Mathematical Models and Problems of Fractional-Differential Dynamics of Some Relaxation Filtration Processes
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MATHEMATICAL MODELS AND PROBLEMS OF FRACTIONAL-DIFFERENTIAL DYNAMICS OF SOME RELAXATION FILTRATION PROCESSES
V. M. Bulavatsky
UDC 517.9:519.6
Abstract. The author constructs fractional-differential mathematical models to describe the dynamics of geofiltration processes under pressure relaxation. The models are based on the concepts of the generalized Caputo and Hilfer derivatives, as fractional-order derivatives of a function with respect to another function. Within the framework of these models, analytical solutions of some filtration boundary-value problems, including the problem with nonlocal boundary conditions, are obtained. Keywords: mathematical modeling, locally non-equilibrium processes of geofiltration, fractional-differential mathematical models, Caputo and Hilfer derivatives, boundary-value problems, nonlocal boundary conditions. INTRODUCTION Mathematical modeling of the dynamics of geofiltration processes under complicated conditions is an important direction in geomathematics, geoinformatics, and geomechanics, which develops mainly within the framework of classical problem statements on the basis of standard methods and approaches of the continuum theory [1–4]. The majority of well-known mathematical models of transfer processes in geoporous media is based on classical transfer laws, which are inadequate in case of a substantial deviation of the system from equilibrium state [1, 5]. Moreover, classical transfer models impose rather rigid constraint on processes such as infinite velocity of the propagation of perturbations, which contradicts modern physical representations. Attempts of theoretical account for non-equilibrium effects (in particular, memory effects) in case of nonstationary filtration in a porous medium resulted in the relaxation filtration theory, which is apparently presented in detail in the well-known study [4]. An efficient modern approach in the description of transfer processes in systems for which it is important to take into account nonlocal space–time properties is related to using the apparatus of integro-differentiation of non-integer order [6–11]. For example, in [10], mathematical models are constructed and solutions are obtained to some boundary-value filtration problems on modeling the fractional-differential dynamics of relaxation filtration processes in porous and fractured-porous rock mass of finite thickness. And in [11], the problem of modeling the fractional-differential dynamics of relaxation filtration process under nonlocal boundary conditions is formulated and solved. Noteworthy is also the study [12] on mathematical modeling of the fractional-differential dynamics of relaxation processes of convection diffusion of soluble substances in underground filtration flows. In the present paper, we will consider new fractional-differential mathematical models of geofiltration processes with regard for pressure relaxation, which are based on the concepts of Caputo and Hilfer derivatives of a function with V. M. Glushkov Institute of Cybernetics, National Academy of Sc
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