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The first boundary value problem for a one-dimensional nonlinear heat equation is considered, where the heat conductivity coefficient and the power function of heat sources have a power-law dependence on temperature. For a numerical analysis of this problem, it is proposed to use the method of two-s...
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System research and information technologies| _version_ | 1866302762601414656 |
|---|---|
| author | Gybkina, Nadiia Sidorov, Maxim Vasylyshyn, Kostiantyn |
| author_facet | Gybkina, Nadiia Sidorov, Maxim Vasylyshyn, Kostiantyn |
| author_sort | Gybkina, Nadiia |
| baseUrl_str | http://journal.iasa.kpi.ua/oai |
| collection | OJS |
| datestamp_date | 2022-06-20T14:19:48Z |
| description | The first boundary value problem for a one-dimensional nonlinear heat equation is considered, where the heat conductivity coefficient and the power function of heat sources have a power-law dependence on temperature. For a numerical analysis of this problem, it is proposed to use the method of two-sided approximations based on the method of Green’s functions. After replacing the unknown function, the boundary value problem is reduced to the Hammerstein integral equation, which is considered as a nonlinear operator equation in a semi-ordered Banach space. The conditions for the existence of a single positive solution of the problem and the conditions for two-sided convergence of successive approximations to it are obtained. The developed method is programmatically implemented and researched in solving test problems. The results of the computational experiment are illustrated by graphical and tabular information. The conducted experiments confirmed the efficiency and effectiveness of the developed method that allowed recommending its practical use for solving problems of system analysis and mathematical modeling of nonlinear processes. |
| doi_str_mv | 10.20535/SRIT.2308-8893.2021.4.09 |
| first_indexed | 2025-07-17T10:27:30Z |
| format | Article |
| fulltext |
N. Gybkina, M. Sidorov, K. Vasylyshyn, 2021
Системні дослідження та інформаційні технології, 2021, № 4 115
UDC 517.927.4 : 519.624.2
DOI: 10.20535/SRIT.2308-8893.2021.4.09
APPLICATION OF TWO-SIDED APPROXIMATIONS METHOD
TO SOLUTION OF FIRST BOUNDARY VALUE PROBLEM
FOR ONE-DIMENSIONAL NONLINEAR HEAT CONDUCTIVITY
EQUATION
N. GYBKINA, M. SIDOROV, K. VASYLYSHYN
Abstract. The first boundary value problem for a one-dimensional nonlinear heat
equation is considered, where the heat conductivity coefficient and the power func-
tion of heat sources have a power-law dependence on temperature. For a numerical
analysis of this problem, it is proposed to use the method of two-sided approxima-
tions based on the method of Green’s functions. After replacing the unknown func-
tion, the boundary value problem is reduced to the Hammerstein integral equation,
which is considered as a nonlinear operator equation in a semi-ordered Banach
space. The conditions for the existence of a single positive solution of the problem
and the conditions for two-sided convergence of successive approximations to it are
obtained. The developed method is programmatically implemented and researched
in solving test problems. The results of the computational experiment are illustrated
by graphical and tabular information. The conducted experiments confirmed the ef-
ficiency and effectiveness of the developed method that allowed recommending its
practical use for solving problems of system analysis and mathematical modeling of
nonlinear processes.
Keywords: nonlinear thermal conductivity, positive solution, Green’s function, two-
sided iterative method, equation with isotonic operator.
INTRODUCTION
System studies of various objects and processes require the use of the method of
mathematical modeling and the apparatus of computational mathematics. In par-
ticular, the problem of mathematical modeling of nonlinear stationary heat con-
duction processes leads to the need of developing the effective numerical methods
for solving initial, boundary value and initial boundary value problems for qua-
silinear differential equations with a coefficient nonlinearly dependent on temperature
[1–4]. Today there are many methods of numerical analysis of these problems.
Among them are methods of similarity theory, methods of finite differences, finite
elements, boundary integral equations [1, 5–8] or successive approximations with
two-sided convergence [9, 10]. The methods of the last group allow to build two
sequences of functions, which, respectively, from the bottom and top approach
the desired solution of the problem. Due to this fact, in the implementation of
these methods, we have a convenient a posteriori estimation of the approxima-
tions error, and hence a convenient criterion for the end of iterations. This makes
the methods of two-sided approximations more attractive than other methods used
to solve boundary value problems for stationary equations.
The purpose of the work is to develop on the base of the Green’s function
method the method of two-sided approximations for solving the first boundary
N. Gybkina, M. Sidorov, K. Vasylyshyn
ISSN 1681–6048 System Research & Information Technologies, 2021, № 4 116
value problem for a nonlinear one-dimensional equation of thermal conductivity
and to study its work in solving test problems.
