Analytical solution framework for fractional differential equations using Laplace transforms and Mittag-Leffler functions

Abstract

Fractional differential equations have emerged as an important mathematical tool for modeling dynamical phenomena in which memory and hereditary effects play a significant role. Unlike classical differential equations, their fractional-order counterparts involve nonlocal operators, making the construction of exact analytical solutions considerably more challenging. This paper presents a unified analytical framework based on the combined use of Mittag-Leffer functions and the Laplace transform for obtaining closed-form solutions of fractional differential equations. The study first reviews the Grünwald-Letnikov, Riemann-Liouville, and Caputo definitions of fractional derivatives together with their essential mathematical properties. The analytical characteristics of the one-parameter and two-parameter Mittag-Leffer functions are then examined, and the corresponding Laplace transform relations required for solution development are established. These theoretical results are subsequently employed to derive exact solutions for representative fractional differential equations. The analysis confirms that Mittag-Leffer functions naturally generalize the exponential function and accurately characterize the behavior of fractional-order systems. In particular, for the one-term Caputo fractional differential equation ^C₀ D^α_x y(x) + λy(x) = h(x), an analytical solution is obtained in terms of the Mittag-Leffer function and its associated convolution integral. Furthermore, the integration of Laplace transform techniques with Mittag-Leffer functions provides a systematic and rigorous procedure for solving both homogeneous and non-homogeneous fractional models. The proposed framework contributes to the theoretical analysis of fractional differential equations and offers a practical foundation for future investigations in applied mathematics, science, and engineering.