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An equation commonly used to describe solute transport in aquifers is called an advection-dispersion equation. It has been observed that in case of diffusion processes, where the diffusion takes place in a nonhomogeneous medium, the traditional Fick’s law is not satisfied and the classical advection-diffusion may not be adequate without detailed, decimeter-scale, information of the connectivity of high and low hydraulic conductivity sediments. The fractional advection-dispersion equation is presented as a useful approach for the description of transport dynamics in complex systems, which are governed by anomalous diffusion and non-exponential relaxation patterns. Fractional advection-dispersion equations are nonlocal, they describe transport affected by hydraulic conditions at a distance. Space-fractional advection-dispersion arise when velocity variations are heavy tailed and describe particle motion that accounts for variation in the flow field over the entire system. Time-fractional advection-dispersion equations arise as a result of power law particle residence time distributions and describe particle motion with memory in time. Due to vast range of applications of the fractional advection-dispersion equation, we have done a lot to find numerical solution and fundamental solution for this equation. Some authors have discussed the numerical approximation for the fractional advection-dispersion equation. The research on the analytical solution of boundary value problem for space-fractional advection-dispersion equation is relatively new and still at an early stage of development.

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.

In recent time there is a very great interest in the study of differential equations of fractional order, in which the unknown function is under the symbol of fractional derivative. It is due to the development of the theory of fractional integro-differential theory and application of it in different fields.The fractional integrals and derivatives of fractional integro-differential equations are widely used in modern investigations of theoretical physics, mechanics, and applied mathematics. The fractional calculus is a very powerful tool for describing physical systems, which have a memory and are non-local. Many processes in complex systems have nonlocality and long-time memory. Fractional integral operators and fractional differential operators allow describing some of these properties. The use of the fractional calculus will be helpful for obtaining the dynamical models, in which integro-differential operators describe power long-time memory by time and coordinates, and three-dimensional nonlocality for complex medium and processes.Differential equations of fractional order appear when we use fractal conception in physics of the condensed medium. The transfer, described by the operator with fractional derivatives at a long distance from the sources, leads to other behavior of relatively small concentrations as compared with classic diffusion. This fact redefines the existing ideas about safety, based on the ideas on exponential velocity of damping. Fractional calculus in the fractal theory and the systems with memory have the same importance as the classic analysis in mechanics of continuous medium.In recent years, the application of fractional derivatives for describing and studying the physical processes of stochastic transfer is very popular too. Many problems of filtration of liquids in fractal (high porous) medium lead to the need to study boundary value problems for partial differential equations in fractional order.In this paper the authors first considered the boundary value problem for stationary equation for mass transfer in super-diffusion conditions and abnormal advection. Then the solution of the problem is explicitly given. The solution is obtained by the Fourier’s method.The obtained results will be useful in liquid filtration theory in fractal medium and for modeling the temperature variations in the heated bar.

The use of fractional derivatives for describing and studying the physical processes of stochastic transport has become one of the most popular fields of physics in the recent years, many of the problems of fluid flow in highly-porous (fractal) environments also lead to the need to study boundary value problems for the equations of fractional order.The paper considers one of the boundary value problems for one-dimensional differential equation of fractional order. Using the Fourier method, the solution to this problem was explicitly written. The author also studied the qualitative properties of the solutions of the boundary value problem. It was proved that, in the case of going to infinity, the limit of the decisions recorded in the form of the function and the limit of the derivative of this solution tend to zero.The results can find application in the theory of fluid flow in a fractal environment and in order to simulate changes in temperature.Fractional integrals and derivatives of fractional integral-differential equations find wide application in contemporary studies of theoretical physics, mechanics and applied mathematics. Fractional calculus is a very powerful tool for describing the physical systems, which have memory and are non-local. Many processes in complex systems are non-locality and have long-term memory. The fractional integral operators and the fractional differential operators allow describing some of these properties. The use of fractional calculus will be useful for obtaining the dynamic models, in which integraldifferential operators describe the power of long-term memory and time coordinate and three-dimensional nonlocality for medium and complex processes.