The construction of two-sided approximations methods for solving boundary
value problems for partial differential equations is based on the use of the theory
of nonlinear operators in semi-ordered spaces. The theory of linear semi-ordered
spaces was built by L.V. Kantorovich in the second half of the 30’s XX cen-
tury [11]. Further development of the methods of this theory is associated with the
work of M.A. Krasnoselsky [12], H. Amann [13], V.I. Opoytsev [14], N.S. Kur-
pel, B.A. Shuvar [15], A.I. Kolosov [16]. In [12] the question of the existence of
positive solutions of equations with monotone operators was investigated, and in
[14] the solvability of equations with operators that have a generalized property of
monotonicity (so-called heterotonic or mixed monotone operators) was investi-
gated. These works laid the theoretical foundations for the development of two-
sided iterative schemes, but the iterations themselves were considered by the au-
thors as an aid to prove the theorems of the existence of fixed points of operators
and did not lead to computational results.
Works [9, 10] are devoted to the development of two-sided iterative schemes
for solving boundary value problems for partial differential equations as means of
applied mathematics with bringing them to computational implementation. In this
case, only the first boundary value problem for a semilinear elliptic equation with
the Laplace operator and power or exponential nonlinearity was mainly considered.
This work continues the research started in [9, 10] and aims to generalize
them and extend them to ordinary nonlinear differential equations.
So, we will consider the problem of finding a solution to a nonlinear bound-
ary value problem, which is a mathematical model of thermal conductivity in a
rod of length l , when the thermal conductivity coefficient has a power-law de-
pendence on temperature and when there are heat sources in the rod distributed
according to the power law:
)()( Tf
dx
dT
Tk
dx
d
, lx 0 , (1)
0)( xT , lx 0 , (2)
0)()0( lTT , (3)
where TkTk 0)( is heat conductivity coefficient, 00 k is the value of the
thermal conductivity in a linear medium, TTf )( is heat source power func-
tion, 0 , 0 are parameters of nonlinearity of the medium, 0 is con-
stant, which characterizes the power of heat sources.
To analyze the boundary value problem (1) – (3), we apply the methods of
the theory of nonlinear operators in semi-ordered spaces [12, 14].
MATERIALS AND METHODS
Let us present some definitions and facts from the theory of nonlinear operators in
semi-ordered spaces, which will be used below [12, 14].
Let be a real Banach space, and be a zero element of the space . A
cone is a closed convex set for which the following conditions hold:
a) if u and u , then u for any ;0
b) if u and u , then u .
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An arbitrary cone allows one to introduce semi-ordering in the space
according to the rule: wv , if vw . Elements u (i.e., u ) are
called positive. The set of elements wv, of a semi-ordered space, which con-
sists of those u for which wuv , is called a conical segment.
Normal cones are an important class of cones for applications of the theory
of semi-ordered spaces in computational mathematics. A cone is called normal
if there exists a number 0)( N such that wNv )( follows from wv .
In this case, one says that the norm is semi-monotonous. If 1)( N , then the
cone is called acute, and it is said that the norm is monotonic.
Let us present the definition of some classes of operators in spaces with a cone.
An operator :T is said to be positive if it leaves an invariant cone
, i.e., )(uT for any u .
Let some non-zero element 0u be fixed. Denote by )( 0uK the set of
those elements u for which we can specify such 0)( u , 0)( u
that 00 uuu .
An important subclass of positive operators is the so-called 0u -positive op-
erators. A positive operator T , which translates the nonzero elements of the cone
into )( 0uK , is called an 0u -positive operator. Therefore, for an 0u -positive
operator T for any u , u there are such 0)( u , 0)( u that
00 )( uuTu .
The operator :T , which acts in , is called monotonic (isotonic), if
from vu , , vu follows )()( vTuT .
An operator :T is called concave if it is 0u -positive for any
)( 0uKu and )1,0(
)()( uTuT , (4)
and in (4) equality is impossible.
A concave operator T is called 0u -concave if it is 0u -positive and for any
)( 0uKu and )1,0( there is a 0),( u such that
)()1()( uTuT .
Consider the equation
)(uTu (5)
with a positive nonlinear operator :T , and the Banach space is semi-
ordered by a cone . We are interested in the conditions under which this equa-
tion will have at least one nonzero solution u in the cone . This solution
u , u is called the positive solution of equation (5). In the case when
)(T , we are talking about another, different from , solution in the cone .
Let us first consider the question of the existence of a fixed point in an iso-
tonic operator.
