Home > ajphysmat.uz > Vol. 1 > No. 1 (2019)

## Scientific Bulletin. Physical and Mathematical Research

#### Article Title

THE SOLUTION EXISTENCE AND UNIQUENESS OF THE NONLOKAL BOUNDARY VALUE PROBLEM WITH INTEGRAL CONDITION

#### 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.

#### First Page

77

#### Last Page

85

#### References

1. Canon, J.R. (1963) The solution of the heat equation subject to the specification of energy. Quarterly of Applied Mathematics.Vol. 21. Issue 2. pp. 155-160. 2. Ionkin, N.I. (1977) Resheniye odnoy krayevoy zadachi teorii teploprovodnosti s neklassicheskim krayevym usloviyem [The solution of one boundary value problem of theory heat with nonclassical boundary value problem]. Differensialnye uravneniya. Vol. 13. Issue 2. pp. 294-304. 3. Akbarova, M.Kh. (1992) Ob odnoy nolokalnoy zadache s integralnym usloviyem dlya smeshanno-parabolicheskogo uravneniya [On a nonlocal problem with integral condition for the mixed-parabolic equation]. Doklady Akademii Nauk Respubliki Uzbekistan. pp.6-9. 4. Akbarova, M.Kh. (2016) Nonlocal problem with discontinuous bounding conditions for linear parabolic equations of mixed type. The second Inter Conference on “Application of Mathematics and Informatics in Natural Sciences an Engineering” Dedicated to the Centenary of Andro Bitsadze. I. Javakhishvili Tbilisi State University, L. Vekua Institute of Applied Mathematics. P. 9. 5. Tikhonov, A.N., Samarskiy A.A. (1953) Uravneniya matematicheskoy fiziki [Equation of mathematical physics]. Moscow: Glavizdat. 6. Kerefov, A.A. (1974) Ob odnoy krayevoy zadache Jevre dlya parabolicheskogo uravneniya so znakoperemennym razryvom pervogo roda u koeffisiyenta pri proizvodnoy po vremeni [On a boundary value problem Jevre for a parabolic equation with alternating first-type discontinuity of coefficient at the time derivative time] Differensialnye uravneniya. Vol. 10. Issue 1. pp. 69-77. 7. Fridman, A. (1968) Uravneniye s chastnymi proizvodnymi parabolicheskogo tipa [Partial differensial equations of parabolic type]. Moskov: Mir. 8. Urinov, A.K. (2015) Parabolik tipdagi differensial tenglamalar uchun chegaraviy masalalar [A boundary value problems for differential equations of parabolic type]. Tashkent: Mumtoz so`z. 9. Vladimirov, V.S. (1971) Uravneniya matematicheskoy fiziki [Equation of mathematical physics]. Moscow: Nauka. 10. Salohiddinov, M. (2007) Integral tenglamalar [Integral equations]. Tashkent: Yangiyul polugraph service.

#### Recommended Citation

Akbarova, S. X. and Akbarova, M. X.
(2019)
"THE SOLUTION EXISTENCE AND UNIQUENESS OF THE NONLOKAL BOUNDARY VALUE PROBLEM WITH INTEGRAL CONDITION,"
*Scientific Bulletin. Physical and Mathematical Research*: Vol. 1
:
No.
1
, Article 10.

Available at:
https://uzjournals.edu.uz/adu/vol1/iss1/10