Abstract
A locally one-dimensional scheme for a fractional tracer transport equation of order is considered. An a priori estimate is obtained for the solution of the scheme, and its convergence is proved in the uniform metric.
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References
O. Yu. Dinariev, “Flow in a fractured medium with fractal fracture geometry,” Fluid Dyn. 25 (5), 704–708.
V. L. Kobelev, Ya. L. Kobelev, and E. P. Romanov, “Non-Debye relaxation and diffusion in fractal space,” Dokl. Phys. 43, 752–753 (1998).
V. L. Kobelev, Ya. L. Kobelev, and E. P. Romanov, “Self-maintained processes in the case of nonlinear fractal diffusion,” Dokl. Phys. 44, 752–753 (1999).
A. Yu. Kochubei, “Fractional order diffusion,” Differ. Uravn. 26, 660–670 (1990).
V. Kh. Shogenov, S. K. Kumykova, and M. Kh. Shkhanukov-Lafishev, “Generalized transport equations and fractional derivatives,” Dop. Nats. Akad. Nauk Ukr., No. 12, 47–55 (1997).
R. R. Nigmatullin, “The realization of generalized transfer equation in a medium with fractional geometry,” Phys. Status Solidi B 133, 425–430 (1986).
V. M. Goloviznin, V. P. Kisilev, I. A. Korotkii, and Yu. P. Yurkov, Preprint IBRAE-2002-01 (Nuclear Safety Institute, Russian Academy of Sciences, Moscow, 2002).
V. M. Goloviznin, V. P. Kisilev, and I. A. Korotkii, Preprint IBRAE-2003-12 (Nuclear Safety Institute, Russian Academy of Sciences, Moscow, 2003).
A. A. Alikhanov, “Boundary value problems for the diffusion equation of the variable order in differential and difference settings,” Appl. Math. Comput. 219, 3938–3946 (2002).
J. Bangti, R. Lazarov, J. Pasciak, and Z. Zhou, “Error analysis of a finite element method for the space-fractional parabolic equation,” Soc. Ind. Appl. Math. 52 (5), 2272–2294 (2014).
J. Bangti, R. Lazarov, and Z. Zhou, “An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data,” IMA J. Numer. Anal. 36, 197–221 (2015).
J. Bangti, R. Lazarov, J. Pasciak, and Z. Zhou, “A Petrov–Galerkin finite element method for fractional convection-diffusion equations,” IMA Soc. Ind. Appl. Math. 54 (1), 481–503 (2016).
J. Feder, Fractals (Plenum, New York, 1988; Mir, Moscow, 1991).
V. Kh. Shogenov, A. A. Akhubekov, and R. A. Akhubekov, “Fractional differentiation method in the theory of Brownian motion,” Izv. Vyssh. Uch. Zaved. Severo-Kavkaz. Reg., No. 1, 46–50 (2004).
A. K. Bazzaev and M. Kh. Shkhanukov-Lafishev, “Locally one-dimensional schemes for the diffusion equation with a fractional time derivative in an arbitrary domain,” Comput. Math. Math. Phys. 56 (1), 106–115 (2016).
I. P. Mazin and S. M. Shmeter Clouds: Structure and Physics of Formation (Gidrometeoizdat, Leningrad, 1963) [in Russian].
V. A. Ashabokov and A. V. Shapovalov, Convective Clouds: Numerical Models and Results of Simulation under Natural Conditions and Artificial Forcing (Kabardino-Balkar. Nauchn. Tsentr Ross. Akad. Nauk, Nalchik, 2008) [in Russian].
A. A. Samarskii, The Theory of Difference Schemes (Nauka, Moscow, 1977; Marcel Dekker, New York, 2001).
M. M. Lafisheva and M. Kh. Shkhanukov-Lafishev, “Locally one-dimensional difference schemes for the fractional order diffusion equation,” Comput. Math. Math. Phys. 48 (10), 1875–1884 (2008).
A. A. Samarskii and A. V. Gulin, Stability of Difference Schemes (Nauka, Moscow, 1973) [in Russian].
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Original Russian Text © B.A. Ashabokov, Z.V. Beshtokova, M.Kh. Shkhanukov-Lafishev, 2017, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2017, Vol. 57, No. 9, pp. 1517–1529.
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Ashabokov, B.A., Beshtokova, Z.V. & Shkhanukov-Lafishev, M.K. Locally one-dimensional difference scheme for a fractional tracer transport equation. Comput. Math. and Math. Phys. 57, 1498–1510 (2017). https://doi.org/10.1134/S0965542517090044
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DOI: https://doi.org/10.1134/S0965542517090044