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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.
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.
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 ...
By applying a Laplace transform to the LTI system above, the transfer function becomes = () = = =For general orders and this is a non-rational transfer function. Non-rational transfer functions cannot be written as an expansion in a finite number of terms (e.g., a binomial expansion would have an infinite number of terms) and in this sense fractional orders systems can be said to have the ...
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 ...
1 Fractional step methods. 2 Strang ... In applied mathematics Strang splitting is a numerical method for solving differential equations that are decomposable into a ...
We obtain the distribution of the property i.e. a given two dimensional situation by writing discretized equations of the form of equation (3) at each grid node of the subdivided domain. At the boundaries where the temperature or fluxes are known the discretized equation are modified to incorporate the boundary conditions.
The convection–diffusion equation describes the flow of heat, particles, or other physical quantities in situations where there is both diffusion and convection or advection. For information about the equation, its derivation, and its conceptual importance and consequences, see the main article convection–diffusion equation. This article ...