A conical segment 00 , wv is said to be invariant for the isotonic operator
T if the inequalities hold
00 )( vvT , 00 )( wwT . (6)
N. Gybkina, M. Sidorov, K. Vasylyshyn
ISSN 1681–6048 System Research & Information Technologies, 2021, № 4 118
Therefore, the invariance of the conical segment 00 , wv means that
0000 ,),( wvwvT .
Let the isotonic operator T have an invariant conic segment 00 , wv .
Consider the iterative process
)( )()1( kk uTu , ...,2,1,0 k . (7)
The following statement takes place [12, 14].
Theorem 1 (about the existence of a fixed point in an isotonic operator). Let
the isotonic operator T have an invariant conic segment 00 , wv , the cone
is normal, and the operator T is completely continuous. Then successive ap-
proximations (7) coincide to v at 0
)0( vu and to w at 0
)0( wu . Points v
and w are fixed points of the operator T , and if
00 , wvu is any other
fixed point of T , then the inequalities wuv hold. If it is known in ad-
vance that the fixed point
00 , wvu is unique, then the iterative process (7)
coincides with u from any initial point 00
)0( , wvu .
So, consider an iterative scheme
)( )()1( kk vTv , )( )()1( kk wTw , ...,2,1,0 k , (8)
0
)0( vv , 0
)0( ww .
As we can see, scheme (8) is a set of two independent iterative processes,
and therefore, its computational implementation can be performed using parallel
calculations.
The condition that guarantees the existence of a single fixed point u of the
isotonic operator T is its 0u -concavity. Then for the 0u -concave operator T the
iterative process (8) will coincide bilaterally to this fixed point, i.e.
0
)0()1()()()1()0(
0 ............ wwwwuvvvv kk .
Thus, the application of the method of two-sided approximations to find a
solution to equation )(uTu with an isotonic operator T is as follows:
1) construct a cone segment 00 , wv invariant for the isotonic operator T
using conditions (6);
2) using the 0u -concavity condition, draw a conclusion about the existence
of a unique on 00 , wv solution u of equation (5) and two-sided convergence
to it of successive approximations (8);
3) set the accuracy 0 and, using the iterative scheme (8), construct the
sequences }{ )(kv , }{ )(kw ;
4) if 2)()( kk vw , then write the approximate solution of equation (5)
2
)()(
)(
kk
k wv
u
, in this case )(kuu , i.e. )(kuu with accuracy .
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TWO-SIDED APPROXIMATIONS METHOD FOR SOLVING
ONE-DIMENSIONAL BOUNDARY VALUE PROBLEMS FOR NONLINEAR
EQUATION OF THERMAL CONDUCTIVITY BASED ON THE USE
OF GREEN FUNCTION
Making the substitution
1
1
0
1
u
k
T in problem (1) – (3), where )(xu is a
new unknown function, we obtain for the function u the boundary value problem
)(uFu , ),0( lx , (9)
0)( xu , ),0( lx , (10)
0)0( u , 0)( lu , (11)
where
1)( uuF ,
1
0
1
k
.
The function )(uF is continuous and positive for 0u .
For problem (9) – (11) the Green’s function ),( sxG has the form
.,
)(
,0,
)(
),(
lxs
l
xls
sx
l
slx
sxG (12)
As one can see, 0),( sxG for all lsx ,0 . Then the problem (9) – (11)
is equivalent to Hammerstein’s integral equation
l
dssuFsxGxu
0
))((),()( . (13)
Consider the equation (13) in the Banach space ],0[ lC of functions continu-
ous on the segment ],0[ l . The norm in ],0[ lC is introduced by a rule
)(max
],0[
xuu
lx
. Select in ],0[ lC the cone ]},0[,0)(:],0[{ lxxulCu
of nonnegative functions. Cone in ],0[ lC is normal (and even acute) [12]. In
the space ],0[ lC , using the cone , one can introduce semi-ordering using the
rule:
for ],0[, lCvu vu , if uv ,
i.e.
vu , if )()( xvxu for all ],0[ lx .
If there is a classical solution to problem (9) – (11), that is, a function
],0[),0(2 lClCu satisfying the conditions (10), (11) and equation (9), then
we can conclude that this function also satisfies the integral equation (13). If the
classical solution is absent, then the definition of the generalized solution of prob-
lem (9) – (11) can be based on the integral equation (13). Namely, we can intro-
N. Gybkina, M. Sidorov, K. Vasylyshyn
ISSN 1681–6048 System Research & Information Technologies, 2021, № 4 120
duce the definition of the solution (generalized) of the boundary value problem
(9)–(11) as a function
u , which is the solution to the integral equation (13).
Connect the equation (13) with the nonlinear integral operator T acting in
],0[ lC by the rule
l
dssuFsxGxuT
0
))((),())(( . (14)
Let us investigate some properties of the operator T of the form (14).
