Limitations and Future Directions

Fractional derivatives (FDs) play a crucial role in representing non-local phenomena in both spatial and temporal domains within physics. In space, the long-range effects of the flow are captured by the algebraic decay of the FD kernel. Additionally, FDs, in time, account for non-Markovian effects and thus provide a means to incorporate memory. These effects, spatial non-locality and memory have been observed in diverse fluid mechanical and physical contexts. In this concluding chapter, we explore the prospective future applications and challenges stemming from using FD-based models. It sheds light on how FD-based models could offer valuable insights into various physical phenomena. Moreover, this chapter emphasizes the current limitations and pitfalls in the existing theoretical foundation based on FDs. This comprehensive assessment aims to provide a roadmap for further research and applications of FD models in advancing our general understanding of complex physical systems.