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5. Взаимодействие различных видов
транспорта: примеры и расчеты / Н. В. Правдин, В. Я. Негрей, В. А. Подкопаев. - М. : Транспорт, 1989. - 208 с.
6. Методика и указания по определению мощности и размещению предпортовых станций и районных парков порта / В. Я. Болотный. -М. : Транспорт, 1976. - 47 с.
7. Оптимизация планирования и управления транспортными системами / Е. М. Васильева, Р. В. Игудин, В. Н. Лившиц и др. -М. : Транспорт. 1987. - 208 с.
8. Управление эксплуатационной работой на железнодорожном транспорте : учебник для студентов вузов ж.-д. транспорта. В 2-х томах. Т. 1 / В. И. Ковалев, А. Т. Осьминин. - М. : ГОУ
«Учебно-методический центр по образованию на железнодорожном транспорте», 2009. - 263 с. -ISBN 978-5-89035-548-5.
9. Применение теории массового обслуживания для расчета устройств станций : курс лекций / Н. Н. Шабалин. - М. : МИИТ, 1968. - 88 с.
10. Рыбин П. К. Маневровое обслуживание морских портов и его влияние на путевое развитие портовых станций : дис. ... канд. техн. наук : 05.22.08 : защищена 17.04.2003 : утв. 11.07.2003 /Рыбин Петр Кириллович. - СПб., 2003. - 192 с. - Библиогр.: с. 184-192.
11. Теория вероятностей / Е. С. Вентцель. -М. : Высшая школа, 2006. - 576 с. - ISBN 5-06005688-0.
UDK 621.313.12 K. K. Kim
Petersburg State Transport University
GROUP PROPERTY OF THE EQUATION OF THE INDUCTION
FOR THE CONTINUOUS MEDIUM WITH NONLINEAR CONDUCTIVITY
In this paper based on the theory of the Lee groups we learned about the group property of the equation of the induction for the continuous medium with the nonlinear conductivity of type I *17
S = C | j (C is the function of nuclear temperature, j is the current density in the medium, у is the
constant) and found all the possible essentially various invariant solutions. The example of such a medium is plasma consisting of noble gases (argon, helium) with the admixture of alkaline metals (cesium, potassium). The exchange of energy between electrons and heavy particles is caused basically by the elastic collisions in such plasmas. Therefore, the considerable declination of the electron temperature from the nuclear temperature is possible at rather weak electric fields as well. In the plasma a thermodynamic balance is set up at the electron temperature which defines its conductivity.
рlasma, group property, nonlinear conductivity, invariant, subgroup, operator.
Introducthion
As it is shown in [1], the equation for the conductivity of argon-potassium plasma when there is no external magnetic field can be approximated by the dependence
s = C|jj7, where C and g are the constants,j is the current density in the medium. At
wete > 1 (where we is the cyclotron frequency
of dadra^ Te = , Ve = Vea + Vei , V ea ,
Vei are the average frequencies of collisions
between electrons and atoms and between electrons and ions respectevely) and if there
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is an external magnetic field there arises the ionizing instability and the effective conductivity of plasma drops in the direction of an operating current. It influences on the numerical values of С and g.
1 The initial equations
Let’s use the equations of induction in the following form
rotrotA = m0s
ЗА Л
----+ v, x rotA ,(1)
Зt 1 )
where A is the vector potential satisfying the conditions B = rotA and divA = 0. We will express the conductivity through the vector potential
s = C1-g
ЗА
-----+ v, x rotA
Зt 1
g
i-g
(2)
Let’s introduce the dimensionless magnitudes for convenience:
a * = A, 0' =A, v; = ^, tx = 12P0,
Sa
m0S0f0
An
(3)
0 0J0 Ai, rot' = —-rot.