Firstly, the operator T of the form (14) is positive, i.e., it leaves cone an
invariant: )(T . Indeed, the Green’s function is continuous and nonnega-
tive in the square lsx ,0 . Then, taking into account condition (10) for any
],0[ lCu , the integrand in (14) is continuous and nonnegative at lsx ,0 ,
and therefore, the function ))(( xuT is continuous and nonnegative on the interval
],0[ l , that is, from u it follows that )(uT .
Secondly, the operator T of the form (14) is an 0u -positive operator with
)(
2
1
),()(
0
0 xlxdssxGxu
l
, (15)
because for the Green’s function of the form (12) there is an estimate
)()(),()()( 00 xussxGxus , lsx ,0 ,
where },min{
2
)(
2
sls
l
s ,
l
s
2
)( .
Then, if u , u , then there is an inequality
)())((),()( 0
0
0 xudssuFsxGxu
l
, (16)
where 0))(()(
0
l
dssuFs ,
l
dssuFs
0
))(()( .
So, if u , u , then )()( 0uKuT , that is, )(uT , and there are
such 0)( u , 0)( u , that 00 )( uuTu .
Since 0 , 0 , the function )(uF grows monotonically by u , from
which it follows that the operator )(uT of the form (14) will be isotonic.
Obviously, the operator T is completely continuous.
Find the conditions under which the isotonic operator T of the form (14)
will be 0u -concave with the function )(0 xu of the form (15). As it is known [12],
the definition of 0u -concavity will be fulfilled under the condition: for any posi-
tive number u for any )1,0(
)()( uFuF . (17)
For the function
1)( uuF the condition (17) takes the form
11)( uu ,
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wherefrom
1
1
1
,
that is,
1 . (18)
Therefore, if condition (18) is satisfied, the operator T of the form (14) will
be 0u -concave.
Thus, the following statement is held.
Lemma 1. The operator T of the form (14), where ),( sxG is the Green’s
function (12) of the problem (9) – (11), which acts in the space ],0[ lC , semi-
ordered by the cone of nonnegative functions, has the following properties:
a) it is a positive operator;
b) it is an 0u -positive operator, where the function )(0 xu is defined by
equality (15);
c) it is an isotonic operator;
d) if inequality (18) holds, it is an 0u -concave operator, where function
)(0 xu has the form (15).
In cone we distinguish the invariant conical segment 00 , wv by
conditions (6), which for the operator T , determined by equation (14), take the form
)())((),( 0
0
0 xvdssvFsxG
l
for all ],0[ lx , (19)
)())((),( 0
0
0 xwdsswFsxG
l
for all ],0[ lx . (20)
Let us form an iterative process according to the scheme:
l
kk dssvFsxGxv
0
)()1( ))((),()( , ...,2,1,0 k , (21)
l
kk dsswFsxGxw
0
)()1( ))((),()( , ...,2,1,0 k , (22)
)()( 0
)0( xvxv , )()( 0
)0( xwxw . (23)
Taking into account the invariance of the cone segment 00 , wv and the
isotonicity of the operator T , we can conclude that the sequence )}({ )( xv k does
not decrease along the cone , and the sequence )}({ )( xw k does not increase
along the cone . In addition, from the normality of the cone and the
complete continuity of the operator T follows the existence of the boundaries
)(xv and )(xw of these sequences. Thus, the chain of inequalities is true
0
)0()1()()()1()0(
0 ............ wwwwwvvvvv kk .
There are two possible cases: wv and wv . If the operator T is 0u -
concave, then only the second case is possible, and then wvu : is the only
N. Gybkina, M. Sidorov, K. Vasylyshyn
ISSN 1681–6048 System Research & Information Technologies, 2021, № 4 122
fixed point of the operator T on the conical segment 00 , wv , and hence u is
the unique on 00 , wv positive solution of the boundary value problem (9)–(11).
Thus, the following theorem holds.
Theorem 2. Let 00 , wv be an invariant conic segment for an isotonic
operator T of the form (14) and condition (18) holds. Then the iterative process
(21) – (23) converges according to the norm of the space ],0[ lC to the unique on
00 , wv continuous positive solution u of the boundary value problem (9) –
(11), and a chain of inequalities takes place
0
)0()1()()()1()0(
0 ............ wwwwuvvvv kk . (24)
For the approximate solution of the boundary value problem (9) – (11) on
the k th iteration we take the function
2
)()(
)(
)()(
)( xvxw
xu
kk
k
. (25)
The advantage of the constructed two-sided iterative process is that on each
k th iteration we have a convenient a posteriori error estimate for the approximate
solution (25):
))()((max
2
1 )()(
],0[
)( xvxwuu kk
lx
k
.
