A boundary value problem for one-dimensional fractional advection-dispersion equation
Fluid, gas and plasma mechanics
Аuthors*, **, ***
Moscow State University of Civil Engineering, MSUCE, 26, Yaroslavskoe shosse, Moscow, 129337, Russia
In last time, there is very great interest to the study of differential equations of fractional order, in which the unknown function is under the symbol of fractional derivative. It is due to development of the theory of fractional integro-differential theory and application of those ones in different fields.
The fractional integrals and derivatives fractional integro-differential equations are wide using 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 allows to describe some of this properties. Using of the fractional calculus will be helpful for obtaining of dynamical models, in which integro-differential operators described power long-time memory by time and coordinates, and three-dimensional nonlocality for complex medium and processes.
Differential equations of fractional order appears when we use fractal conception in physics of condensed medium. The transfer, describing by the operator with fractional derivatives at long distance from sources, leads to other behavior of relatively small concentrations as compared with classic diffusion. This fact redefines the existing ideas about safety, based on ideas on exponential velocity of damping. Fractional calculus in the fractal theory and systems with memory have the same importance, as a classic analysis in mechanics of continuous medium.
In last years, the application of fractional derivatives for describing and studying of physical processes of stochastic transfer is very popular too. Many problems of filtration of liquids in fractal (high porous) medium leads to the need to study boundary value problems for partial differential equations in fractional order.
In this paper considered first boundary value problem for stationary equation for mass transfer in super-diffusion conditions and abnormal advection. Then solution of the problem given explicitly. The solution obtained by the Fourier’s method. Also, studied some properties of this solution.
Obtained results will be useful in liquid filtration theory in fractal medium and for modeling of temperature variations in heated bar.
Keywords:equation of fractional order, fractional derivative, Mittag-Leffler function, eigenvalues, eigenfunctions, the Fourier coefficients
Nahushev A. M. Drobnoe ischislenie i ego primenenie (Fractional calculus and its application), Moscow, Fizmatlit, 2003, 272 p.
Dzhrbashhjan M.M. Izvestija AN Armjanskoj SSR. Serija «Matematika», 1970, Vol. 5, no.2, pp. 71-96.
Dzhrbashhjan M. M. Integral’nye preobrazovaniya i predstavleniya funktsii v kompleksnoi oblasti (Integral transforms and representations of functions in the complex domain), Moscow, Nauka, 1966, 672 p.
Tihonov A. N., Samarskij A. A. Uravneniya matematicheskoi fiziki (Equations of Mathematical Phsics), Moscow, Izd-vo MGU, 1999, 799 p.
Aleroev T. S., Kirane M., Tang Y.-F. Boundary-value problems for differential equations of fractional order. Journal of Mathematical Sciences. Nov. 2013, Volume 194, Issue 5, pp. 499-512.
Samko S.G., Kilbass A.A., Marichev O.I. Integraly i proizvodnye drobnogo poryadka i nekotorye ikh prilozheniya (Integrals and derivatives of fractional order, and some applications), Minsk, Nauka i tehnika, 1987, 688 p.
Aleroev T.S., Aleroeva H. T.A problem on the zeros of the Mittag-Leffler function and the spectrum of a fractional-order differential operator. Electron. J. Qual. Theory Diff. Equ., no. 25, 18 p. (2009).
Hasambiev M.V., Aleroev T.S. Vestnik MGSU, no.6, 2014, pp. 71-76.
Aleroev T.S., Kirane M. Malik S.A. Determination of a source term for a time fractional diffusion equation with an integral type over-determining condition. Electronic Journal of Differential Equations, Vol. 2013 (2013), No. 270, pp. 1–16. ISSN: 1072-6691. //URL: http://ejde.math.txstate.edu/
Aleroev T.S., Aleroeva H.T. Izvestiya vuzov.Matematika, 2014, no.10, pp. 3-12.
Popov A.Ju., Sedleckij A.M. Sovremennaya matematika. Fundamental’nye napravleniya. Vol. 40. RUDN, 2011, pp. 3-171.