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It has been found that equations describing normal diffusion are not capable of characterizing some complex diffusion processes, for instance, diffusion process in inhomogeneous or heterogeneous medium, e.g. porous media. Fractional diffusion equations were introduced in order to characterize anomalous diffusion phenomena.
Fractal derivatives were created for the study of anomalous diffusion, by which traditional approaches fail to factor in the fractal nature of the media. A fractal measure t is scaled according to t α. Such a derivative is local, in contrast to the similarly applied fractional derivative. Fractal calculus is formulated as a generalization of ...
Anomalous diffusion processes in complex media can be well characterized by using fractional-order diffusion equation models. [ 60 ] [ 61 ] The time derivative term corresponds to long-time heavy tail decay and the spatial derivative for diffusion nonlocality.
Fick's first law relates the diffusive flux to the gradient of the concentration. It postulates that the flux goes from regions of high concentration to regions of low concentration, with a magnitude that is proportional to the concentration gradient (spatial derivative), or in simplistic terms the concept that a solute will move from a region of high concentration to a region of low ...
An equivalent formulation of the CTRW is given by generalized master equations. [5] A connection between CTRWs and diffusion equations with fractional time derivatives has been established. [ 6 ] Similarly, time-space fractional diffusion equations can be considered as CTRWs with continuously distributed jumps or continuum approximations of ...
In statistical mechanics and information theory, the Fokker–Planck equation is a partial differential equation that describes the time evolution of the probability density function of the velocity of a particle under the influence of drag forces and random forces, as in Brownian motion. The equation can be generalized to other observables as ...
It is described by the differential equations ... The dynamics has been described as deterministic fractional Brownian motion, and exhibits anomalous diffusion. [2] [3]
Anomalous diffusion is one more dynamic system where fractional-order systems play significant role to describe the anomalous flow in the diffusion process. Viscoelasticity is the property of material in which the material exhibits its nature between purely elastic and pure fluid.
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