Investigation of Three-Dimensional Fluid Filtration Problems with Sources on the Boundaries

• Published in: Differential equations Vol. 57; no. 9; pp. 1214 - 1230
• Main Author: Piven’, V. F.
• Format: Journal Article
• Language: English
• Published: Moscow Pleiades Publishing 01.09.2021; Springer; Springer Nature B.V
• Subjects: Boundary conditions; Boundary value problems; Difference and Functional Equations; Differential equations; Filtration; Inhomogeneous media; Integral and Integro-Differential Equations; Integral equations; Mathematical models; Mathematics; Mathematics and Statistics; Ordinary Differential Equations; Partial Differential Equations; Smooth boundaries; Soil layers; Soil water; Three dimensional flow; Three dimensional models
• ISSN: 0012-2661; 1608-3083
• DOI: 10.1134/S001226612109010X

We state and study the first and second boundary value problems and the transmission problem with complicated boundary conditions containing singular functions for three-dimensional filtration flows in an inhomogeneous medium. The flow sources are arbitrary discrete sources and are located both on the boundaries and outside the boundaries. When boundaries are modeled by canonical surfaces (a plane and a sphere), the solutions of the problem can be presented in closed form. In the case of arbitrary closed smooth boundary surfaces with a point sink (source) located on them, the generalized double layer potential is used; this permits one to reduce the transmission problem and the second boundary value problem to singular and hypersingular integral equations, respectively. The problems studied here are mathematical models of three-dimensional filtration processes in heterogeneous media, which are of interest, for example, for the practice of developing natural oil-bearing (water-bearing) soil layers.