### Qualitative properties of one-dimensional fractional differential advection-diffusion equation

Pages 28-33

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.

DOI: 10.22227/1997-0935.2014.7.28-33

- Benson D.A., Wheatcraft S.W., Meerschaert M.M. Application of a Fractional Advection-Dispersion Equation. Water Resources Research. 2000, vol. 36, no. 6, pp. 1403—1412. DOI: http://dx.doi.org/10.1029/2000WR900031.
- Malamud M.M., Oridoroga L.L. On Some Questions of the Spectral Theory of Ordinary Differential Equations of Fractional Order. Dopov. NAN Ukr. 1998. No. 9. Pp. 39—47.
- Nakhushev A.M. Drobnoe ischislenie I ego primenenie [Fractional Calculation and its Application]. Moscow, 2003, 272 p.
- Bechilova A.R. O skhodimosti raznostnykh skhem dlya uravneniya diffuzii drobnogo poryadka [On the Convergence of Difference Schemes for Fractional Diffusion Equation of Order]. Nelineynye kraevye zadachi matematicheskoy fiziki i ikh prilozhenie [Nonlinear Boundary Value Problems of Mathematical Physics and Their Application]. Kiev, 1996, pp. 42—43.
- Nakhusheva V.A. Differentsial'nye uravneniya matematicheskikh modeley nelokal'nykh protsessov [Differential Equations of Mathematical Models of Non-Local Processes]. Moscow, 2006, 174 p.
- Khasambiev M.V. Ob odnoy kraevoy zadache dlya mnogomernogo drobnogo differentsial'nogo uravneniya advektsii-diffuzii [A Boundary Value Problem for a Multidimensional Fractional Differential Advection-Diffusion Equation]. Nelokal'nye kraevye zadachi i rodstvennye problemy sovremennogo analiza [Nonlocal Boundary Value Problems and Relevant Problems of Modern Analysis]. Terskol, 2013, pp. 79—85.
- Pskhu A.V. Kraevye zadachi dlya differentsial'nykh uravneniy s chastnymi proizvodnymi drobnogo i kontinual'nogo poryadka [Boundary Value Problems for Differential Equations with Fractional and Continuous Derivatives]. Nalchik, 2005, 186 p.
- Samarskiy A.A. Teoriya raznostnykh skhem [The Theory of Difference Schemes]. Moscow. Nauka Publ., 1983, 616 p.
- Aleroev T.S. Kraevye zadachi dlya differentsial'nykh uravneniy drobnogo poryadka [Boundary Value Problems for Differential Fraction Equations]. Doklady Adygskoy (Cherkesskoy) Mezhdunarodnoy akademii nauk [Reports of Circassian International Academy of Sciences]. 2013, no. 1, pp. 9—14.
- Pshu A.V. Uravneniya v chastnykh proizvodnykh drobnogo poryadka [Equations of Quotient Fractional Derived Numbers]. Ìoscow. Nauka Publ., 2005. 200 p.