About Dirichlet boundary value problem for the heat equation in the infinite angular domain

Jenaliyev, Muvasharkhan; Amangaliyeva, Meiramkul; Kosmakova, M.T.; Ramazanov, M.I.

doi:10.1186/s13661-014-0213-4

Cited by 11 publications

(13 citation statements)

References 7 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%

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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%

“…In [6], along with the direct problem, the conjugate boundary-value problem for the heat equation in the weighted functional class was also studied, and it was established that the posed boundary value problem is Noetherian problem.…”

Section: Introductionmentioning

confidence: 99%

Constructing the fundamental solution to a problem of heat conduction

Kosmakova¹,

Tanin²,

Tuleutaeva³

2020

Bul. Kar. Univ. - Math.

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“…Remark. Singular homogeneous integral equations were considered in works [1][2][3][4]. Their kernels were also incompressible, but kernels had an another form.…”

Section: Resultsmentioning

confidence: 99%

“…Hence, the characteristic part of equation (1) is the second term of the kernel (2). Using relations:…”

Section: Incompressibility Of An Integral Operator and Reducing The Imentioning

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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“…Partial differential equations play an important role for the development of models in heat conduction and investigated in various aspects (see, for example, literature() and the references therein). To realize the physical changes, some models need to be expressed as free or moving boundary problems.…”

Section: Introductionmentioning

confidence: 99%

An approach for solving an inverse spherical two‐phase Stefan problem arising in modeling of electric contact phenomena

Sarsengeldin

Erdogan

Nauryz

et al. 2017

Math Methods in App Sciences

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In this paper, a model problem that can be used for mathematical modeling and investigation of arc phenomena in electrical contacts is considered. An analytical approach for the solution of a two‐phase inverse spherical Stefan problem where along with unknown temperature functions heat flux function has to be determined is presented. The suggested solution method is obtained from a new form of integral error function and its properties that are represented in the form of series whose coefficients have to be determined. Using integral error function and collocation method, the solution of a test problem is obtained in exact form and approximately.

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About Dirichlet boundary value problem for the heat equation in the infinite angular domain

Cited by 11 publications

References 7 publications

Constructing the fundamental solution to a problem of heat conduction

Constructing the fundamental solution to a problem of heat conduction

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

An approach for solving an inverse spherical two‐phase Stefan problem arising in modeling of electric contact phenomena

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