Advanced Applications of Fractional Calculus and Integral Transformations in Mathematical Analysis

Abstract

This research paper explores the fundamental relationships between fractional calculus and integral transformations, with particular emphasis on their applications in solving complex mathematical problems. The study examines various aspects of fractional derivatives, including the Riemann-Liouville and Caputo derivatives, and their interactions with classical integral transforms such as Laplace, Fourier, and Mellin transforms. Through detailed analysis of mathematical frameworks and practical applications, this paper demonstrates the significance of these tools in modeling real-world phenomena and solving fractional differential equations.