scholarly journals To the solution of the Solonnikov-Fasano problem with boundary moving on arbitrary law x = γ(t).

2021 ◽  
Vol 101 (1) ◽  
pp. 37-49
Author(s):  
M.T. Jenaliyev ◽  
◽  
M.I. Ramazanov ◽  
A.O. Tanin ◽  
◽  
...  
Keyword(s):  
Integral Equation ◽  
Integral Operator ◽  
Boundary Value Problem ◽  
Heat Equation ◽  
Volterra Integral Equation ◽  
Boundary Value ◽  
Nonzero Solution ◽  
Initial Moment ◽  
Homogeneous Integral Equation

In this paper we study the solvability of the boundary value problem for the heat equation in a domain that degenerates into a point at the initial moment of time. In this case, the boundary changing with time moves according to an arbitrary law x = γ(t). Using the generalized heat potentials, the problem under study is reduced to a pseudo-Volterra integral equation such that the norm of the integral operator is equal to one and it is shown that the corresponding homogeneous integral equation has a nonzero solution.

Heat Transfer in Composite Solids with Heat Generation

1979 ◽  
Vol 101 (1) ◽  
pp. 137-143 ◽  
Author(s):  
L. Feijoo ◽  
H. T. Davis ◽  
D. Ramkrishna
Keyword(s):  
Heat Transfer ◽  
Integral Equation ◽  
Integral Operator ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Adjoint Operator ◽  
The Self ◽  
Basis Set ◽  
Inner Product ◽  
Steady State Heat Transfer

Steady-state heat transfer problems have been considered in a composite solid comprising two materials, one, a slab, which forms the bulk of the interior and the other, a plate, which forms a thin layer around the boundary. Through the use of appropriate Green’s functions, it is shown that the boundary value problem can be converted into a Fredholm integral equation of the second kind. The integral operator in the integral equation is shown to be self-adjoint under an appropriate inner product. Solutions have been obtained for the integral equation by expansion in terms of eigenfunctions of the self-adjoint integral operator, from which the solution to the boundary value problem is constructed. Two problems have been considered, for the first of which the eigenvalues and eigenvectors of the self-adjoint operator were analytically obtained; for the second, the spectral decomposition was obtained numerically by expansion in a convenient basis set. Detailed numerical computations have been made for the second problem using various types of heat source functions. The calculations are relatively easy and inexpensive for the examples considered. These examples, we believe, are sufficiently diverse to constitute a rather stringent test of the numerical merits of the eigenvalue technique used.

scholarly journals On the singular Volterra integral equation of the boundary value problem for heat conduction in a degenerating domain

2021 ◽  
Vol 31 (2) ◽  
pp. 241-252
Author(s):  
M.I. Ramazanov ◽  
N.K. Gulmanov
Keyword(s):  
Integral Equation ◽  
Heat Conduction ◽  
Boundary Value ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Initial Moment ◽  
Volterra Type ◽  
Equivalent Regularization ◽  
Type Integral ◽  
Homogeneous Integral Equation

In this paper, we consider a singular Volterra type integral equation of the second kind, to which some boundary value problems of heat conduction in domains with a boundary varying with time are reduced by the method of thermal potentials. The peculiarity of such problems is that the domain degenerates into a point at the initial moment of time. Accordingly, a distinctive feature of the integral equation under study is that the integral of the kernel, as the upper limit of integration tends to the lower one, is not equal to zero. This circumstance does not allow solving this equation by the method of successive approximations. We constructed the general solution of the corresponding characteristic equation and found the solution of the complete integral equation by the Carleman–Vekua method of equivalent regularization. It is shown that the corresponding homogeneous integral equation has a nonzero solution.

scholarly journals On a boundary value problem for the heat equation and a singular integral equation associated with it

2021 ◽  
Vol 399 ◽  
pp. 126009
Author(s):  
Meiramkul Amangaliyeva ◽  
Muvasharkhan Jenaliyev ◽  
Sagyndyk Iskakov ◽  
Murat Ramazanov
Keyword(s):  
Integral Equation ◽  
Singular Integral Equation ◽  
Boundary Value Problem ◽  
Heat Equation ◽  
Singular Integral ◽  
Boundary Value

scholarly journals Application of thermal potentials to the solution of the problem of heat conduction in a region degenerates at the initial moment

Filomat ◽  
2018 ◽  
Vol 32 (3) ◽  
pp. 825-836
Author(s):  
Alexey Kavokin ◽  
Adiya Kulakhmetova ◽  
Yuriy Shpadi
Keyword(s):  
Heat Transfer ◽  
Integral Equation ◽  
Heat Conduction ◽  
Liquid Metal ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Constructive Method ◽  
Initial Time ◽  
Initial Moment ◽  
Electric Contacts

In this paper, the boundary value problem for the heat equation in the region which degenerates at the initial time is considered. Such problems arise in mathematical models of the processes occurring by opening of electric contacts, in particular, at the description of the heat transfer in a liquid metal bridge and electric arcing. The boundary value problem is reduced to a Volterra integral equation of the second kind which has a singular feature. The class of solutions for the integral equation is defined and the constructive method of its solution is developed.

scholarly journals To solving the fractionally loaded heat equation

2021 ◽  
Vol 101 (1) ◽  
pp. 65-77
Author(s):  
M.T. Kosmakova ◽  
◽  
S.A. Iskakov ◽  
L.Zh. Kasymova ◽  
◽  
...  
Keyword(s):  
Integral Equation ◽  
Boundary Value Problem ◽  
Heat Equation ◽  
Fractional Derivative ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Continuous Functions ◽  
Uniqueness Of Solutions ◽  
Loaded Term ◽  
Differential Part

In this paper we consider a boundary value problem for a fractionally loaded heat equation in the class of continuous functions. Research methods are based on an approach to the study of boundary value problems, based on their reduction to integral equations. The problem is reduced to a Volterra integral equation of the second kind by inverting the differential part. We also carried out a study the limit cases for the fractional derivative order of the term with a load in the heat equation of the boundary value problem. It is shown that the existence and uniqueness of solutions to the integral equation depends on the order of the fractional derivative in the loaded term.

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