About one boundary-value problem arising in modeling dynamics of groundwater

. Modeling the movement of moisture in the soil is of great importance for assessing the impact of agricultural land on surface water bodies and, consequently, on the natural environment and humans. This is because huge volumes of pollutants from the fields (pesticides, mineral fertilizers, nitrates, and nutrients contained in them) are transferred to reservoirs by filtering moisture. Different methods solve all these tasks. The method of natural analogies is based on the analysis of graphs of fluctuations in groundwater level. To apply this method on irrigated lands, it is necessary to have a sufficiently studied irrigated area with similar natural, organizational and economic conditions. The successful application of this method, based on the fundamental theory of physical similarity, mainly depends on the availability of a sufficiently close comparison object, which is quite rare in practice. Physical modeling is often used to construct dams and other hydraulic structures. Previously, the method of electrical modeling was also widely used. It was further found that nonlocal boundary conditions arise in the problems of predicting soil moisture, modeling fluid filtration in porous media, mathematical modeling of laser radiation processes, and plasma physics problems, as well as mathematical biology.


Introduction
At present, boundary value problems for equations of mixed type have become an important part of the modern theory of partial differential equations.One of the main problems in the theory of partial differential equations is the study of mixed-type equations, which is of theoretical and practical interest.In 1959, I.N.Vekua pointed out the importance of the problem of equations of mixed type in connection with problems in the theory of infinitesimal bendings of surfaces.The problem of the outflow of a supersonic jet from a vessel with flat walls is reduced to the Tricomi problem for the Chaplygin equation (a degenerate equation of mixed type).There are several works by F. Tricomi, S. Gelderstedt, A. V. Bitsadze, M. S. Salakhitdinov, T.D. Dzhuraev and their students in which the main mixed boundary value problems are studied, and new correct problems are posed for the equations of the elliptic-hyperbolic, parabolic-hyperbolic types of the first kind, i.e., equations for which the degeneracy line is not a characteristic.
In recent years, a large number of papers have appeared devoted to the study of equations of composite and mixed-composite types.Correct boundary value problems for equations of mixed-composite type, the main part of which contains an elliptic-hyperbolic operator, were first formulated by A.V. Bitsadze (see [1], [2]).These problems and some of their generalizations have now been studied in detail.
We note that the results of all the above works were obtained for equations of the first kind, and for equations of the second kind of the third order, boundary value problems have not been previously studied.
Therefore, the study of boundary value problems for mixed-type equations of the second kind seems very relevant and little studied.We note the works [3][4][5][6].
In this paper, we study a local boundary value problem for equations of mixed composite type of the second kind, i.e., for an equation where the line of degeneracy is a characteristic., , , , ,

Statement of the problem
Note that this problem is in the case 0 m studied in [2] and in the case 0 1 m considered in [1].
Without loss of generality, we can assume that where , cos 2 Q are continuous and integrable on (0, 1), where Green's function of the first boundary value problem for the heat equation has the form [7][8]: x y e a t y z x y y at y Based on (10), (11) and that in 1 : y BD The last equality can be expressed as follows , 0 , ; 1 ,  12) is an integral Fredholm equation of the second kind, the solvability of which follows from the uniqueness of the solution to the problem and is determined by the formula Calculating the derivative x z w w , then letting x tend to zero, taking into account (9) and the Dirichlet transformation, we have We extend the first and second integrals on the right side of (13) concerning t to (0, 1) those., , , , Taking into account the gluing condition and excluding y y z x Q , 0 from ( 14) and ( 15), we have The study of equation (16) shows that it is an integral Fredholm equation of the second kind with a weak singularity.Its unique solvability follows from the uniqueness of the solution to the problem.Solutions of the integral equation ( 16) can be written using the resolvent as is the resolvent of equation (16).
Subordinating (7) to the conditions on the characteristics of OC, BC .,

\ \
and taking into account (17), i.e., on the 0 : [ OC from (7) and denoting The latter system has a solution, which proves the existence of a solution to the Dirichlet problem.

Studies on the smoothness of given functions
It can be seen that if we use from (18), we can find ‫,)ݕ(ܶ‬ using the fractional operator, we rewrite ܰ(ߟ) in the following form: Let us present some auxiliary expansions of the Green's function involved inside the integral as a kernel , where * \ is a continuous function.Now from ( 22), we will study the function ^> @ It can be seen that the right side of equation ( 23) has a weak feature.Therefore, we cannot immediately differentiate it.To avoid this, we will first integrate by parts and then differentiate.Repeating this process five times and putting the result obtained in (22) ^K K . The study of this expression shows that the existence of the integral depends on the continuity of the function belonging to the kernel.To do this, we will do the following: Insofar as .From (20), it can be seen that the function ) ( ) (

Conclusions
Thus, with the help of energy integrals, the uniqueness of the solution of the boundary value problem for the homogeneous equation of parabolic -hyperbolic type of the third order of the second kind is proved.Necessary and sufficient conditions for the existence of a generalized solution to the formulated problem are found.An explicit representation of the solution to the problem under study is obtained.The results obtained and the developed method makes it possible to further investigate similar boundary value problems for a homogeneous parabolic-hyperbolic type equation of the third order of the second kind.

1 D , and in the domain 2 D
of equation(2) in the domain is a generalized solution of the class R. Denote derivatives in equation(1), and the smoothness of the function is given by the definition of a generalized solution of the class R of equation(1).Dirichlet problem.Required to define a function a generalized solution of equation (1) of class R in the domain 2 D , and in the domain 1 D is regular; c) the gluing condition is satisfied on the degeneracy line ௫ continuous up to the transition line both on the left and on the right; e) satisfies the boundary conditions > @

. 1 D 2 D3 2 D
Based on (3) and boundary conditions, the Dirichlet problem is reduced to the definition of a regular solution in the domain , a generalized solution of the class R in the domain Uniqueness of solutions to the problem We will prove the uniqueness of the problem under consideration by the method of energy integrals.In the domain of ‫ܦ‬ ଵ we have the equation 0 Let us show that the second integral of the left side of the equality is equal to zero.To do this, we use Green's formula, and since ‫ܦ‬ ଶ and applying Green's formula to the right side of equality, we have Let us divide the first integral into three parts, i.e., integrating by parts, respectively; we have Conferences 365, 01016(2023)   https://doi.org/10.1051/e3sconf/202336501016CONMECHYDRO uniqueness of the Cauchy problem in the hyperbolic domain we obtain 0 , which was to be proved.

4 2 D
Existence of a solution to the problemIt is known that the solution of the Cauchy problem for the equation 0 has the form

2 D
z x y is the generalized solution of the Cauchy problem for the equation 0 2 z L in the domain of from the class ܴ ଶ then has representation(7

For ( 21 )
to take place, it is necessaryФ ଶ was a continuous function, then from the representation Ф ଶ ‫)ݕ(‬ it easily follows that

2 \.
From the definition of integro- differential operators of fractional order ߙ > 0 those.

2 \
above results, we will formulate the following theorem: continuous functions, then the solution of the Dirichlet problem exists and is unique. x