We state and study the first and second boundary value problems and the transmission problem for complex potentials of plane filtration flows in porous media. In the general case, the flow sources are arbitrary discrete sources that can be located both on the boundaries and outside the medium boundaries. If the boundaries are canonical (modeled by a straight line or a circle), then the solutions of the problems are obtained in closed form. In the case of arbitrary smooth boundaries with a point source (sink) located on them, we use a complex double layer potential with density characterized by an isolated singularity of the logarithmic type. This permits reducing the problems to boundary integral equations of the second kind of the Fredholm type whose right-hand sides are continuous functions of the coordinates. The problems under study are mathematical models of filtration flows in boundary value problems that are of interest, for example, in the practice of groundwater (oil) extraction from natural soil layers.

中文翻译：

---

边界处有源的平面平行过滤流问题

我们陈述和研究了多孔介质中平面过滤流的复势的第一和第二边值问题以及传输问题。在一般情况下，流源是任意离散源，可以位于边界上和介质边界外。如果边界是规范的（由直线或圆建模），则问题的解以封闭形式获得。在具有位于其上的点源（汇）的任意平滑边界的情况下，我们使用具有密度的复数双层势，其特征在于对数类型的孤立奇点。这允许将问题简化为第二类 Fredholm 类型的边界积分方程，其右手边是坐标的连续函数。