On a homogeneous parabolic problem in an infinite corner domain

Jenaliyev, Muvasharkhan; Ramazanov, M.I.

doi:10.1063/1.4959699

Cited by 9 publications

(13 citation statements)

References 2 publications

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“…Further, on the basis of the integral representation of the solution of the boundary value problem in the form of a sum of thermal potentials, we will reduce the study of the original problem to the study of the Volterra integral equation of the second kind, following [21] and [1][2][3][4][5][6].…”

Section: Resultsmentioning

confidence: 99%

“…and to conditions (2) and 3, we obtain in the domain {r, r > 0} the boundary-value problem for the ordinary differential equation:…”

Section: Function Of the Thermal Instantaneous Point Sourcementioning

confidence: 99%

“…Problem II. In the domain Ω ∞ = {(r, t) : 0 < r < ∞; t > 0} we consider the boundary value problem for equation (1) under boundary conditions (2), (3) and and initial condition u(r, 0) = 0, r > 0.…”

Section: Introductionmentioning

confidence: 99%

“…where ∂Ω 1 is the lateral surface of the cone. Earlier [1][2][3][4] we studied a homogeneous problem for the heat equation in the angular domain G = {(x; t) : t > 0, 0 < x < t} (as the domain Ω):…”

Section: Introductionmentioning

confidence: 99%

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Constructing the fundamental solution to a problem of heat conduction

Kosmakova¹,

Tanin²,

Tuleutaeva³

2020

Bul. Kar. Univ. - Math.

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In this article, we discuss auxiliary initial-boundary value problems which will subsequently be used to solve boundary-value problem of heat conduction with axial symmetry in a degenerating domain. One of the problems is posed with homogeneous boundary conditions in order to construct a fundamental solution that is used to determine thermal potentials. The initial condition contains the Dirac function. The solution to the problems is found explicitly using the Laplace integral transformation. The boundary value problem is also considered in the absence of axial symmetry. It is shown that this problem splits into families of boundary-value problems similar to the problems considered above. In conclusion, we state the boundary value problem of heat conduction with axial symmetry in a degenerating domain, and its fundamental solution, found above, is written out.

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Section: Resultsmentioning

confidence: 99%

“…and to conditions (2) and 3, we obtain in the domain {r, r > 0} the boundary-value problem for the ordinary differential equation:…”

Section: Function Of the Thermal Instantaneous Point Sourcementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Constructing the fundamental solution to a problem of heat conduction

Kosmakova¹,

Tanin²,

Tuleutaeva³

2020

Bul. Kar. Univ. - Math.

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“…In this connection, the weight classes of the solution existence differ from the class of the solution existence for the equation considered in this work. We also note that boundary value problems for a spectrally loaded parabolic equation reduce to this kind of singular integral equations, when the load line moves according to the law x = t [5-10] and problems for essentially loaded equation of heat conduction [11][12][13][14][15].…”

Section: Resultsmentioning

confidence: 99%

On the integral equation of an adjoint boundary value problem of heat conduction

Kosmakova¹,

Romanovski²,

Orumbayeva³

et al. 2019

Bul. Kar. Univ. - Math.

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An integral equation is considered, to which a nonhomogeneous first boundary value problem with an adjoint heat conduction operator is reduced. The problem is set in an infinite plane angle, that is, a boundary of the domain moves with a constant velocity, and the domain degenerates to a point at the initial moment of time. The incompressibility of the integral operator for the equation under study is shown. Using the relations for an independent variable, the equation under study is equivalently reduced to a certain simplified equation. With the help of replacements for independent variables, the equation is reduced to an integral equation with a difference kernel. By applying the Laplace transform, the obtained equation is reduced to an ordinary first-order differential equation (linear). Its solution is found. By using the inverse Laplace transform, a solution of the nonhomogeneous integral equation under study is obtained in the form of a convergent series in some domain.

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