On existence of solution to boundary value problem with integral condition for one model equation
Abstract
This article proves the existence of a solution to a boundary value problem for a model equation describing the dynamics of moisture in heterogeneous soil and groundwater.The movement of water in the upper horizons of the heterogeneous soil, even without raising the ground to the surface of the earth, can cause a redistribution of readily soluble salts in them and secondary salinization under an irrational irrigation regime, or vice versa, desalinization of soils with correctly selected rates and timing of irrigation.In general, a long-term forecast of the water and salt regime of soils is a necessary component of any irrigation reclamation project.When designing dams, especially bulk dams, it is necessary to determine the position of the depression curve, filtering through the body of the dam moisture, which determines the structure's durability.Mathematical modelling uses analytical methods based on the hydrodynamic laws of groundwater movement.As a result, a boundary value problem for a mixed equation of the second kind in a bounded elliptic domain with local fractional differential boundary conditions is formulated and studied.The principle of the extremum method was used to prove the solvability of the problem posed.In this case, the question of the solvability of the problem was equivalently reduced to the question of the solvability of a singular integral equation.As a result, an existence theorem for the problem posed was formulated and proven.