Therefore, if the accuracy 0 is given, then the iterative process should be
carried out before the inequality
2))()((max )()(
],0[
xvxw kk
lx
is fulfilled and
with the accuracy we can assume that )()( )( xuxu k .
Then the function
1
1
)(
0
)( )(
1
)( xu
k
xT kk
can be considered an approximate solution to the original problem.
Since there is an inequality (16), which means that )()( 0uKuT for any
u , u , then the ends of the invariant conical segment 00 , wv can be
found in the form )()( 00 xuxv , )()( 00 xuxw , where 0 , and the
function )(0 xu is determined by equality (14). Then inequalities (19), (20) take
the form
)())((),( 0
0
0 xudssuFsxG
l
for all ],0[ lx , (26)
)())((),( 0
0
0 xudssuFsxG
l
for all ],0[ lx . (27)
Inequalities (25), (26) can be reduced to the form
m
1
1
, M
1
1
, (28)
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where
1
+
2
3
1
+12 1
3
1
1
12
lm , (29)
1
1,
1
2,
2
1
2 1
3
1
12
lM . (30)
Here
0
1)( dtetz tz is Euler’s gamma function, ),,( baz
z
ba dttt
0
11 )1( is Euler’s incomplete beta function.
Since the value of )(max)())()((max 0
],0[
00
],0[
xuxvxw
lxlx
should be as
small as possible for faster convergence of iterations, in the practical implementa-
tion of the iterative process (21)–(23) one should take the largest and the
smallest , satisfying inequalities (27), i.e.
1
1
)(
m ,
1
1
)(
M . (31)
EXPERIMENT, RESULTS AND DISCUSSIONS
The computational experiment in problem (1)–(3) was carried out for the values
of 1l , 10 k , 1 and 5,2 , which corresponds to the case of an experi-
ment with a source of energy from thermonuclear reactions in the study of the
thermal conductivity of hydrogen plasma [2]. The condition for the convergence
of the proposed method of two-sided approximations for finding a positive solu-
tion is the condition 5,31 . Making the substitution
7/2
2
7
uT , where
)(xu is a new unknown function, for the function u we obtain a boundary value
problem of the form (9) – (11).
Consider the case where 6,1 (condition 1 , obviously, is satisfied).
By formulas (29), (30) we find m and M : 30871,0m , 35218,0M , and by
formulas (31) we obtain that 32951,0 , 42000,0 . So, provided that )(0 xu
has the form (15) at 1l , the invariant cone segment will have the form
00 , uu . The iterative process (21) – (23) converged with accuracy 510
to the solution of the problem for the function )(xu in 9 iterations.
Fig. 1 shows the graphs of the upper )()( xw k (dotted line) and lower )()( xv k
approximations (dashed line), 9...,,1,0 k , Fig. 2 shows a graph of the
approximate solution )()9( xu , and Fig. 3 shows a graph of the approximate
solution )()9( xT . Tables 1 and 2 show the values of the approximate solutions
)()9( xu and )()9( xT on the grid with step 1,0 , respectively.
N. Gybkina, M. Sidorov, K. Vasylyshyn
ISSN 1681–6048 System Research & Information Technologies, 2021, № 4 124
Fig. 1 clearly demonstrates the two-sided character of the convergence of the
constructed iterative sequences )}({ )( xv k and )}({ )( xw k in accordance with the
chain of inequalities (24): at each k th iteration, the unknown exact solution
)(xu of problem (9)–(11) is above the approximation )()( xv k and below the ap-
proximation )()( xw k
T a b l e 1 . The value of the approximate solution )()9( xu
x 0,0 0,1 0,2 0,3 0,4 0,5
)()9( xu 0,000000 0,017262 0,031830 0,042761 0,049515 0,051798
x 0,6 0,7 0,8 0,9 1,0
)()9( xu 0,049515 0,042761 0,031830 0,017262 0,000000
Table 2. The value of the approximate solution )()9( xT
x 0,0 0,1 0,2 0,3 0,4 0,5
(9) ( )T x 0,000000 0,448497 0,534180 0,581191 0,606059 0,613914
x 0,6 0,7 0,8 0,9 1,0
(9) ( )T x 0,606059 0,581191 0,534180 0,448497 0,000000
The dependence of the norm of solutions )(xu and )(xT at 5,2
depending on the value of the parameter was also investigated numerically.
