Computer modeling of the dynamics of migration processes of soluble substances in the case of groundwater filtration with free surface on the base of the fractional derivative approach

TitleComputer modeling of the dynamics of migration processes of soluble substances in the case of groundwater filtration with free surface on the base of the fractional derivative approach
Publication TypeJournal Article
Year of Publication2018
AuthorsBogaenko, VA, Bulavatsky, VM
Abbreviated Key TitleDopov. Nac. akad. nauk Ukr.
DOI10.15407/dopovidi2018.12.021
Issue12
SectionInformation Science and Cybernetics
Pagination21-29
Date Published12/2018
LanguageRussian
Abstract

The mathematical modeling of the fractional differential dynamics of the process of anomalous convective diffusion of soluble substances is conducted for the case of flat-vertical steady state groundwater filtration with free surface. Within the framework of the model with a generalized Caputo—Gerasimov fractional derivative, the corresponding non-linear boundary-value problem is posed, a finite-difference method for its approximated solution is given, and the results of computer experiments are described.

Keywordsdynamics of convective and diffusive processes, finite-difference solutions, fractional differential mathematical models, generalized Caputo—Gerasimov derivative, mathematical and computer modeling, non-linear boundary-value problems, steady state flat-vertical groundwater filtration
References: 
  1. Lavryk, V. I., Filchakova, V. P. & Yashyn, A. A. (1990). Conformal mappings of physical topological models. Kiev: Naukova Dumka (in Russian).
  2. Liashko, I. I., Demchenko, L. I. & Mystetskyj, G. E. (1991). Numerical solution of heat and mass transfer problems in porous media. Kiev: Naukova Dumka (in Russian).
  3. Mystetskyj, G. E. (1985). Hydroconstruction. Automation of computations of mass transfer in soils. Kiev: Budivelnyk (in Russian).
  4. Polubarinova-Kochina, P. Ia. (1977). Theory of groundwater movement. Moscow: Nauka (in Russian).
  5. Bulavatskyj, V. M., Kryvonos, Iu. G. & Skopetskyj, V. V. (2005). Non-classical mathematical models of heat and mass transfer. Kyiv: Naukova Dumka (in Ukrainian).
  6. Bohaienko, V. A., Bulavatskyj, V. M., Skopetskyj, V. V. (2008). Parallel algorithm for computing filtrational convective pollutants diffusion from aquiferous strata. Upravliaiushchie sistemy i machyny, No. 5, pp. 18-23 (in Russian).
  7. Vlasiuk, A. P. & Ostanchuk, O. P. (2015). Mathematical modelling of salt solutions movement in the case of ground water filtration in soil massifs. Rivne: NUVGP (in Ukrainian).
  8. Bulavatsky, V. M. (2012). Mathematical modeling of dynamics of the process of filtration convection diffusion under the condition of time nonlocality. J. Automation and Information Science, 44, No. 2, pp. 13-22. doi: https://doi.org/10.1615/JAutomatInfScien.v44.i4.20
  9. Vlasiuk, A. P. & Martyniuk, P. M. (2008). Mathematical modelling of soil consolidation in the case of salt solutions filtration in non-isothermal conditions. Rivne: NUVGP (in Ukrainian).
  10. Bulavatskyj, V. M. & Kryvonos, Iu. G. (2014). Mathematical models with control functions for studying fractional differential dynamics of geomigration processes. Problemy upravlenija i informatiki, No. 3, pp. 138-147 (in Russian).
  11. Almeida, R. A. (2017). Caputo fractional derivative of a function with respect to another function. Communications in Nonlinear Science and Numerical Simulation, 44, pp. 460-481. doi: https://doi.org/10.1016/j.cnsns.2016.09.006
  12. Abramovitz, M. & Stegun, I.A. (1965). Handbook of Mathematical Functions. New York: Dover.
  13. Podlubny, I. (1999). Fractional differential equations. New York: Academic Press.
  14. Kilbas, A. A., Srivastava, H. M. & Trujillo, J. J. (2006). Theory and applications of fractional differential equations. Amsterdam: Elsevier.
  15. Samarskii, A. A. (2001). The Theory of Difference Schemes. New York: CRC Press. doi: https://doi.org/10.1201/9780203908518