UDC 517.9: 519.6
1 V.M. Glushkov Institute of Cybernetics, National Academy of Sciences of Ukraine, Kyiv, Ukraine
 v_bulav@ukr.net
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SOME TWO-DIMENSIONAL BOUNDARY-VALUE PROBLEMS OF FILTRATION
DYNAMICS FOR MODELS WITH PROPORTIONAL CAPUTO DERIVATIVE
Abstract. Closed solutions are obtained for some two-dimensional non-stationary boundary-value problems of filtration dynamics in fractured-porous formations, 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.
Keywords: mathematical modeling, fractional-differential dynamics of filtration processes, fractured-porous media, non-classical models, proportional Caputo derivative, Riemann–Liouville derivative, two-dimensional boundary-value problems, inverse problems, problems with non-local conditions, closed-form solutions.
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