Super-Exponential Approximation of the Riemann-Liouville Fractional Integral via Gegenbauer-Based Fractional Approximation Methods

TL;DR

This work introduces the GBFA method for high-precision approximation of the left Riemann-Liouville fractional integral (RLFI) by combining Gegenbauer-based interpolation with precomputable fractional-order shifted Gegenbauer integration matrices (FSGIMs). The approach leverages tunable SG parameters to achieve super-exponential convergence for smooth inputs and provides rigorous error bounds that decay rapidly with the polynomial degree $n$. Numerical results show GBFA outperforming standard quadrature tools (e.g., MATLAB's integral, MATHEMATICA's NIntegrate) by orders of magnitude in accuracy and delivering efficiency via precomputation, especially for repeated RLFI evaluations across varying $\alpha\in(0,1)$. Practical parameter guidelines and stability considerations further enhance the method's applicability, with future work targeting non-smooth cases, adaptive strategies, and extensions to broader fractional-order operators.

Abstract

This paper introduces a Gegenbauer-based fractional approximation (GBFA) method for high-precision approximation of the left Riemann-Liouville fractional integral (RLFI). By using precomputable fractional-order shifted Gegenbauer integration matrices (FSGIMs), the method achieves super-exponential convergence for smooth functions, delivering near machine-precision accuracy with minimal computational cost. Tunable shifted Gegenbauer (SG) parameters enable flexible optimization across diverse problems, while rigorous error analysis verifies a fast reduction in approximation error when appropriate parameter choices are applied. Numerical experiments demonstrate that the GBFA method outperforms MATLAB's integral, MATHEMATICA's NIntegrate, and existing techniques by up to two orders of magnitude in accuracy, with superior efficiency for varying fractional orders 0 < α < 1. Its adaptability and precision make the GBFA method a transformative tool for fractional calculus, ideal for modeling complex systems with memory and non-local behavior, where understanding underlying structures often benefits from recognizing inherent symmetries or patterns.

Figures (9)

• Figure 1: Workflow of the GBFA method for RLFI approximation. The main path (solid arrows) shows the standard procedure: (1) interpolate the input function at SGG nodes, (2) apply the RLFI operator with variable transformation, (3) approximate the integrals of SG polynomials using SG quadrature, (4) construct the FSGIM, and (5) compute the final approximation. The dashed arrow indicates an alternative path: precomputing the FSGIM for direct evaluation and repeated use. The tunable parameters $\lambda$ (interpolation) and $\lambda_q$ (quadrature) enable optimization across different problems.
• Figure 2: Behavior of the bounding constant $\vartheta_{\alpha, \lambda}$ as $\alpha$ varies from $0.1$ to $5$, shown for eight representative values of $\lambda$ ($-0.49$, $-0.4$, $-0.2$, $-0.1351$, $0$, $0.5$, $1$, and $5$). Each curve corresponds to a distinct $\lambda$ value: dark red ($\lambda = -0.49$), green ($\lambda = -0.4$), olive ($\lambda = -0.2$), bright red ($\lambda = -0.1351$), dark green ($\lambda = 0$), blue ($\lambda = 0.5$), magenta ($\lambda = 1$), and orange ($\lambda = 5$).
• Figure 3: (Top) Comparison of the functions $T_2$ (blue curve) and $T_3$ (red curve) over the interval $\mathbb{R}_{-1/2}^-$. (Bottom) The product $T_2 T_3$ (purple), showing the combined behavior of the two functions. All plots demonstrate the dependence on the parameter $\lambda$ in the negative domain near zero.
• Figure 4: Logarithmic absolute errors of the RLFI approximations for the power function $f$, computed using the GBFA method. The fractional order is set to $\alpha = 0.5$, and approximations are evaluated at $t = 0.5$. Gegenbauer interpolant degrees match the function's degrees for $n = 3:2:11$. The figure presents errors under different conditions: Top-left: Varying $\lambda$ with fixed $\lambda_q = 0.5$ and $n_q = 4$. Top-right: Varying $\lambda_q$ with fixed $\lambda = 0.5$ and $n_q = 4$. Bottom-left: Varying $n_q$ with $\lambda = \lambda_q = 0.5$. Bottom-right: Comparison between RLIM and MATLAB's integral function using $n_q = 12$ and $\lambda = \lambda_q = 0.5$. "Error" refers to the difference between the true RLFI value and its approximation. Missing colored lines indicate zero error (approximation exact within numerical precision).
• Figure 5: 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 $n_q = 12$.

Theorems & Definitions (18)

• Theorem 4.1
• proof
• Theorem 4.2
• proof
• Theorem 4.3
• proof
• Theorem 4.4
• proof
• Theorem 4.5: Asymptotic Total Truncation Error Bound
• proof