Finite difference approximations for fractional advection-dispersion flow equations

TL;DR: It is proved that the difference scheme is uniquely solvable, stable, and convergent with order $O(\tau^2+h^4)$, where $\tau$ is the time step size and $h=\max\{h_1,h_2\}$ are space grid sizes in the x and y direction.

TL;DR: In this paper, the steady state fractional advection dispersion equation (FADE) on bounded domains in ℝd is discussed and a theoretical framework for the variational solution of FADE is presented.

TL;DR: In this paper, a review of the finite difference methods for fractional differential equations is presented, which mainly include the fractional kinetic equations of diffusion or dispersion with time, space and time-space derivatives.

TL;DR: The implicit finite difference scheme with the shifted Grunwald formula is employed to discretize fractional diffusion equations and the spectrum of the preconditioned matrix is proven to be clustered around 1 if diffusion coefficients are constant; hence the convergence rate of the proposed iterative algorithm is superlinear.

TL;DR: The Riemann-Liouville Fractional Integral Integral Calculus as discussed by the authors is a fractional integral integral calculus with integral integral components, and the Weyl fractional calculus has integral components.

TL;DR: Fractional integrals and derivatives on an interval fractional integral integrals on the real axis and half-axis further properties of fractional integral and derivatives, and derivatives of functions of many variables applications to integral equations of the first kind with power and power-logarithmic kernels integral equations with special function kernels applications to differential equations as discussed by the authors.

TL;DR: In this article, the authors consider the specific effects of a bias on anomalous diffusion, and discuss the generalizations of Einstein's relation in the presence of disorder, and illustrate the theoretical models by describing many physical situations where anomalous (non-Brownian) diffusion laws have been observed or could be observed.