In this paper the mathematical models in the form of nonlocal problems for the two-dimensional heat equation are considered. Relation of a nonlocal problem and a boundary value problem, which describe the same physical heating process, is investigated. These problems arise in the study of the temperature distribution during annealing of the movable wire and the strip by permanent or periodically operating internal and external heat sources. The first and the second nonlocal problems in the mobile area are considered. Stability and convergence of numerical algorithms for the solution of a nonlocal problem with piecewise monotone functions in the equations and boundary conditions are investigated. Piecewise monotone functions characterize the heat sources and heat transfer conditions at the boundaries of the area that is studied. Numerous experiments are conducted and temperature distributions are plotted under conditions of internal and external heat sources operation. These experiments confirm the effectiveness of attracting non-local terms to describe the thermal processes. Expediency of applying nonlocal problems containing nonlocal conditions – thermal balance conditions – to such models is shown. This allows you to define heat and mass transfer as the parameters of the process control, in particular heat source and concentration of the substance.

1.
M. V.
Zagirnyak
,
V. P.
Lyashenko
,
T. A.
Grigorova
, and
D.
Miljavec
(
2011
)
Modeling the sintering of powder parts
,
Powder Metallurgy and Metal Ceramics
49
(
11
),
737
741
.
2.
V.
Lyashenko
, and
T.
Hryhorova
, “
Generalized mathematical model of thermal diffusion in powder metallurgy
”, in
Application of Mathematics in technical and Natural Sciences (AMiTaNS’14)
,
AIP CP1629
, edited by
M. D.
Todorov
,
American Institute of Physics
,
Melville, NY
,
2014
, pp.
85
93
.
3.
V.
Lyashenko
, and
E.
Kobilskaya
, “
Control of heat source in a heat conduction problem
”, in
Application of Mathematics in technical and Natural Sciences (AMiTaNS’14)
,
AIP CP1629
, edited by
M. D.
Todorov
,
American Institute of Physics
,
Melville, NY
,
2014
, pp.
94
101
.
4.
R.
Vengerov
,
Termophysics Mines. Mathematical Models
,
Donbass
,
Donetsk
,
2012
, pp.
200
315
.
5.
N. E.
Benouar
, and
N. I.
Yurchuk
(
1991
)
Mixed problem with an integral condition for parabolic equations with the Bessel operator
,
Diferentsial’nye Uravneniya
, pp.
2094
2098
.
6.
A.
Bouziani
(
1996
)
Mixed problem with boundary integral conditions for a certain parabolic equation
,
J. Appl. Math. Stochastic Anal.
9
,
323
330
.
7.
V. S.
Il’kiv
,
V. N.
Polishchuk
, and
B. I.
Ptashnik
, “A nonlocal boundary-value problem for a system of pseudodifferential equations,” in
Methods for Investigation of Differential and Integral Operators
,
Naukova Dumka
,
Kiev
,
1989
, pp.
75
79
.
8.
L. I.
Komarnyts’ka
and
B. I.
Ptashnyk
, “A problem with nonlocal conditions for a partial differential equation unsolved with respect to the leading time derivative,” in
Boundary-Value Problems with Various Degenerations and Singularities
,
Ruta
,
Chernivtsi
,
1990
, pp.
86
95
.
9.
B. I.
Ptashnyk
,
M. M.
Symotyuk
, and
N. M.
(
2011
)
A problem with nonlocal conditions for quasilinear hyperbolic equations
,
Nonlinear Boundary-Value Problems
11
,
161
167
.
10.
N. I.
Ionkin
(
1977
)
Solution of a boundary-value problem in heat conduction with a nonclassical boundary condition
,
Diff. Eqs.
13
(
2
),
294
304
.
11.
V.
Gulin
,
N. I.
Ionkin
, and
V. A.
Morozova
, “Difference schemes for nonlocal problems,” in
Russian Mathematics Izvestiya VUZ. Matematika
,
Editorial M. V. Lomonosov State University
,
Moscow
,
2005
, pp.
36
46
.
12.
A.
Bouziani
(
1999
)
Strong solution for a mixed problem with nonlocal condition for a certain pluriparabolic equations
,
Hiroshima Math. J.
27
,
373
390
.
13.
I.I.
Novikov
,
The Theory of Heat Treatment of Metals
,
Metallurgiya
,
Moscow
,
1978
.
14.
O. A.
,
Troitsky
,
V. I.
,
Stashenko
,
V. G.
Ryzhkov
,
V. P.
Lyashenko
, and
E. B.
Kobilskaya
(
2011
)
Electroplastic wire drawing and new technologydevelopment light wire
,
Problems of Atomic Science and Technology
4
,
111
117
.
15.
O. M.
Alifanov
,
Inverse Problems of Heat Transfer
,
Moscow
,
Machinery
,
1988
, pp.
107
121
.
16.
A.A.
Samarskiy
and
P.N.
Vabishchevich
,
Computational Heat Transfer
,