2p 0 2p
v
0
e
0
Here A 0 is the length of a characteristic wave, v0 is the phase speed of this wave corresponding to the characteristic frequency f0. Besides that,
A
K0
2р/Л
-,(4)
i. e. the choice of the unit value A0 for the vector potential at the set conductivitys0 has
not been made arbitrarily. Taking into account (3) and (4) we receive:
In this paper we consider that these magnitudes are known and they have properties of the quasiconstants.
-rot 'rot'A' =
=e
ЗА' *
------v, x rotA
3t 1
!-1 ,
ЗА
- v' x rotA'
3t 1
,(5)
1
n =
1-g
from the equations of (1) and (2). Further, we consider the configurations having a cylindrical symmetry. Supposing that v1 = exv, = const andfrom the equations (5)
we shall receive:
З2 A _^i_3_ З'2 Зр p Зр
(PA) =
= e0
ЗА ЗА
----+ v, —
Зt З'
n-1
( ЗА ЗА )
v Зt З' )
(6)
here and everywhere A = Аф, index «'» is rejected.
In relation to the problems considered below we suppose that the boundary conditions look like А |р = фг- (' -1), or
|p = = y. (' -1) by which running waves
Зр ‘
of the type sin (' -1) can be understood in a
specific case. Therefore, we shall search for the solutions of the equation (6) in the class of functions in which the variables ' and t are only in the form 0 = ' -1. Then we can rewrite the equation (6)
З2 А З_ ,_З_ З02 Зр р Зр
(РА) = de0 ^n
(-ЗА У IЗ0 ^,
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Г
5 =
sign
V
ЭА
ЭО
s = vi “I, (7)
(sign s)” \
Using the general method of receiving the invariant-group solutions (Я-solutions) [2], we define the Я-solutions of the equation (7) by one-parameter subgroups.
2 The basic group
Let's replace the equation (7) by an equivalent system
du1 du2
dX1 +dXT ~
du3 Эх1
3 \
5e 0 sn
1
- u
1 d
(u3 X2)-
du
Эх1
0,
u2
= 0,
(8)
0,
■■A.
x2 dx2
x1 =0, x2 = p, u3
and calculate the infinitesimal operators that generate the basic group. Supposing that this group is one-parametrical, let’s write down the infinitesimal operators like this:
X = p d | d • Sx dx' Su duk ’
i = 1, 2; k = 1, 2, 3,
we write down (8) as y^pf + yk = 0, where p = 1, 2, 3 is the equation number, plk =
duk
=—-, uk are the decision variables (k = 1, 2,
dx'
3), x' are the independent variables (i = 1, 2), then the condition of invariance looks like this:
yk'+ Xyk +
dx
+
" yk' -yk' ^+Xyk'Л
V
due Te dxJ
dX x
P,
J
-yk'
du
pkpe=o.
e rj
Hence, we find the following conditions of invariance for the equations (8):
XP1 + XP2 “5e0S"n (ul )” 4u = 0,
X3 — x1 = 0
Sp1 Su
XP 2 — X
X2
u
(10)
22
(xz)
X
3 _L_
4 x2
= 0.
(9) here
here the coordinates X'x and Xk are
unknown.
The coordinates XI and Xt should satisfy the conditions of the invariance of the equations (8) concerning the operator X. If
XP1 = A (XI)—p1a(Xx)—Pi A(X2),
XP1 = A(X2)—p2A(XI)—p3A(X2),
Xp2 = A(X3)—pfA(Xx)—p2A(X2), <n>
XP 2 = A (XI)—p2A(Xx)—p2A(X2),
э э
i=1, 2, i = 1, 2, 3.
