In this paper, the one-dimensional α-fractional diffusion equation is revisited. This equation is a particular case of the timeand space-fractional diffusion equation with the quotient of the orders of the timeand space-fractional derivatives equal to one-half. First, some integral representations of its fundamental solution including the Mellin-Barnes integral representation are derived. Then ...

In this paper we reconsider the classical nonlinear diffusivity equation of real gas in an heterogenous porous medium in light of the recent studies about the generalized fractional equation of conservation of mass. We first recall the physical meaning of the fractional conservation of mass recently studied by Wheatcraft and Meerschaert [22] and then consider the implications in the classical m...

In this article, a new application to find the exact solutions of nonlinear partial time-space fractional differential Equation has been discussed. Firstly, the fractional complex transformation has been implemented to convert nonlinear partial fractional differential Equations into nonlinear ordinary differential Equations. Afterwards, the (Gʹ/G)-expansion method has been implemented, to celeb...

the immiscible displacement of oil by water through a porous and permeable reservoir rock can be described by the use of a fractional flow curves (fw versus sw). water flooding project parameters can be obtained from the fractional flow curve. however, developing a representative fractional flow curve for a specific reservoir can be quite challenging when fluid and special core analysis data is...

A fractional wave equation replaces the second time derivative by a Caputo derivative of order between one and two. In this paper, we show that the fractional wave equation governs a stochastic model for wave propagation, with deterministic time replaced by the inverse of a stable subordinator whose index is one half the order of the fractional time derivative.

Subdiffusive fractional equations are not structurally stable with respect to spatial perturbations to the anomalous exponent (Phys. Rev. E 85, 031132 (2012)). The question arises of applicability of these fractional equations to model real world phenomena. To rectify this problem we propose the inclusion of the random death process into the random walk scheme from which we arrive at the modifi...

In this article, the (G ′ /G)-expansion method has been implemented to find the travelling wave solutions of nonlinear evolution equations of fractional order. For this, the fractional complex transformation method has been used to convert fractional order partial differential equation to ordinary differential equation. Then, (G ′ /G)-expansion method has been implemented to celebrate the serie...

Abstract. We develop a new infinite dimensional gluing method for fractional elliptic equations. In Part I, as a model problem, we construct a solution of the fractional Allen–Cahn equation vanishing on a rotationally symmetric surface which resembles a catenoid and has sub-linear growth at infinity. In Part II, we construct counterexamples to De Giorgi Conjectures to fractional Allen-Cahn equa...