Bulavatsky V.M.
Some two-dimensional boundary-value problems of filtration dynamics for models with proportional Caputo derivative

Closed-form solutions are obtained for some two-dimensional non-stationary boundary-value problems of filtration dynamics in fractured-porous rocks, posed within the framework of fractional-differential mathematical models. These mathematical models are constructed using the generalized (proportional) Caputo derivative with respect to the time variable and Riemann–Liouville derivatives with respect to geometric variables. Along with the direct problem, we also consider a two-dimensional inverse boundary-value problem for determining the unknown source function that only depends on geometric variables. Conditions for the existence of regular solutions of the considered problems are given. For a separate case of only time nonlocality of the filtration process, a boundary-value problem with nonlocal boundary conditions is solved. © 2022, Springer Science+Business Media, LLC, part of Springer Nature.

Keywords: closed-form solutions, fractional-differential dynamics of filtration processes, fractured-porous media, inverse problems, mathematical modeling, non-classical models, problems with nonlocal conditions, proportional Caputo derivative, Riemann–Liouville derivative, two-dimensional boundary-value problems, Boundary value problems, Dynamics, Fracture, Porous materials, Caputo derivatives, Classical modeling, Closed form solutions, Filtration process, Fractional differential, Fractional-differential dynamic of filtration process, Fractured porous media, Mathematical modeling, Non-classical model, Non-local conditions, Problem with nonlocal condition, Proportional caputo derivative, Riemann-Liouville derivatives, Two-dimensional boundary value problems, Inverse problems

Cite:
Bulavatsky V.M. (2022). Some two-dimensional boundary-value problems of filtration dynamics for models with proportional Caputo derivative. Cybernetics and Systems Analysis, 58 (4), 70–81. doi: https://doi.org/10.1007/s10559-022-00499-2 http://jnas.nbuv.gov.ua/article/UJRN-0001335521 [In Ukrainian].