n}(t)=\sum_{k=1}^{n}a_{i,k}\frac{t^{kx}}{\Gamma(k\alpha+1)},i=1,2,3,4.\]  
For numerical considerations, choose \(v=2\) such that the corresponding forcing term of Equation (13) is \(\varphi(t)=2+4\sqrt{\frac{t}{\pi}}+t^{2}\). In this sense, the equivalent system can be given by  
\[\left\{ \begin{array}{l}D_{0^{+}}^{\alpha}x_{1}(t)=x_{2}(t),\\ D_{0^{+}}^{\alpha}x_{2}(t)=x_{3}(t),\\ D_{0^{+}}^{\alpha}x_{3}(t)=x_{4}(t),\\ D_{0^{+}}^{\alpha}x_{4}(t)+x_{4}(t)+x_{1}(t)=2+4\sqrt{\frac{t}{\pi}}+t^{2}, \end{array} \right.\]  
whereas the \(nth\)-residual functions are \(\textit{Resx}_{1}^{n}(t)=D_{0^{+}}^{\alpha}x_{1}^{n}(t)-x_{2}^{n}(t)\), \(\textit{Resx}_{2}^{n}(t)=D_{0^{+}}^{\alpha}x_{2}^{n}(t)-x_{3}^{n}(t)\), \(\textit{Resx}_{3}^{n}(t)=D_{0^{+}}^{\alpha}x_{3}^{n}(t)-x_{4}^{n}(t)\) and \(\textit{Resx}_{4}^{n}(t)=D_{0^{+}}^{\alpha}x_{4}^{n}(t)+x_{4}^{n}(t)+x_{1}^{n }(t)-\left(2+4\sqrt{\frac{t}{\pi}}+t^{2}\right)\). Thus, using the procedures of the RPS algorithm [25; 26; 27; 28], the 4 th RPS approximate solution of FBTEs (13) and (14) can be given by \(\omega_{4}(t)=\frac{2t^{kx}}{\Gamma(4x+1)}\). Consequently, the RPS solution at \(\alpha=1/2\) will be \(\omega(t)=t^{2}\), which is fully compatible with the exact solution investigated earlier in [32]. To show the accuracy of the method, some numerical results of the RPS solutions are given for inputs between 0 and 1 with a step of 0.1 in Table 1, which displays the comparison between the results obtained by RPS with those obtained by the application of reproducing kernel algorithm (RKA) [33], variational iteration method (VIM) [34], genetic algorithm method (GAM), pattern search technique (PST), and Podlubny matrix approach (PMA) that developed in [35]. From this table, it can be observed that the results obtained by the RPS approach correspond well to those obtained in [33; 34; 35]. Figure 1 shows the behavior of the exact and RPS solutions for different values of \(\alpha\), where \(\alpha\in\{0.5,0.45,0.4,0.35,0.3\}\) and \(t\in[0,1]\). The RPS solutions are in good agreement with each other and with the exact solution. **Example 2**. : _We consider the following special case of FBTE [29; 31]:_  
\[\omega^{\prime\prime}\left(t\right)+D_{0^{+}}^{3/2}\omega(t)+\omega(t)=t+1,\ 0<t<1, \tag{15}\]  
_with the initial conditions_  
\[\omega(0)=1,\ \omega^{\prime}(0)=1. \tag{16}\]  
The values of the assuming parameter in this example are \(A=B=C=1\), where the exact solution is given as \(\omega(t)=t+1\).