New methodologies in fractional and fractal derivatives modeling

This paper surveys the latest advances of the first author's group on the three new methodologies of fractional and fractal derivatives modeling to meet the increasing and challenging demands in scientific and engineering communities. Firstly, the structural fractal was proposed as a generalization of the Euclidean distance. Using the structural metric, the structural derivative approach was derived as a significant extension of the global fractional calculus and the local fractional derivative approaches to tackle the perplexing modeling problems. The classical derivative describes the change rate of a certain physical variable with respect to time or space, which rarely takes into account the significant influence of mesoscopic time-space metric of a complex system on its physical behaviors. The structural function plays a central role in this new strategy as a kernel transform of underlying time-space structural metric of physical systems. Secondly, we employed the fundamental solution or probability density function of statistical distribution which can describe the problem of interest to construct the implicit calculus governing equation. The 'implicit' suggests that the explicit calculus expression of this governing equation is difficult to derive and not required. The fundamental solution or potential function of calculus governing equation and corresponding boundary conditions are sufficient to do numerical simulation. We call this strategy the implicit calculus equation modeling. Thirdly, based on the implicit calculus equation modeling approach, we introduced the concept of fundamental solution on fractal and consequently defined the fractal differential operator to describe various mechanical behaviors of fractal materials. Fractal calculus operator significantly extends the application scope of the classical calculus modeling approach under the framework of continuum mechanics. This is also a step-forward advance of the fractal derivative proposed earlier by the first author. To demonstrate the structural derivative application, we applied the inverse Mittag-Leffler function as the structural function to model ultraslow diffusion of a random system of two interacting particles. On the other hand, this paper uses the fractional Riesz potential as the fundamental solution to establish the implicit calculus equation of fractional Laplacian modeling the power law behaviors of steady heat conduction in multiple phase material. Finally, by using the singular boundary method, we made numerical simulation of the fractal Laplacian equation for phenomenological modeling potential problems in fractal media. Numerical experiments show that all the three new methodologies are feasible mathematical tools to describe complex physical behaviors.