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

Elgindy, Kareem T.

doi:10.3390/a18070395

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

by

Kareem T. Elgindy

^1,2

¹

Department of Mathematics and Sciences, College of Humanities and Sciences, Ajman University, P.O. Box 346 Ajman, United Arab Emirates

²

Nonlinear Dynamics Research Center (NDRC), Ajman University, P.O. Box 346 Ajman, United Arab Emirates

Algorithms 2025, 18(7), 395; https://doi.org/10.3390/a18070395

Submission received: 24 May 2025 / Revised: 17 June 2025 / Accepted: 25 June 2025 / Published: 27 June 2025

(This article belongs to the Collection Feature Papers in Randomized, Online and Approximation Algorithms)

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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 confirms rapid error decay under optimal settings. 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.

Keywords: Riemann–Liouville fractional integral; shifted Gegenbauer polynomials; pseudospectral methods; super-exponential convergence; fractional-order integration matrix

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MDPI and ACS Style

Elgindy, K.T. Super-Exponential Approximation of the Riemann–Liouville Fractional Integral via Gegenbauer-Based Fractional Approximation Methods. Algorithms 2025, 18, 395. https://doi.org/10.3390/a18070395

AMA Style

Elgindy KT. Super-Exponential Approximation of the Riemann–Liouville Fractional Integral via Gegenbauer-Based Fractional Approximation Methods. Algorithms. 2025; 18(7):395. https://doi.org/10.3390/a18070395

Chicago/Turabian Style

Elgindy, Kareem T. 2025. "Super-Exponential Approximation of the Riemann–Liouville Fractional Integral via Gegenbauer-Based Fractional Approximation Methods" Algorithms 18, no. 7: 395. https://doi.org/10.3390/a18070395

APA Style

Elgindy, K. T. (2025). Super-Exponential Approximation of the Riemann–Liouville Fractional Integral via Gegenbauer-Based Fractional Approximation Methods. Algorithms, 18(7), 395. https://doi.org/10.3390/a18070395

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

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

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