Related papers: Beyond Classical Diffusion: Fractional Derivatives…
Field equations with time and coordinates derivatives of noninteger order are derived from stationary action principle for the cases of power-law memory function and long-range interaction in systems. The method is applied to obtain a…
It is shown how Adler's trace dynamics can be applied to stochastic mechanics and other complex classical dynamical systems. Emergent non-commutivity due to the fractal nature of sample trajectories is closely related to the fact that the…
In this chapter we provide an introduction to fractional dissipative partial differential equations (PDEs) with a focus on trying to understand their dynamics. The class of PDEs we focus on are reaction-diffusion equations but we also…
Fractional diffusion equations for three-dimensional lattice models based on fractional-order differences of the Grunwald-Letnikov type are suggested. These lattice fractional diffusion equations contain difference operators that describe…
The partial differential equation of Gaussian diffusion is generalized by using the time-fractional derivative of distributed order between 0 and 1, in both the Riemann-Liouville (R-L) and the Caputo (C) sense. For a general distribution of…
This note shows how classical tools from linear control theory can be leveraged to provide a global analysis of nonlinear reaction-diffusion models. The approach is differential in nature. It proceeds from classical tools of contraction…
We review statistical properties of models generated by the application of a (positive and negative order) fractional derivative operator to a standard random walk and show that the resulting stochastic walks display slowly-decaying…
The extraction of natural gas from the earth has been shown to be governed by differential equations concerning flow through a porous material. Recently, models such as fractional differential equations have been developed to model this…
In a fractional Cauchy problem, the usual first order time derivative is replaced by a fractional derivative. The fractional derivative models time delays in a diffusion process. The order of the fractional derivative can be distributed…
In this paper we will present some alternative types of mathematical description and methods of solution of the fractional-order dynamical system in the state space. We point out the difference in the true sense of the name "state" space…
A numerical study of the role of anomalous diffusion in front propagation in reaction-diffusion systems is presented. Three models of anomalous diffusion are considered: fractional diffusion, tempered fractional diffusion, and a model that…
We prove that any given function can be smoothly approximated by functions lying in the kernel of a linear operator involving at least one fractional component. The setting in which we work is very general, since it takes into account…
Numerical methods for fractional calculus attract increasing interests due to its wide applications in various fields such as physics, mechanics, etc. In this paper, we focus on constructing high-order algorithms for Riesz derivatives,…
We generalize the method of Van Hove so as to deal with the case of non-ordinary statistical mechanics, that being phenomena with no time-scale separation. We show that in the case of ordinary statistical mechanics, even if the adoption of…
Many physical, biological, and engineered systems exhibit memory effects that challenge Markovian models. Fractional calculus provides nonlocal operators to capture hereditary dynamics. This survey connects modeling, analysis, and…
Starting from the model of continuous time random walk, we focus our interest on random walks in which the probability distributions of the waiting times and jumps have fat tails characterized by power laws with exponent between 0 and 1 for…
The study of systems with memory requires methods which are different from the methods used in regular dynamics. Systems with power-law memory in many cases can be described by fractional differential equations, which are…
As fractional diffusion equations can describe the early breakthrough and the heavy-tail decay features observed in anomalous transport of contaminants in groundwater and porous soil, they have been commonly employed in the related…
This paper introduces the use of statistical distributions based on transport differential equations for clear distinction of transport modes within transient kinetic experiments. More specifically,novel techniques are developed for the…
This is a review of statistical inference methodology for stochastic differential equations driven by fractional Brownian motion, otherwise called fractional diffusions. The first section reviews the theory needed to rigorously define them.…