There is growing interest in fractional calculus as a powerful mathematical tool for modeling complex systems, especially anomalous diffusion. Classical calculus is based on differentiations and integrations leading Anomalous diffusion, unlike normal diffusion, is characterized by non-linear Mean Squared Displacement (MSD) growth and non-Gaussian behavior It is increasingly popular in the study of anomalous diffusion since it incorporates discrete and continuous approaches to dynamic development. For complex media such as biological tissues and turbulent plasmas, the system is more non-local, meaning every point on the middle picture has fractional derivatives and the system's past momentarily alters the current dynamics. Although theory permits recalling, this isn't a simple thing to duplicate in application.

Fractional calculus models significantly improve the description of anomalous diffusion by integrating memory and hereditary proportions. This is made possible by non-local fractional derivatives, which account for the impact of a system's past states on its present dynamics. The long-range temporal and spatial correlations in intricate media such as porous materials, biological tissues, and turbulent plasmas are described naturally by these fractional derivatives. Following the unexpected nature of the jump length competencies, attention has shifted to evaluating the spatial diffusion of biological particles more accurately versus the consecutive nature of the inducing factors. In comparison to isotropic medium, the difference in particle path time was attributed to a fractal topography.

One influential model relies on replacing the classical first-order time derivative with a fractional derivative having an orderof α 0< α

Fractional calculus models improve the description of anomalous diffusion in physics by using non-integer order derivatives that better capture memory effects and irregular particle movement. Unlike regular diffusion (described by standard calculus), anomalous diffusion involves particles spreading faster or slower than expected. Fractional models can accurately describe these complex behaviors by accounting for long-range interactions and history-dependent dynamics, making them more suitable for real-world systems like porous materials or biological tissues.Demander à ChatGPT

Fractional calculus models significantly enhance the description of anomalous diffusion processes in physics because they generalize classical models by incorporating non-locality and memory effects, which are essential features of anomalous transport phenomena.

As a novice that is learning from this thread I have my own question. Back in the old days, before fractional calculus became popular, people used integral equations, or integral-differential equations, to deal with non-local or memory effects. What advantage do fractional equations have over the old method? Let me guess. I'm guessing that a lot of theorems were proven regarding fractional derivatives and these theorems make the same old problem (an integral equation representing memory effects) easier to solve by expressing it as a fractional derivative equation and using those theorems. Is my guess correct? Or is there something more to fractional derivatives that make them applicable to cases that cannot be described by integral-differential equations?