Fractional differential equations (FDEs) have provided theoretical and practical assets concerning complex and memory-dependent phenomena that would not be able to be captured by classical derivatives merely. This upper hand has yielded applicability in various fields among which engineering, biology, epidemiology, physics, applied sciences, finance, social sciences, and so on can be enumerated. While the applicability and success of FDE models in application-related fields depend on different factors, it can be stated that reliability of the methods employed for obtaining their solutions is at the core. As another benefit, FDEs allow for the introduction of memory effects into mathematical models of physical systems. This makes them important for the purpose of forecasting the behaviors and patterns of physical systems that have non-local or memory effects besides providing a flexible scheme through the use of non-integer orders of differentiation, bringing about more accurate predictions in physical system behaviors.

Furthermore, difference equations, as a subset of nonlinear dynamic systems, can provide discrete-time models. Within this scope, fractional equations extend differential and difference equations, allowing for more accurate modeling of systems that have memory and non-local effects amid nonlinear dynamic systems. With these aspects, difference equations and fractional equations are tools for modeling nonlinear dynamic systems. While difference equations provide discrete-time models, fractional equations can provide a way to combine memory and non-local effects. On the other hand, nonlinear dynamic systems provide the theoretical framework for comprehending complex behaviors these equations can produce. The more profound uses of these concepts can be seen in modeling anomalous diffusion and viscoelastic materials in physics, disease dynamics in biology, modeling fluctuations in stock market in finance, signal processing and control systems in engineering, to name some. Thus, the aim of our special session is to show a way and provide insights into the leveraging of these equations, which can ensure engineers, researchers and practitioners engaged in different areas to better predict, analyze and optimize the behavior of complex systems, patterns of dynamical systems and mechanisms underlying control.