Iterations were carried out until four significant digits were clarified in the
norm of the approximate solution )(ku . The corresponding graphs of the
Fig. 1. Graphs of )()( xv k and )()( xw k , 9,8,7,6,5,4,3,2,1,0k
02 04 06 08
2
3
4
0,2 0,4 0,6 0,8
0,04
0,03
0,02
0,01
x
w(k)(x), v(k)(x),
0,2 0,4 0,6 0,8
0,04
0,03
0,02
0,01
x
u (x)
Fig. 2. Graph of )()9( xu
0,2 0,4 0,6 0,8
0,05
0,04
0,03
0,02
0,01
x
T (x)
Fig. 3. Graph of )()9( xT
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dependence of the norms on the parameter are shown in Fig. 4 and 5. As we
can see, with increasing parameter , the norms of solutions )(xu and )(xT go
to zero.
The rate of convergence of the iterative process was estimated by the value
k
k
1 , where ))()((max )()(
]1,0[
xvxw kk
x
k
, ...,2,1k . It is established that the
process coincides with the speed of geometric progression. Estimation of the de-
nominator q of this progression depending on is given in Table. 3. As one can
see, with increasing , the value of q approaches unity, which indicates decel-
eration in the convergence of iterations.
T a b l e 3 . The value of the estimate of the rate of convergence q for different
values of
0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0
q 0,029 0,058 0,086 0,116 0,145 0,172 0,200 0,229 0,258 0,286
1,1 1,2 1,3 1,4 1,5 1,6 1,7 1,8 1,9 2,0
q 0,315 0,343 0,372 0,401 0,430 0,458 0,487 0,515 0,544 0,573
2,1 2,2 2,3 2,4 2,5 2,6 2,7 2,8 2,9 3,0
q 0,602 0,631 0,659 0,688 0,715 0,744 0,773 0,802 0,829 0,858
3,1 3,2 3,3 3,4
q 0,887 0,916 0,944 0,999
CONCLUSIONS
The problem of construction of two-sided approximations to the positive solution
of the first boundary value problem for a nonlinear one-dimensional equation of
thermal conductivity is solved in the work.
The scientific novelty of the results obtained lies in the fact that the method
of two-sided approximations for solving nonlinear operator equations with an iso-
tonic operator was further developed in terms of its application to boundary value
problems for the nonlinear one-dimensional heat equation. The developed method
has a few advantages, such as a convenient a posteriori estimation of the error of
the approximate solution and a simple computational algorithm. This distin-
guishes it from other numerical methods for solving boundary value problems for
nonlinear ordinary differential equations of the second order and makes it attrac-
tive for application in engineering practice.
0,5 1,0 1,5 2,0 2,5 3,0
0,10
0,08
0,06
0,04
0,02
γ
||u(x)||
Fig. 4. Graph of the dependence of the
norm of solution ( )u x on the parameter
Fig. 5. Graph of the dependence of the
norm of solution ( )T x on the parameter
||T(x)||
0,5 1,0 1,5 2,0 2,5 3,0
0,06
0,04
0,02
γ
N. Gybkina, M. Sidorov, K. Vasylyshyn
ISSN 1681–6048 System Research & Information Technologies, 2021, № 4 126
The practical significance of the results obtained lies in the fact that the pro-
posed method has shown itself well in solving test problems, allows fast software
implementation, which will allow carrying out highly invariant computational
experiments when solving practical problems of mathematical modeling of non-
linear processes.
The limited use of the method can be associated with the conditions imposed on
the behavior of nonlinearities included in the equations of the boundary value problem.
Prospects for further research are the extension of the method of two-sided
approximations developed in this work to boundary value problems for ordinary
differential equations with other types of nonlinearities, in particular, exponential
ones, as well as to initial boundary value problems for quasilinear parabolic equa-
tions, using semi-discrete methods (for example, the Rothe line method).
