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Open AccessArticle
Super-Exponential Approximation of the Riemann–Liouville Fractional Integral via Gegenbauer-Based Fractional Approximation Methods
by
Kareem T. Elgindy
Kareem T. Elgindy
Dr. Kareem T. Elgindy is an Associate Professor in the Department of Mathematics and Sciences at He [...]
Dr. Kareem T. Elgindy is an Associate Professor in the Department of Mathematics and Sciences at Ajman University, UAE. He holds a Ph.D. in Applied and Computational Mathematics from Monash University, Australia, where his dissertation was nominated for the prestigious Mollie Holman Doctoral Medal. Dr. Elgindy's academic journey includes a master's degree in Scientific Computations from Assiut University, Egypt, and a bachelor's degree in Mathematics with first-class honors from the same institution, where he ranked as the top graduate in the Faculty of Science. His research focuses on numerical analysis, numerical fractional calculus, fractional optimal control theory, mathematical biology, and nonlinear programming, with numerous publications in high-impact ISI journals. Dr. Elgindy has held academic positions at King Fahd University of Petroleum & Minerals (KFUPM), Xiamen University Malaysia (XMUM), and Assiut University, and was granted the title of Visiting Scholar at the California Institute of Technology (Caltech) under the Fulbright Egyptian Visiting Scholar Award. A former member of the Australian Mathematical Society (AustMS), Dr. Elgindy remains actively engaged in the academic community as an editor for the International Journal of Mathematics and Mathematical Sciences and through his contributions to teaching, research mentorship, and professional service.
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 (registering DOI)
Submission received: 24 May 2025
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Revised: 17 June 2025
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Accepted: 25 June 2025
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Published: 27 June 2025
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 . 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.
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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
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