## 6 Conclusions and Future Works  
This study presents the GBFA method, a powerful approach for approximating the left RLFI. By using precomputable FSGIMs, the method achieves super-exponential convergence for smooth functions, delivering near machine-precision accuracy with minimal computational effort. The tunable Gegenbauer parameters \(\lambda\) and \(\lambda_{q}\) enable tailored optimization, with sensitivity analysis showing that \(\lambda_{q}<\lambda\) accelerates convergence for \(n_{q}\sim n\). The strategic parameter selection guideline outlined in Section 4.1 improves the GBFA method's versatility, ensuring high accuracy for a wide range of problem types. Rigorous error bounds confirm rapid error decay, ensuring robust performance across diverse problems. Numerical results highlight the GBFA method's superiority, surpassing MATLAB's integral, MATHEMATICA's NIntegrate, and prior techniques like trapezoidal [29], spline-based methods [30], and the neural network method [31] by up to several orders of magnitude in accuracy, while maintaining lower computational costs. The FSGIM's invariance for fixed points improves efficiency, making the method ideal for repeated fractional integrations with varying \(\alpha\in\boldsymbol{\Omega_{1}^{\circ}}\). These qualities establish the GBFA method as a versatile tool for fractional calculus, with significant potential to advance modeling of complex systems exhibiting memory and non-local behavior, and to inspire further innovations in computational mathematics. The error bounds derived in this study are asymptotic in nature, meaning they are most accurate for large values of \(n\) and \(n_{q}\). While these bounds provide valuable insights into the theoretical convergence behavior of the method, they may not be tight or predictive for small-to-moderate values of \(n\) and \(n_{q}\). For practitioners, this implies that while the asymptotic bounds guarantee eventual convergence, careful numerical experimentation is recommended to determine the optimal values of \(n\), \(n_{q}\), \(\lambda\), and \(\lambda_{q}\) for a given problem. The numerical experiments presented in Section 5 demonstrate that the method performs exceptionally well even for small and moderate values of \(n\) and \(n_{q}\), but the asymptotic bounds should be interpreted as theoretical guides rather than precise predictors for all cases. To address this limitation, future work could focus on deriving non-asymptotic error bounds or developing heuristic strategies for parameter selection in practical scenarios. Additionally, adaptive algorithms could be explored to dynamically adjust \(n\) and \(n_{q}\) based on local error estimates, further improving the method's robustness for real-world applications. While the current work focuses on \(0<\alpha<1\), the extension of the GBFA method to \(\alpha>1\) is the subject of ongoing research by the author and will be addressed in future \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Quantity** & **Value** & **Relative Error** & **CPU Time** \\ \hline Exact Value & 2.218878969089873 & — & — \\ GBFA Method Approximation & 2.218878969089874 & 2.001412497195449 \(\times 10^{-16}\) & 0.003125 s \\ (\(\lambda=\lambda_{q}=0.5,n=3,n_{q}=4\)) & MATLAB’s integral Function Approximation & 2.218878973119478 & 1.816054080462689 \(\times 10^{-9}\) & 0.015625 s \\ (\(\text{RelFl}=\text{AbsTol}=10^{-15}\)) & & & \\ MATHEMATICA NIntegrate Function Approximation & 2.2188789399346960 & 1.3139598073477670 \(\times 10^{-8}\) & 0.0098979 s \\ (\(\text{PrecisionGoal}=\text{AccuracyGoal}=15\)) & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of the exact 0.5 th-order RLFI of \(f(t)=2t^{3}+8t\) over the interval \([0,0.5]\), the GBFA method, MATLAB’s integral function, and MATHEMATICA NIntegrate function approximations. The table also includes the CPU times for running the three numerical integration routines. All relative error approximations are rounded to 16 significant digits. The CPU times were computed in seconds (s). publications.