\begin{table}
\begin{tabular}{c c c c c} \hline \multicolumn{5}{c}{**Functions and Special Functions**} \\ \hline \(W:\) Lambert W function & \(\Gamma(\cdot):\) Gamma function & \(\Gamma(\cdot_{\cdot}):\) upper incomplete gamma function & \(\lceil\cdot\rceil:\) ceiling function & \(\lfloor\cdot\rfloor:\) floor function \\ \(E_{n,q}(z):\) two-parameter Mittag-Leffler function & & & \\ \({}_{1}F_{1}(a;G;z)\): The confluent hypergeometric function defined as \(\sum\limits_{n=0}^{\infty}\frac{\mathrm{d}\cdot\mathrm{d}}{\mathrm{d}^{2}\eta}\) where \(q\) is the Pochhammer symbol & \\ \({}_{2}F_{1}(a,b,c,z)\): The Gauss hypergeometric function defined as \(\sum\limits_{n=0}^{\infty}\frac{\mathrm{d}\cdot\mathrm{d}\cdot\mathrm{d}}{ \mathrm{d}^{2}\eta}\) & \\ \(G_{1}^{\lambda}:\) with-degree Gegenbauer polynomial with index \(\lambda>-1/2\) & \\ \(G_{2}^{\lambda}:\) with-degree SG polynomial with index \(\lambda>-1/2\) defined on \(\Omega_{1}\) & \\ \((x)_{m}:\) The generalized falling factorial \(\frac{\Gamma(x+1)}{\Gamma(x-\pi+1)}\forall x\in C,m\in Z_{0}^{\lambda}\) & \(\delta_{n\kappa}\): The Kronecker delta with integer indices \(m\) and \(n\) & \(\mathrm{supp}(f):\) support of function \(f\) \\ \(f^{*}:\) complex conjugate of \(f\) & \(\mathbb{I}_{\gamma\geq k}:\) indicator (characteristic) function \(\begin{cases}1&ifj\geq k,\\ 0&\text{otherwise}.\end{cases}\) \\ \hline \multicolumn{5}{c}{**Notation and Shorthands**} \\ \hline \multicolumn{5}{c}{**Function Spaces**} \\ \hline \(\mathfrak{C}:\) set of all complex-valued functions & \(\mathfrak{F}:\) set of all real-valued functions & \(\mathrm{Def}(\mathbf{\Omega}):\) space of all functions defined on \(\Omega\) \\ \(\mathrm{C}^{\mathrm{d}}(\mathbf{\Omega}):\) space of \(k\) times continuously differentiable functions on \(\Omega\) & \\ \(L^{p}(\mathbf{\Omega}):\) Banach space of measurable functions \(u\in\mathrm{Def}(\mathbf{\Omega})\) with \(\left\|u\right\|_{L^{p}}=(T_{\Omega}|u|^{p})^{1/p}<\infty\) & \\ \(L^{\infty}(\mathbf{\Omega}):\) space of all essentially bounded measurable functions on \(\Omega\) & \\ \(\mathfrak{F}^{k\ell^{p}}(\mathbf{\Omega}):\) Sobolev space of weakly differentiable functions with integrable weak derivatives up to order \(k\) & \\ \hline \end{tabular}
\end{table} 
Table 1: _Cnt._  
\begin{table}
\begin{tabular}{c c} \hline \multicolumn{3}{c}{**Integris and Derivatives**} \\ \hline \(\mathcal{I}_{h}^{(t)}h:\int_{0}^{h}h(t)\,dt\) & \(\mathcal{I}_{h}^{(t)}h:\int_{0}^{h}h(t)\,dt\) & \(\mathcal{I}_{h}^{(t)}h:\int_{0}^{h}h(t)\,(. )\) & \(\mathcal{I}_{h}^{(t)}h\text{i}(t)\}:\int_{0}^{h}h(u(t))\,dt\) \\ \(\mathcal{I}_{h}^{(t)}h\text{i}(t)\}:\int_{0}^{h}h(u(t))\,dt\) & \(\mathcal{I}_{h}^{(t)}h:\int_{0}^{h}h(x)\,dx\) & \(\mathcal{I}_{h}^{(t)}f:\) The left RLFI defined by \(\frac{1}{|\alpha|}\int_{0}^{t}(t-\tau)^{-\alpha-1}f(\tau)\,d\tau\) \\ \(\partial_{x}:d/dx\) & \(\partial_{x}^{h}:d^{\prime}/dx^{\prime}\) & \(\mathcal{I}_{h}^{(t)}f:\) -other Caputo fractional derivative of \(f\) at \(x\) \\ \hline \multicolumn{3}{c}{**Norms and Metrics**} \\ \hline \(\|f\|_{1^{\prime\prime}(\mathbf{1})}:\mathbf{1}^{n}\) norm: \(\text{supp}_{x\in\Omega}\,|f(x)|=\inf\{M\geq 0:[:f(x)|\leq M\,\forall x\in \Omega\}\quad\|\cdot\|_{1}:\mathbf{1}_{+}\text{norm}\quad\|\cdot\|_{2}:\text{ Euclidean norm}\) \\ \hline \multicolumn{3}{c}{**Vectors and Matrices**} \\ \hline \(\mathbf{t}_{N}:[\,t_{N\lambda\text{\tiny{}}}\!,\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\! This paper is organized as follows: Section 2 presents the GBFA framework. Section 3 analyzes computational complexity. Section 4 provides a detailed error analysis. Section 4.1 provides actionable guidelines for selecting the tunable parameters \(\lambda\) and \(\lambda_{q}\), balancing accuracy and computational efficiency. Section 5 evaluates numerical performance. Section 6 presents the conclusions of this study with future works of potential methodological extensions to non-smooth functions encountered in fractional calculus. Appendix A lists all acronyms used in the paper.