REDUCTION OF THE FRACTIONAL DIFFERENTIAL MODEL OF FLAT-RADIAL FLUID FLOW IN A FRACTAL MEDIUM TO A SECOND-KIND FREDHOLM EQUATION

Over the last decade, interest in the theory of mass transfer in fractal media stems from geophysical and experimental data demonstrating the non-linearity of porous media, including their block structure and fractal properties. In such media, the filtration process, even for homogeneous mixtures, is modelled by non-integer-order differential equations, complicating theoretical research. In this paper, a solution is constructed for the fractional-order non-stationary diffusion equation describing the plane-radial motion of single-phase fluids in fractal porous media. To solve the reduced spectral problem for the derivative of the Laplace transform function, a Volterra integral equation of the second kind with a regular kernel is derived. For this integral equation, a resolvent in the form of a rapidly converging series is obtained by successive substitution. A necessary condition for the convergence of the series is found, and an analytical expression for the derivative of the Laplace transform of the desired pressure function is obtained. By integrating the resulting analytical expression for the Laplace transform with respect to the spatial variable, an integral representation is found. Finally, by applying the inverse Laplace transform, an expression for the dynamic distribution of the desired pressure function is obtained. The resulting expression, firstly, allows one to study the problem of unsteady fluid flow in a radial medium of fractal nature, and secondly, also answers the question of the existence and uniqueness of the problem under consideration. Using numerical integration methods, one can calculate the values of the pressure function at any point r and any time t, with an error estimate. Furthermore, the obtained results allow one to evaluate the solution found using approximate methods. © 2025 Earth Science Division, Azerbaijan National Academy of Sciences. All rights reserved.