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Received 27.09.2021
INFORMATION ON THE ARTICLE
Nadiia V. Gybkina, ORCID: 0000-0002-2564-6903, Kharkiv National University of Ra-
dio Electronics, Ukraine, e-mail: nadiia.gybkina@nure.ua
Maxim V. Sidorov, ORCID: 0000-0001-8022-866X, Kharkiv National University of
Radio Electronics, Ukraine, e-mail: maxim.sidorov@nure.ua
Kostiantyn V. Vasylyshyn, ORCID: 0000-0002-6123-8683, Kharkiv National University
of Radio Electronics, Ukraine, e-mail: kostiantyn.vasylyshyn@nure.ua
ЗАСТОСУВАННЯ МЕТОДУ ДВОБІЧНИХ НАБЛИЖЕНЬ ДО
РОЗВ’ЯЗАННЯ ПЕРШОЇ КРАЙОВОЇ ЗАДАЧІ ДЛЯ ОДНОВИМІРНОГО
НЕЛІНІЙНОГО РІВНЯННЯ ТЕПЛОПРОВІДНОСТІ / Н.В. Гибкіна, М.В. Сидо-
ров, К.В. Василишин
Анотація. Розглянуто першу крайову задачу для одновимірного нелінійного
рівняння теплопровідності, де коефіцієнт теплопровідності та функція потуж-
ності теплових джерел степенево залежать від температури. Для числового
аналізу цієї задачі запропоновано використати метод двобічних наближень на
основі методу функцій Гріна. Після заміни невідомої функції крайова задача
зведена до інтегрального рівняння Гаммерштейна, яке розглянуто як нелінійне
операторне рівняння у напівупорядкованому банаховому просторі. Отримано
умови існування єдиного додатного розв’язку задачі та умови двобічної збіж-
ності до нього послідовних наближень. Розроблений метод програмно реалізо-
вано та досліджено під час розв’язання тестових задач. Результати обчислюва-
льного експерименту проілюстровано графічною та табличною інформаціями.
Проведені експерименти підтвердили працездатність та ефективність розроб-
леного методу і дозволяють рекомендувати його для використання на практиці
для розв’язання задач системного аналізу та математичного моделювання не-
лінійних процесів.
Ключові слова: нелінійна теплопровідність, додатний розв’язок, функція Грі-
на, двобічний ітераційний метод, рівняння з ізотонним оператором.
ПРИМЕНЕНИЕ МЕТОДА ДВУСТОРОННИХ ПРИБЛИЖЕНИЙ К РЕШЕНИЮ
ПЕРВОЙ КРАЕВОЙ ЗАДАЧИ ДЛЯ ОДНОМЕРНОГО НЕЛИНЕЙНОГО УРАВ-
НЕНИЯ ТЕПЛОПРОВОДНОСТИ / Н.В. Гибкина, М.В. Сидоров, К.В. Василишин
Аннотация. Рассмотрена первая краевая задача для одномерного нелинейного
уравнения теплопроводности, в которой коэффициент теплопроводности и
функция мощности тепловых источников являются степенными функциями
температуры. Для численного анализа этой задачи предложено использовать
метод двусторонних приближений на основе метода функций Грина. После
замены искомой функции краевая задача сведена к интегральному уравнению
Гаммерштейна, рассматриваемому как нелинейное операторное уравнение в
полуупорядоченном банаховом пространстве. Получены условия существова-
ния единственного положительного решения задачи условия двусторонней
сходимости к нему последовательных приближений. Разработанный метод
программно реализован и исследован при решении тестовых задач. Результаты
вычислительного эксперимента проиллюстрированы графической и табличной
информациями. Проведенные эксперименты подтвердили работоспособность
и эффективность разработанного метода, что позволяет рекомендовать его для
использования на практике при решении задач системного анализа и матема-
тического моделирования нелинейных процессов.
Ключевые слова: нелинейная теплопроводность, положительное решение, функция
Грина, двусторонний итерационный метод, уравнение с изотонным оператором.
|
| id | journaliasakpiua-article-240131 |
| institution | System research and information technologies |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2025-07-17T10:27:30Z |
| publishDate | 2021 |
| publisher | The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" |
| record_format | ojs |
| resource_txt_mv | journaliasakpiua/3a/04bc20dbc59067d6838273249f1b9a3a.pdf |
| spelling | journaliasakpiua-article-2401312022-06-20T14:19:48Z Application of two-sided approximations method to solution of first boundary value problem for one-dimensional nonlinear heat conductivity equation Применение метода двусторонних приближений к решению первой краевой задачи для одномерного нелинейного уравнения теплопроводности Застосування методу двобічних наближень до розв’язання першої крайової задачі для одновимірного нелінійного рівняння теплопровідності Gybkina, Nadiia Sidorov, Maxim Vasylyshyn, Kostiantyn додатний розв’язок нелінійна теплопровідність функція Гріна двобічний ітераційний метод рівняння з ізотонним оператором nonlinear thermal conductivity positive solution Green’s function two-sided iterative method equation with isotonic operator нелинейная теплопроводность положительное решение функция Грина двусторонний итерационный метод уравнение с изотонным оператором The first boundary value problem for a one-dimensional nonlinear heat equation is considered, where the heat conductivity coefficient and the power function of heat sources have a power-law dependence on temperature. For a numerical analysis of this problem, it is proposed to use the method of two-sided approximations based on the method of Green’s functions. After replacing the unknown function, the boundary value problem is reduced to the Hammerstein integral equation, which is considered as a nonlinear operator equation in a semi-ordered Banach space. The conditions for the existence of a single positive solution of the problem and the