Fractal Diffusion In An Isotropous Medium

This paper provides a comprehensive examination of fractal diffusion in an isotropic medium. This process has a fascinating temporal memory and an extremely high initial response, which are unquestionably characterized by the Caputo fractional derivative. Its spatio-symmetric property is described by the Riesz fractional derivative. To illustrate the diffusion mechanism, we present an example of a toxic gas leak in a subway station. The numerical results are extremely promising and challenging. They clearly indicate that a MEMS-based gas sensor can track a toxic gas leak immediately due to its extremely fast diffusion at the initial time. This is of great importance for public safety. Furthermore, the diffusion process is primarily contingent upon the density of the air, which we can easily control. The diffusion pattern in an isotropic medium exhibits fractal behavior, which is extremely helpful for designing an emergency system for evacuating people from the center of toxic gas leaks by changing air density or temperature. This paper presents a rigorous mathematical concept for monitoring systems and early warning systems for toxic gas leaks.