## 6 Conclusions and Future Works
Exploring further adaptations to operators such as the right RLFI presents exciting avenues for future work. The super-exponential convergence of spectral and pseudospectral methods, including those using Gegenbauer polynomials and Gauss nodes, typically degrades to algebraic convergence when applied to functions with limited regularity. This degradation is a well-established phenomenon in approximation theory, with the rate of algebraic convergence directly related to the function's degree of smoothness. Specifically, for a function possessing \(k\) continuous derivatives, theoretical analysis predicts a convergence rate of approximately \(O(N^{-k})\), where \(N\) represents the number of collocation points [42]. To improve the applicability of the GBFA method for non-smooth functions, several methodological extensions can be incorporated: (i) Modal filtering techniques offer one promising approach, effectively dampening spurious high-frequency oscillations without significantly compromising overall accuracy. This process involves applying appropriate filter functions to the spectral coefficients, thereby regularizing the approximation while preserving accuracy in smooth solution regions. When implemented within the GBFA framework, filtering the coefficients in the SG polynomial expansion can potentially recover higher convergence rates for functions with isolated non-smoothness. The tunable parameters inherent in SG polynomials may provide valuable flexibility for optimizing filter characteristics to address specific types of non-smoothness. (ii) Adaptive interpolation strategies represent another valuable extension, employing local refinement near singularities or implementing moving-node approaches to more accurately capture localized features. By strategically concentrating computational resources in regions of limited regularity, these methods maintain high accuracy while effectively accommodating non-smooth behavior. Within the GBFA method, this approach could be realized through non-uniform SG collocation points with increased density near known singularities. (iii) Domain decomposition techniques offer particularly powerful capabilities by partitioning the computational domain into subdomains with potentially different resolutions or spectral parameters. This approach accommodates irregularities while preserving the advantages of the GBFA within each smooth subregion. Domain decomposition proves especially effective for problems featuring isolated singularities or discontinuities, allowing the GBFA method to maintain super-exponential convergence in smooth subdomains while appropriately addressing non-smooth regions through specialized treatment. (iv) For fractional problems involving weakly singular or non-smooth solutions with potentially unbounded derivatives, graded meshes provide an effective solution. These non-uniform discretizations concentrate points near singularities according to carefully chosen distributions, often recovering optimal convergence rates despite the presence of singularities. The inherent flexibility of SG parameters makes the GBFA method particularly amenable to such adaptations. (v) Hybrid spectral-finite element approaches represent yet another viable pathway, combining the high accuracy of spectral methods in smooth regions with the flexibility of finite element methods near singularities. Such hybrid frameworks effectively balance accuracy and robustness for problems with limited regularity. The GBFA method could be integrated into these frameworks by utilizing its super-exponential convergence where appropriate while delegating singularity treatment to more specialized techniques. These theoretical considerations and methodological extensions can significantly expand the potential applicability of the GBFA method to non-smooth functions commonly encountered in fractional calculus applications.