Comparative analysis of fractional-order and classical ODE models in explaining real-world dynamics

Abstract

Fractional calculus extends classical differential operators to non-integer orders, enabling the explicit modelling of memory and hereditary effects that are often absent in ordinary differential equation (ODE) formulations. This study evaluated fractional-order models formulated with the Caputo derivative against classical ODEs across four datasets: global population growth, enzyme kinetics, Indian rainfall, and blood glucose regulation. The parameters were estimated using least-squares optimisation, and the performance was evaluated based on the root mean square error (RMSE). In all cases, the fractional-order models achieved lower RMSE values, with improvements ranging from substantial to modest, yet systematic. Importantly, only one additional parameter, the fractional order ($\alpha$), was introduced, preserving the model structure while enhancing accuracy. These results highlight fractional-order modelling as a flexible, interpretable, and computationally feasible framework for modelling complex dynamical systems.