Solvability of Boundary Value Problems With Non-Local Conditions for Multidimensional Hyperbolic Equations

Koshanov, B. D.; Koshanova, G.D.; Azimkhan, G.E.; Segizbayeva, R.U.

doi:10.32014/2020.2518-1726.21

OF THE NATIONAL ACADEMY OFSCIENCES OF THE REPUBLIC OF KAZAKHSTA

2020

DOI: 10.32014/2020.2518-1726.21

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Solvability of Boundary Value Problems With Non-Local Conditions for Multidimensional Hyperbolic Equations

B. D. Koshanov¹,

G.D. Koshanova²,

G.E. Azimkhan³

et al.

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2021

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“…The solvability of various boundary value problems for a biharmonic equation in a multidimensional sphere is studied in [15][16][17][18].…”

mentioning

confidence: 99%

Green's Functions of Some Boundary Value Problems for Byharmonic Operators and Their Correct Constrictions

Koshanov

Baiarystanov

Daurenkyzy

et al. 2021

News of NAN RK. SGTS

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In this paper, a constructive method is given for constructing the Green function of the Dirichlet problem for a biharmonic equation in a multidimensional ball. The need to study boundary value problems for elliptic equations is dictated by numerous practical applications in the theoretical study of the processes of hydrodynamics, electrostatics, mechanics, thermal conductivity, elasticity theory, and quantum physics. The distributions of the potential of the electrostatic field are described using the Poisson equation. When studying the vibrations of thin plates of small deflections, biharmonic equations arise. There are various ways to construct the Green Function of the Dirichlet problem for the Poisson equation. For many types of domains, it is constructed explicitly. And for the Neumann problem in multidimensional domains, the construction of the Green function is an open problem. For the ball, the Green function of the internal and external Neumann problem is constructed explicitly only for the two-dimensional and three-dimensional cases. Finding general correct boundary value problems for differential equations is always an urgent problem. The abstract theory of operator contraction and expansion originates from the work of John von Neumann, in which a method for constructing self-adjoint extensions of a symmetric operator was described and a theory of extension of symmetric operators with finite defect indices was developed in detail. Many problems for partial differential equations lead to operators with infinite defect indices. In the early 80s of the last century, M.O. Otelbaev and his students built an abstract theory that allows us to describe all correct constrictions of a certain maximum operator and separately - all correct extensions of a certain minimum operator, independently of each other, in terms of the inverse operator. In this paper, the correct boundary value problems for the biharmonic operator are described using the Green's function.

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“…The solvability of various boundary value problems for a biharmonic equation in a multidimensional sphere is studied in [15][16][17][18].…”

mentioning

confidence: 99%

Green's Functions of Some Boundary Value Problems for Byharmonic Operators and Their Correct Constrictions

Koshanov

Baiarystanov

Daurenkyzy

et al. 2021

News of NAN RK. SGTS

Self Cite

View full text Add to dashboard Cite

show abstract

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Solvability of Boundary Value Problems With Non-Local Conditions for Multidimensional Hyperbolic Equations

Cited by 1 publication

References 12 publications

Green's Functions of Some Boundary Value Problems for Byharmonic Operators and Their Correct Constrictions

Green's Functions of Some Boundary Value Problems for Byharmonic Operators and Their Correct Constrictions

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