## 5 Further Numerical Simulations  
**Example 1. **_To demonstrate the accuracy of the derived numerical approximation formulas, we consider the power function \(f(t)=t^{N}\), where \(N\in\mathbb{Z}^{+}\), as our first test case. The RLFI of \(f\) is given analytically by_  
\[\mathcal{I}_{t}^{\pi}f=\frac{N!}{\Gamma(N+\alpha+1)}t^{N+\alpha}.\]  
Figure 4 displays the logarithmic absolute errors of the RLFI approximations computed using the GBFA method, with fractional order \(\alpha=0.5\) evaluated at \(t=0.5\). The figure consists of four subplots that investigate (i) the effects of varying the parameters \(\lambda\), \(\lambda_{q}\), and \(n_{q}\), and (ii) a comparative analysis between the GBFA method and MATLAB's integral function with tolerance parameters set to \(\texttt{RelTol}=\texttt{AbsTol}=10^{-15}\). Our numerical experiments reveal several key observations:  
* Variation of \(\lambda\) while holding other parameters constant (\(\forall_{s}\,n,n_{q}\)) shows negligible impact on the error. The error reaches near machine epsilon precision at \(n=3\), consistent with Theorems 1 and 3, which predict the collapse of both interpolation and quadrature errors when \(f^{(n+1)}\equiv 0\) and \(n_{q}>n\). * For \(n\geq 5\), the total error reduces to pure quadrature error since \(f^{(n+1)}\equiv 0\) while \(n_{q}<n\). * Variation of \(\lambda_{q}\) significantly affects the error, with \(\lambda_{q}\leq\lambda\) generally yielding higher accuracy. * Increasing either \(n\) or \(n_{q}\) while fixing the other parameter leads to exponential error reduction. The GBFA method achieves near machine-precision accuracy with parameter values \(\lambda=\lambda_{q}=0.5\) and \(n_{q}=12\), outperforming MATLAB's integral function by nearly two orders of magnitude. The method demonstrates remarkable stability, as evidenced by consistent error trends for \(\lambda_{q}\leq\lambda\), with nearly exact approximations obtained for \(n_{q}\geq 12\) in optimal parameter ranges. Figure 5: Logarithmic absolute errors of the RLFI approximations for the exponential function \(g\), computed using the GBFA method. The fractional order is set to \(\alpha=0.5\), and the approximations are evaluated at \(t=0.5\) using a 13 th-degree Gegenbauer interpolant for \(k=-2,-1,1,2\), except for the second-right subplot, where \(n\) varies. The figure presents errors under different conditions: **(Left)**: Varying \(\lambda\) with fixed \(\lambda_{q}=0.5\) and \(n_{q}=4\). **(Second-left)**: Varying \(\lambda_{q}\) with fixed \(\lambda=0.5\) and \(n_{q}=4\). **(Middle)**: Varying \(n_{q}\) with \(\lambda=\lambda_{q}=0.5\). **(Second-right)**: Varying \(n\) with \(\lambda=\lambda_{q}=0.5\) and \(n_{q}=12\). **(Right)**: Comparison between RLIM and MATLAB’s integral function using \(n_{q}=12\) and \(\lambda_{1}=\lambda_{2}=0.5\). “Error” refers to the difference between the true RLFI value and its approximation. Figure 6 compares further the computation times of the GBFA method and MATLAB's integral function, plotted on a logarithmic scale. The GBFA method demonstrates significantly lower computational times compared to MATLAB's integral function. This highlights the efficiency of the GBFA method, which achieves high accuracy with minimal computational cost. **Example 2**. : _Next, we evaluate the \(0.5\)th-order RLFI of \(g(t)=e^{kt}\), where \(k\in\mathbb{R}_{\Theta}\), over \(\mathbf{\Omega}_{0.5}\). The analytical solution on \(\mathbf{\Omega}_{t}\) is_  
\[\mathcal{J}_{t}^{a}g=\frac{t^{a}e^{kt}}{\Gamma(\alpha+1)}\,{}_{1}F_{1}(\alpha; \alpha+1;-kt).\]  
Figure 5 presents the logarithmic absolute errors of the GBFA method, with subplots analyzing (i) the impact of \(\lambda\), \(\lambda_{q}\), \(n\), and \(n_{q}\), and (ii) a performance comparison against MATLAB's integral function (tolerances \(10^{-15}\)). Some of the main findings include  
* Similar to **Example 1**, varying \(\lambda\) (with fixed small \(n\), \(n_{q}\)) has minimal effect on accuracy, whereas \(\lambda_{q}\leq\lambda\) consistently improves precision.