Linear Fractional Differential Equations for Modeling Nonlinear Hydro Environmental Phenomena

In recent years, the adoption of fractional calculus in hydro-environmental science has seen significant growth. This study investigates how linear fractional differential equations can model different types of nonlinear water dynamics, such as diffusion in porous media and flows that follow Bernoulli’s principle. The order of the fractional derivative in a linear fractional differential equation dominates the behavior of its mild solution, such as decay, oscillation, asymptotic stability, and regularity in local and global time. The explicit mild solutions to some elementary problems are given by Mittag-Leffler functions. We provide two instances that are pertinent to the hydrological environment. First, the causality between rainfall intensity and soil moisture at a given site is established using an upper boundary linear autoregressive model with exogenous input based on observed data. That model is regarded as a discrete counterpart of a linear fractional differential equation with an order between 0 and 1. Second, a linear fractional differential equation with an order between 1 and 2 approximates the nonlinear ordinary differential equation governing the storage volume of a water reservoir. Then, based on the state-of-the-art knowledge of linear fractional differential equations, we address the implication and further development of the proposed approaches..