## 5 Further Numerical Simulations
* Exponential error decay occurs when increasing either \(n\) or \(n_{q}\) while holding the other constant. * Near machine-epsilon accuracy is achieved for \(\lambda=\lambda_{q}=0.5\) and \(n_{q}=12\), with the GBFA method surpassing integral by two orders of magnitude. The method's stability is further demonstrated by the uniform error trends for \(\lambda_{q}\leq\lambda\) and its rapid exponential convergence (see Figure 7), underscoring its suitability for high-precision fractional calculus, particularly in absence of closed-form solutions. Figure 6: Comparison of the elapsed computation times (displayed on a logarithmic scale) for the GBFA Method and MATLAB’s integral Method as a function of the polynomial degree \(n\). All experiments were performed with \(t=0.5,\alpha=0.5,\lambda_{1}=\lambda_{2}=0.5\), and \(n_{q}=12\). Notably, this test problem was previously studied in [29; 30] on \(\Omega_{2}\). The former employed an asymptotic expansion for trapezoidal RLFI approximation, while the latter used linear, quadratic, and three cubic spline variants, with all computations performed in 128-bit precision. The methods were tested for grid sizes \(N\in\{40,80,160,320,640\}\) (step sizes \(\Delta x\in\{0.05,0.025,\ldots,0.003125\}\)). At \(N=640\), [PERSON]'s method achieved the smallest error of \(2.34\times 10^{-13}\), whereas [PERSON] and [PERSON] [30] reported their smallest error of \(9.17\times 10^{-13}\) using Cubic Spline Variant 1. With \(\lambda=\lambda_{q}=0.5\) and \(n_{q}=12\), our method attains errors close to machine epsilon (\(\sim 10^{-16}\)), surpassing both methods in [29; 30] by several orders of magnitude, even with significantly fewer computational nodes. Section 4.1 notes the potential instability of SG polynomials as \(\lambda\to-1/2\), a behavior not yet investigated within the GBFA framework. Figure 8 visually confirms this trend: initially, the errors decrease as \(\lambda\) approaches \(-0.5\), since the leading error factor \(\theta_{a,\lambda}\to 0\) as \(\lambda\to-1/2^{+}\), as explained by Theorem 2 and evident when \(\lambda\) transitions from \(-0.499999\) to \(-0.499999\). However, the ill-conditioning of SG polynomials becomes prominent as \(\lambda\) approaches \(-0.5\) more closely, leading to larger errors. This non-monotonic behavior near \(\lambda=-0.5\) underscores the practical stability limits of the method, aligning with the early observations in [40] on the sensitivity of Gegenbauer polynomials with negative parameters. In particular, [PERSON] [40] discovered that for large degrees \(M\), Gegenbauer polynomials \(G_{M}^{\lambda}(x)\) with \(\lambda<0\) exhibit strong sensitivity to small perturbations in \(x\), unlike Legendre or Chebyshev polynomials. For example, it was reported in [40] that evaluating \(G_{150}^{-1/4}(x)\) at \(x=1/2\) (exact: \(5.754509478448837\)) versus \(x=0.5001\) (perturbed by \(10^{-4}\)) introduces an absolute error of \(\approx 0.0122\), whereas Legendre and Chebyshev polynomials of the same degree incur errors of only \(\approx 10^{-4}\). This highlights the risks of using SG polynomials with \(\lambda\approx-1/2\) for high-order approximations, where even minor numerical perturbations can disrupt spectral convergence. These results reinforce the need to avoid such parameter regimes for robust approximations, as prescribed in Section 4.1. Figure 7: The left subplot demonstrates the exponential convergence of the absolute error as \(n\) increases, with \(n_{q}\) fixed at 12. The right subplot illustrates the exponential convergence of the absolute error as the \(n_{q}\) parameter increases, with \(n\) fixed at 13. Both subplots show the absolute error on a logarithmic scale. **Example 3**. : _We evaluate the \(0.5\)th-order RLFI of the function \(f(t)=2t^{3}+8t\) over the interval \(\mathbf{\Omega}_{0.5}\). The analytical solution on \(\mathbf{\Omega}_{t}\) is given by_  
\[\mathfrak{I}_{t}^{0.5}f=\frac{192t^{7/2}+1120t^{3/2}}{105\sqrt{\pi}},\]  
_which serves as the reference for numerical comparison.