„ _ Э e Э
' dx' + Pi due,
According to (8), the substitution of the equations (11) into (10) with the subsequent excluding pk leads to the system of defining equations of the Lee algebra of the basic group of the system (8). These equations take the following form:
n
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эх! эх2 + 13X1 + 2 3X2 U эх2 •+-----~ +1 7 +1
Эх1 Эх2
Эи*
Эи3 х2 Эи3
-&0 s"(i1)”
эхх —3X1+&0 ,w 3X1 + (12)
Эх Эи Эи
+и
1ЭХХ
n „1
+- X,
Эи и
3X1 —ЭХ
Эх1 Эх
эхи, 3X2, 1 зxX
2 + 2&оsn(,/)n 3Xх
х
• +
Эи1 Эи2
+ и
Эи1
:3X 2
Эи3
’х - и+ (13)
Эи3
+3X3 = 0;
2 л 3 7
х Эи
ЭХХ , 3X1
Эх2
+ -
deo sn(u)
к n 3XX
Эи2 зx1x, и3 эx1x
Эи2
2 '_/''Эх | и
и —“ +
Эи3 х2 Эи3
3X2, 3X2
= 0;
n, К n ЭХ
Эх1 Эи1
de0 sn(u)
(14)
Эи1
и
1ЭX2
(15)
Эи3
зx1x, зx.
= 0;
Эи1 Эи2
! зxX + 2 3X2 эхи
и т + и
2
х = 0;
Эи1 Эи1 Эи1
и3 эх2
х2 Эи
(16)
1 = 0; (17)
1 зx1x, 2 3X2 3X
Эх | 2 ^Ъх
и ---г + и
Эи2
и3 ЭX2
-ч 2 "ч 2 2 -ч 2
Эи Эи х Эи
= 0; (18)
0
эхи 1 эxx 2 3X.
-и
и
2 ^ Ъх + и
ЭX2
+
+ и1
3 3X2 л
Эх1 Эх1 Эх1 х2 Эх1
эх: , 3x1 2 3X,+и
—г- и —г—и —г +—< Эи Эи Эи х Эи
— х1 = 0;
ju ’
(19)
Эхи 1 Эхх 2 эх;
Эх2
—и
и
2 ^ Ъх + _и_
ЭХ2
г
+
2
2
+
и
V х‘
и3 ЭХ2 л
Эх2 Эх2
и3 Y Эхи 1 ЭХХ
х
Эх2
+
и
Эи Эи
и
х2 Эи
X
и
,ЭХ2
Эи3 л
S2 1 £3
2 Хх 2 Х;
(х2 )2 х
2 ~ и
+ (20)
= 0.
Hence we find the coordinates XI and Хи of the operator X.
Let’s do the following to solve the equations (12)-(20). Having designated
3
j = X3—«XI—и 2X2+и X2 (21)
x
we rewrite the equations (17)-(20) as
2
2
3
X1 = j . X2 =— ЭФ.
"4 1? *^x "4 2 ’
Эи Эи
X1 =j+ /Ф+1 Ф
Хи х, 1 +и х, 3 + ь 2 ;
Эи V Эи х Эи у
Х2 = ЭФ + 2 ЭФ 1 ( 3 ЭФ + 1 ЭФ ф
X =—2+и—3 —2 и—3+и—1—Ф
Эх Эи х V Эи Эи
(22)
Hence it follows that the coordinates of the operator X depend on the function ф.
In order to define this function let’s give the equations (12)-(16) the following form:
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Э2ф + Э2ф
(Эх1)2 (Эх2)2
г 3 2
+ м + 2и и 2
V х у
Э2ф 1 Э2ф
(Э U3)
■ + 2u
Эх1Эи
+ 2
зЛ
2U
U
2
2з 2U
и
Э2ф
V х у
Эх2Эи3
2
V х у
X
X
U
1 Э2ф 1
2 1Л 3 2
х Эи Эи х
+ &0 sn(u)n
1 Эф/ Л2 Эф 1 Эф Эф и Эф U Эф Ф и ~ + (и ) у ~ ~~и—-—т +-т + ——т + -
12
Эх1Эи
2
23 " 2^ 1 ~\ 2 2 1 2 ~v 3
Эи Эи Эх Эи Эх х Эи х Эи х
21
2 2 1 2 -> 3 2
+
Э2ф , 1 Э2ф , Э2ф ,
+ и-:--г +--—- +
Г
3
-ч 1 ^ 1 м -ч 1 ^ 3 -ч 2^ 2
Эх Эи Эи Эи Эх Эи
Эф 1 - n Эф
+ (1- п)—3 +
Эи3 х2 Эи2
2и и 2 V х
= 0;
Э2ф n Эф
-ч 2-ч 3 1 -ч 1
Эи Эи и Эх
+
(23)
Э2ф Э2ф
дх1ди1 Эх2Эи2
2 и
3\
и
2
х у
Э2ф
+
+ и
1 Э 2ф , и1 Э2ф
1 3 2 1 2 Эи Эи х Эи Эи
2ф
= 0.
