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## Scientific Bulletin. Physical and Mathematical Research

#### Abstract

The theory of the equations of mixed parabolic type is one of the most important parts of the theory of differensial equations being important on solving practical problems. The equations of a mixed parabolic type, in particular, are applied in hydrodynamics, when studying the motion of a fluid with an alternating coefficient The nonlocal boundary value problem with integral condition for equation of mixed parabolic type is studied in this paper. Existence and uniqueness of the solution of the problem are proved. In the limited with right quadrilateral domain of the plain , we consider the equation of mixed parabolic type: (1) where - parabolic part, this is – inverse parabolic part of domain ; Use the following notation: Definition. A regular solution of the equation (1) at domain is dedicated function , which satisfies this equation at the domains and be in class Problem I. Find a function with the following properties: 1) 2) is a regular solution of the equation (1) at ; 3) satisfies the conditions (2) (3) (4) (5) where - given sufficiently smooth functions, (6) (7) (8) (9 ) Th e o r e m. Let the conditions (6) - (9) be met. The solution of the problem I existence and uniqueness. To prove the uniqueness of the solution of the problem with integral condition, use the principle of maximumes: the solution of the problem I at positive maximum and negative minimum in the domain reaches only part of the boundary of this domain. The existence of the solution is proved by equivalent reduction to an existence of a solution of Fredholm integral equation of the second kind, wich will be uniquely solvable due to the uniqueness of the solution of the problem.

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#### References

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