A New Analytical Approach to Fractional Calculus via Mittag-Leffler Functions and the complex Sawi Transform

This research article presents an effective analytical method related to solving fractional differential equations by merging the new integral transform: complex Sawi transform (CXST) with the function of Mittag-Leffler. The fundamental and basic characteristics of the CXST, and its duality to the Laplace transform are discussed, also, its applications in handling some kinds of fractional differential equations is presented. The Mittag-Leffler function, usually is considered as a basic tool in fractional calculus, it helps in deriving accurate solutions for homogeneous and nonhomogeneous fractional problems. Through the presented examples, including Caputo fractional derivative in some equations, we show the efficiency of the method. The obtained results show the strength of the CXST in solving complex fractional problems, that gives authors analytical solutions and presents ta new direction for new future research in fractional calculus.