A physical-mathematical approach to anomalous diffusion may be based on generalized diffusion equations (containing derivatives of fractional order in space or/and time) and related random walk models. By the space-time fractional diffusion equation we mean an evolution equation obtained from the standard linear diffusion equation by replacing the second-order space derivative with a Riesz-Fell...

Fractional Langevin equation to describe anomalous diffusion. Abstract A Langevin equation with a special type of additive random source is considered. This random force presents a fractional order derivative of white noise, and leads to a power-law time behavior of the mean square displacement of a particle, with the power exponent being noninteger. More general equation containing fractional ...

Abstract We develop a fully discrete finite volume element scheme of the two-dimensional space-fractional convection–diffusion equation using method to discretize derivative and Crank–Nicholson for time discretization. also analyze prove stability convergence given scheme. Finally, we validate our theoretical analysis by data from three examples.

Fractional order diffusion equations are generalizations of classical diffusion equations which are used to model in physics, finance, engineering, etc. In this paper we present an implicit difference approximation by using the alternating directions implicit (ADI) approach to solve the two-dimensional space-time fractional diffusion equation (2DSTFDE) on a finite domain. Consistency, unconditi...

This paper deals with the finite element solution of the convection diffusion equation in one and two dimensions. Two main techniques are adopted and compared. The first one includes Petrov-Galerkin based on Lagrangian tensor product elements in conjunction with streamlined upwinding. The second approach represents Bubnov/Petrov-Galerkin schemes based on a new group of exponential elements. It ...

In this paper we present a general framework for designing an analyzing high order in time fractional step schemes for integrating evolutionary convection-diffusion-reaction multidimensional problems. These methods are deduced, for example, by combining standard semidiscretization techniques (upwind) and a special kind of Runge-Kutta type schemes called Fractionary Steps Runge-Kutta methods. We...

This paper deals with the investigation of the solution of an unified fractional reaction-diffusion equation associated with the Caputo derivative as the time-derivative and Riesz-Feller fractional derivative as the space-derivative. The solution is derived by the application of the Laplace and Fourier transforms in closed form in terms of the H-function. The results derived are of general natu...

Based on the recently introduced fractional Taylor’s formula, a fractional heat conduction constitutive equation is formulated by expanding the single-phase lag model using the fractional Taylor’s formula. Combining with the energy balance equation, the derived fractional heat conduction equation has been shown to be capable of modeling diffusion-to-Thermal wave behavior of heat propagation by ...

Time fractional PDEs have been used in many applications for modeling and simulations. Many of these are multiscale contain high contrast variations the media properties. It requires very small time step size to perform detailed computations. On other hand, presence spatial grids, is required explicit methods. Explicit methods advantages as we discuss paper. In this paper, propose a partial met...