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 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.

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.

EXISTENCE OF SOLUTION FOR A VOLTERRA TYPE INTEGRAL EQUATION USING DARBO-TYPE F-CONTRACTION

2021 ◽  
Vol 10 (6) ◽  
pp. 2687-2710
Author(s):  
F. Akutsah ◽  
A. A. Mebawondu ◽  
O. K. Narain
Keyword(s):  
Integral Equation ◽  
Fixed Point ◽  
Metric Spaces ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Existence Of Solution ◽  
Volterra Type ◽  
Complete Metric Spaces ◽  
Type Integral ◽  
New Type

In this paper, we provide some generalizations of the Darbo's fixed point theorem and further develop the notion of $F$-contraction introduced by Wardowski in (\cite{wad}, D. Wardowski, \emph{Fixed points of a new type of contractive mappings in complete metric spaces,} Fixed Point Theory and Appl., 94, (2012)). To achieve this, we introduce the notion of Darbo-type $F$-contraction, cyclic $(\alpha,\beta)$-admissible operator and we also establish some fixed point and common fixed point results for this class of mappings in the framework of Banach spaces. In addition, we apply our fixed point results to establish the existence of solution to a Volterra type integral equation.

A volterra-type integral equation

1989 ◽  
Vol 40 (4) ◽  
pp. 438-442 ◽  
Author(s):  
S. Ashirov ◽  
Ya. D. Mamedov
Keyword(s):  
Integral Equation ◽  
Volterra Type ◽  
Type Integral

The Bloch integral equation and electrical conductivity

1950 ◽  
Vol 202 (1071) ◽  
pp. 466-484 ◽  
Keyword(s):  
Electrical Conductivity ◽  
Integral Equation ◽  
Temperature Variation ◽  
Characteristic Temperature ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Theoretical Treatment ◽  
Temperature Range ◽  
High And Low Temperatures ◽  
Semi Empirical

An account is given of the solution, for effectively the whole temperature range, of the Bloch (1928) integral equation for the electron momentum distribution in a metal in an electric field. Solutions of this equation, from which the temperature variation of the electrical conductivity of the metal may be immediately calculated, have previously been obtained only in the limiting cases of ‘high- and low -temperatures’, corresponding to ( T /θ p )≫ 1 and ≫1, where θ p is the Debye characteristic temperature. As a preliminary to its solution by numerical methods the integral equation is expressed in a non-dimensional form (§2). Solutions are obtained by deriving a high-temperature approximation which is valid over a much wider temperature range than that previously known, and by means of a method of successive approximations (§3). The temperature variation of conductivity is calculated from these solutions, and it is shown that there are significant differences between the results and those obtained from the semi-empirical formula of Grüneisen (1930) (§4). A comparison is made between the calculated and observed temperature variation of conductivity for a number of metals. There are deviations in detail, and a brief discussion is given of secondary factors from which they may arise, but in general the agreement is good, and it is concluded that the theoretical treatment covers satisfactorily the main features of the observed variation (§5). In an appendix it is shown that the approximate relations obtainable by the variational method developed by Kohler (1949) are consistent with the more exact results obtained here.

scholarly journals The ring delamination between bodies under the action of heat sinks distributed along a circle

2017 ◽  
pp. 55-62
Author(s):  
Maryana Mykytyn ◽  
Kristina Serednytska ◽  
Bohdan Monastyrskyy ◽  
Rostyslav Martynyak
Keyword(s):  
Integral Equation ◽  
Singular Integral Equation ◽  
Half Space ◽  
Singular Integral ◽  
Elastic Half Space ◽  
Heat Sinks ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Normal Contact ◽  
Frictionless Contact

The frictionless contact an elastic half-space and a rigid thermo-insulated base with a local delamination between them on a ring domain under the action of heat sinks distributed uniformly along a circle and located in the half-space some distance away from its surface, is considered. The corresponding contact thermos-elasticity problem is reduced to a singular integral equation for a height of a ring gap. The solution of the singular integral equation and the internal and external radius of the ring are numerically determined using the method of collocation and the method of successive approximations. The dependence of the form of gap and normal contact stresses on the distance between the heat sinks and the surface of the half-space and the intensity of the heat sink are analyzed.

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