The fractional Fokker-Planck equation is often used to characterize anomalous diffusion. In this paper, a fully discrete approximation for the nonlinear spatial fractional Fokker-Planck equation is given, where the discontinuous Galerkin finite element approach is utilized in time domain and the Galerkin finite element approach is utilized in spatial domain. The priori error estimate is derived in detail. Numerical examples are presented which are inline with the theoretical convergence rate.
It has been recently found that many models were established with the aid of fractional derivatives, such as viscoelastic systems, colored noise, electrode-electrolyte polarization, dielectric polarization, boundary layer effects in ducts, electromagnetic waves, quantitative finance, quantum evolution of complex systems, and fractional kinetics. In this paper, the asymptotical stability of higher-dimensional linear fractional differential systems with the Riemann-Liouville fractional order and Caputo fractional order were studied. The asymptotical stability theorems were also derived.
Finite element method is used to approximately solve a class of linear time-invariant, time-fractional-order diffusion equation formulated by the non-classical Fick law and a “long-tail” power kernel. In our derivation, “long-tail” power kernel relates the matter flux vector to the concentration gradient while the power-law relates the mean-squared displacement to the Gauss white noise. This work contributes a numerical analysis of a fully discrete numerical approximation using the space Galerkin finite element method and the approximation property of the Caputo time fractional derivative of an efficient fractional finite difference scheme. Both approximate schemes and error estimates are presented in details. Numerical examples are included to validate the theoretical predictions for various values of order of fractional derivatives.
In this paper, Adomian’s method is effectively implemented to determine the (approximate) analytic solution of the fractional-order Navier-Stokes equation. Such a solution is expressed in the form of a series with easily computable components. An illustrative example is presented.
From the available literature, the allometric scaling laws generally exist in biology, ecology, etc. These scaling laws obey power law distributions. A possibly better approach to characterize the power law is to utilize fractional derivatives. In this paper, we establish a fractional differential equation model for this allometry by using the Caputo fractional derivatives.
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