Эи2Эи3
+ (24)
+ 8бо Ли1)" Эф
(Эи1)
3Л
Э ф + Э ф + I 2 U + ^~^ +1 и —2 V х
Эх 2Эи Эх1Эи2
Э2ф
Эи Эи и
и1 Эф , 1 Э 2ф ,
- + и ----------Г +
х2 (Эи1)2
Эи2Эи3
(25)
+ б£оДи1)" Эф
(Эи1)2
Э2ф Э2ф
= 0;
(Эи1) (Эи2)
= 0.
(26)
Let's notice that the equations (14) and (15) turned out to be identical owing to (26).
Differentiating (25) by u1and (24) by U and taking their difference we shall receive
1 Э2ф + 2_ф+
х2 (Эи1)2
Эи 2Эи3
+ 5е0 snn(u)
Кп-1 Э 2ф
(27)
ЭиЭи2
0.
Differentiating (25) byu2 and (24) byu1 and the subsequent adding them give us
\2„ Л2,
1 Э 2ф.+2-ф
2 -ч 1^ 2 -ч 1^ 3
х Эи Эи Эи Эи
+5е0 s"n(ul)n 1—-^т = 0.
(28)
(Эи1)2
Proceeding with the equations (27) and (28) in the similar way we shall receive
Э2ф = 0; ЭиЭи2 (29)
Э 2ф 2 = 0. (Эи1) (30)
From (29) it results that
ф = У (х\ х \ и3 )&( и3) +
+ У 2 (х1, х2, и3 )%2 (и2 ) + У0 (х1, х2, и3)
and owing to (26) and (30) we receive
(31)
2
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С (и1)- иХ c 2 (и2)- и2. (32)
Taking (32) into account and substituting (31) into (25) we receive
d¥i = 2 - 0 + Эу - 0 (33)
-ч 3 л 3 ’-ч^-ч1 ’ ()
Эи Эи Эх Эх
3
i. e. у1 and у2 do not depend on и . Thus,
j = у1 (x1, x2, и1) + у2 (x1, x2, и2) +
+y0 (xX x2, и3).
Эу1 Эу
Эу0
(1 - П Игг + т7 + (1 - n № +
dx dx
Эи3
+(1 - n )у=0
(42)
x
(34)
Further using the equations (24) and taking (33) into account we receive
From (33), (34) and (38) it follows that
y1 = bx1 + d1, у2 - bx + d2, (43)
where b,d1,d2 are the arbitrary constants. Using (40) and (42) we can be convinced that d2= 0.
As to у 0, we receive
„ d. (2 - n) + (1 - n) .
У 0 = d, x 2 + -^------d-Л-------L ьи3, (44)
x
1 - n
Эу_Эу
Bx1 dx2
- 0.
(35)
here d3, d4 are the arbitrary constants.
