Fractional Differential Equations and Their Applications

Fractional differential equations (FDEs) are a more advanced version of classical equations that consist of non-integer order derivatives. This modification of differential equations allows modeling complex systems with memory effects, hereditary properties, and anomalous behavior, which cannot be described using classical integer-order models. Over the last few years, fractional differential equations have found broad applications in various areas of science and engineering including physics, biological research, and finance. FDEs are employed in modeling unusual diffusion behavior in physics such as subdiffusion and superdiffusion in nonhomogeneous materials. In biology, FDEs are used to describe memory-based processes including the dynamics of brain networks and viscoelastic materials. In finance, fractional mathematical models help to explain certain aspects of financial market volatility and temporal dependence in financial time series. The field of fractional calculus has undergone extensive evolution, resulting in numerous definitions of fractional derivatives, including the classical Riemann-Liouville derivative, the modern Caputo derivative, and the more advanced Grünwald-Letnikov derivative. The aim of this paper is to succinctly demonstrate basic properties of finite difference equations (FDEs), introduce representative numerical and analytical methods used to resolve these equations, and comment on the wide applicability of these equations in various areas of science and technology. The advantages of the application of finite difference equations in modeling the processes in the real world, the problems of their computational realization, and the opportunities for their implementation in the course of solving scientific problems will be discussed.