conditions for two-sided convergence of successive approximations to it are obtained. The developed method is programmatically implemented and researched in solving test problems. The results of the computational experiment are illustrated by graphical and tabular information. The conducted experiments confirmed the efficiency and effectiveness of the developed method that allowed recommending its practical use for solving problems of system analysis and mathematical modeling of nonlinear processes. Рассмотрена первая краевая задача для одномерного нелинейного уравнения теплопроводности, в которой коэффициент теплопроводности и функция мощности тепловых источников являются степенными функциями температуры. Для численного анализа этой задачи предложено использовать метод двусторонних приближений на основе метода функций Грина. После замены искомой функции краевая задача сведена к интегральному уравнению Гаммерштейна, рассматриваемому как нелинейное операторное уравнение в полуупорядоченном банаховом пространстве. Получены условия существования единственного положительного решения задачи условия двусторонней сходимости к нему последовательных приближений. Разработанный метод программно реализован и исследован при решении тестовых задач. Результаты вычислительного эксперимента проиллюстрированы графической и табличной информациями. Проведенные эксперименты подтвердили работоспособность и эффективность разработанного метода, что позволяет рекомендовать его для использования на практике при решении задач системного анализа и математического моделирования нелинейных процессов. Розглянуто першу крайову задачу для одновимірного нелінійного рівняння теплопровідності, де коефіцієнт теплопровідності та функція потужності теплових джерел степенево залежать від температури. Для числового аналізу цієї задачі запропоновано використати метод двобічних наближень на основі методу функцій Гріна. Після заміни невідомої функції крайова задача зведена до інтегрального рівняння Гаммерштейна, яке розглянуто як нелінійне операторне рівняння у напівупорядкованому банаховому просторі. Отримано умови існування єдиного додатного розв’язку задачі та умови двобічної збіжності до нього послідовних наближень. Розроблений метод програмно реалізовано та досліджено під час розв’язання тестових задач. Результати обчислювального експерименту проілюстровано графічною та табличною інформаціями. Проведені експерименти підтвердили працездатність та ефективність розробленого методу і дозволяють рекомендувати його для використання на практиці для розв’язання задач системного аналізу та математичного моделювання нелінійних процесів. The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" 2021-12-22 Article Article application/pdf https://journal.iasa.kpi.ua/article/view/240131 10.20535/SRIT.2308-8893.2021.4.09 System research and information technologies; No. 4 (2021); 115-127 Системные исследования и информационные технологии; № 4 (2021); 115-127 Системні дослідження та інформаційні технології; № 4 (2021); 115-127 2308-8893 1681-6048 en https://journal.iasa.kpi.ua/article/view/240131/249618 |
| spellingShingle | додатний розв’язок нелінійна теплопровідність функція Гріна двобічний ітераційний метод рівняння з ізотонним оператором Gybkina, Nadiia Sidorov, Maxim Vasylyshyn, Kostiantyn Застосування методу двобічних наближень до розв’язання першої крайової задачі для одновимірного нелінійного рівняння теплопровідності |
| title | Застосування методу двобічних наближень до розв’язання першої крайової задачі для одновимірного нелінійного рівняння теплопровідності |
| title_alt | Application of two-sided approximations method to solution of first boundary value problem for one-dimensional nonlinear heat conductivity equation Применение метода двусторонних приближений к решению первой краевой задачи для одномерного нелинейного уравнения теплопроводности |
| title_full | Застосування методу двобічних наближень до розв’язання першої крайової задачі для одновимірного нелінійного рівняння теплопровідності |
| title_fullStr | Застосування методу двобічних наближень до розв’язання першої крайової задачі для одновимірного нелінійного рівняння теплопровідності |
| title_full_unstemmed | Застосування методу двобічних наближень до розв’язання першої крайової задачі для одновимірного нелінійного рівняння теплопровідності |
| title_short | Застосування методу двобічних наближень до розв’язання першої крайової задачі для одновимірного нелінійного рівняння теплопровідності |
| title_sort | застосування методу двобічних наближень до розв’язання першої крайової задачі для одновимірного нелінійного рівняння теплопровідності |
| topic | додатний розв’язок нелінійна теплопровідність функція Гріна двобічний ітераційний метод рівняння з ізотонним оператором |
| topic_facet | додатний розв’язок нелінійна теплопровідність функція Гріна двобічний ітераційний метод рівняння з ізотонним оператором nonlinear thermal conductivity positive solution Green’s function two-sided iterative method equation with isotonic operator нелинейная теплопроводность положительное решение функция Грина двусторонний итерационный метод уравнение с изотонным оператором |
| url | https://journal.iasa.kpi.ua/article/view/240131 |
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