Mittag-Leffler Algorithm to Solve Linear Fractional Ordinary Differential Equations

This paper investigates the analytical solution of linear homogeneous fractional ordinary differential equations with constant coefficients using the Mittag-Leffler function method (M-LFM). Fractional ODEs play an important role in modeling dynamic systems in science, engineering, and applied mathematics. The proposed method assumes a Mittag-Leffler series solution, from which the characteristic equation is derived to determine the corresponding characteristic roots. A complete set of fundamental solutions is then constructed, and their linear independence is established using the fractional Wronskian. The method provides a systematic algorithm for obtaining a complete solution corresponding to distinct, repeated, and complex characteristic roots. Furthermore, the paper highlights the relationship between the order of the fractional differential equation and the number of characteristic roots required to construct the general solution. Several illustrative examples are presented to demonstrate the applicability and accuracy of the proposed method. A numerical comparison with the Adomian decomposition method, the homotopy analysis method, the fractional differential transform method, and the generalized fractional Taylor method is also carried out for benchmark problems with known exact solutions. The numerical results indicate that, for the benchmark problems considered, the proposed M-LFM achieves high accuracy with comparatively low computational cost. These findings demonstrate the effectiveness of the proposed method for solving linear homogeneous fractional ordinary differential equations with constant coefficients.