Theoretical Foundations and Properties of Riemann-Liouville Fractional Integration (Comprehensive Journal of Science, 2026)
This paper provides an extensive analytical investigation into the Riemann-LiouvilleR-L) fractional integral, a core operator in fractional calculus. By generalizing the)classical n-fold integration process to an arbitrary real order α > 0, the R-L operator enables the modeling of systems with long-range memory and non-localinteractions. This research provides a rigorous derivation from Cauchy’s repeatedintegration formula and explores fundamental, non-classical properties, includinglinearity, the semigroup property, and its action on diverse function classes. Wepresent detailed mathematical examples demonstrating the emergence of special functions like the Mittag-Leffler function. Furthermore, a comparative analysis between Riemann-Liouville and Caputo formulations is conducted, followedby an exploration of Laplace transforms in the fractional domain. This comprehensive study aims to bridge the gap between abstract mathematical theory andapplied computational modeling, providing a robust framework for understanding fractional-order dynamics .