Fractional-Order Ordinary Differential Equations

Abstract

Fractional-order ordinary differential equations (FOODEs) generalize classical differential models by incorporating non-integer derivatives, enabling the accurate representation of memory and hereditary effects in dynamic systems. Despite substantial theoretical progress, critical gaps remain in addressing boundary value problems with degenerate kernels and in extending classical solution methods to the fractional domain. This study employs a hybrid analytical-numerical approach, integrating the Caputo derivative, Picard iterative schemes, and Mittag-Leffler-based Gronwall inequalities, to establish the existence, uniqueness, and stability of FOODE solutions. Results confirm that fractional Green’s function representations effectively solve boundary problems and that fractional systems exhibit continuous dependence on initial conditions, offering greater modeling flexibility. The implications extend to fields such as viscoelasticity, control theory, and biological diffusion processes, and suggest directions for further research in variable-order systems and stochastic fractional modeling.