Boundary value problem for one-dimensional fractional differential advection-dispersion equation

• Main Authors: Khasambiev Mokhammad Vakhaevich, Aleroev Temirkhan Sultanovich
• Format: Article
• Language: English
• Published: Moscow State University of Civil Engineering (MGSU) 2014-07-01
• Series: Vestnik MGSU
• Subjects: the Mittag - Leffler function; fractional derivative; fractional order; equation
• Online Access: http://vestnikmgsu.ru/files/archive/issues/2014/6/ru/8.pdf

An equation commonly used to describe solute transport in aquifers has attracted more attention in recent years. After a formal study of some aspects of the advection-diffusion equation, basically from the mathematical point of view with the solution of a differential equation with fractional derivative, the main interest to this problem shifted onto physical aspects of the dynamical system, such as the total energy and the dynamical response. In this regard it should be pointed out that the interaction with environment is expressed in terms of stochastic arrow of time. This allows one also to reach a progress in one more issue. Formerly the equation of advection-diffusion was not obtained from any physical principles. However, mainly the success concerns linear fractional systems. In fact, there are many cases in which linear treatments are not sufficient. The more general systems described by nonlinear fractional differential equations have not been studied enough. The ordinary calculus brings out clearly that essentially new phenomena occur in nonlinear systems, which generally cannot occur in linear systems. Due to vast range of application of the fractional advection-dispersion equation, a lot of work has been done to find numerical solution and fundamental solution of this equation. The research on the analytical solution of initial-boundary problem for space-fractional advection-dispersion equation is relatively new and is still at an early stage of development. In this paper, we will take use of the method of variable separation to solve space-fractional advection-dispersion equation with initial boundary data.