Thus, according to (34), (43) and (44) we have
If now to take (33)-(35) into account, the equation (23) breaks up into the following parts
Э 2y 0
j- (bx1 + d 1) и1 + bx2u2 + d3 x2 +
. d4 (2 - n) + (1 - n)7 3
■—2------:------Ьи
(45)
x
1 - n
(Эи3)
ЭУ0
0xX
ЭУ 2
г- 0; (36) Now it is easy to find the coordinates of the operator X by using the equations (22) and (45). It is convenient to write down these
- 0; (37) coordinates in the form
xx -(1 - n) c0 x1+c1;
- 0; (38) -(1 - n) c0 x2;
2
Эx
Э2у 1 Эу
0
1 - 0; (39)
2
1^ 3 2 2
Эx Эи x Эx
Э2y0 ■ 1 Эу-2 --Ъ. - 0; (40)
Хи C0 и ;
- c0 и2+2c2;
X3 -( 2 - n ) C0 и 3 + c2 x 2 +
(46)
■
2 c3 2 x 2 + ^ x
dx Эи x dx (x 2)
Э 2у0. ■ Э 2у0 - 2 э 2у0 +
(dx1) (dx2) x dx Эи
+1 _у^+_уи у0 - 0.
(41)
x2 ^2 (x2)2 Эи3 ^x2)
2
Here сi are the arbitrary constants. We find operators X byequating subsequently the arch of с to a unity when the ones of сt are equal to zero and by introducing the received coordinates into (9).
Thus, for the equations (8) the basic group is generated by the following linearly-independent infinitesimal operators
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X =(1 - п) + (1 - п) х2-^2 +
Эх1 1 Э , 2 Э
Эх 2 Э
+ и—у + и2—2 + (2 — п) и3 3,
Эм1 Эи2 Эи3
Э
э
X. = —, X = 2— + x
э
-2 л у ^3
Эх
X=
Эи2 Эи
1 Э
3
4 2 л 3 '
x Эи
2
(47)
3 The optimum system of one-parameter subgroups
It is necessary to find an optimum system of subgroups of the first order to find all the possible and essentially various invariant solutions of the rank 1.
According to [2] the optimum system is defined as follows below.
The operators (Xa, Xp) named
commutators
( Xa, Xb) = —( Xb, Xa) =
_э_
Эх1
x j x xэха
4>a ^ i Sp
Эх1
Эх1
are made for the operators of the basic group
X.=xaA• xР=хрээт •
Эх Эх
where x1 are both the independent variables and the desired ones.
The operators Ea of the adjoint group are composed of
E‘=( Xa •Xb)aiP •
The transformation in n-dimensional space of the kind written below, i. e.
X' = xi +xa( x ) aa
(48)
corresponds to the operators recording them in the
Ea=X‘a(X)Э~~ • where X•
Ea when form of
Xi are the
coordinates of the points х and x, aa are the
coordinates of the vector a . On this basis the formulas of the transformation (48) by the operators of the adjoint group are concretized and then the formula for each of the operators of the adjoint group of a matrix of the internal automorphisms [2] is concretized too. We define the matrix A of the general automorphism as the product of the matrixes of the internal automorphisms. We consider the operator X = X e'Xi, where Xiare the
operators of the basic group and ei is the coordinate of the vectore.
The automorphism A with the operator X and linear transformation of the vector e with the matrix Ax (the transposed matrix) are equivalent. On this basis we at first search for the possible forms of transformation of the vector e and then according to them we determine the corresponding optimum system of subgroups.
Owing to this sequence it is possible to find an optimum system of the basic groups generated by the operators (47).
The basic group g4 is generated by the
operators (47). Let’s prepare a table of commutators of these operators
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Xl X2 X3 X4
Xl 0 -(l-n)X2 -X3 -(3-2n)X4
X2 (l-n)X2 0 0 0
X3 X3 0 0 0
X4 (3-2n)X4 0 0 0
The commutator (Xl, X2), for example,
is equal to the value in the point of crossing the Xrrow with the X2-column.
Using the data of this table let’s find the operators of the adjoint group of the Lee algebraZ4
E =-(l - ”) X^ -
-X3 —----(3 - 2n) X4
dX 3 E2 = (l - n) X.
dX 4 ’
d
dX /
(49)
d
E
3
= X
d ,
dX 1’
E4 =( 3 - 2n) X4
d
dX 1
By the similar way we find the matrixes of the internal automorphisms for other operators of the adjoint group as well
A =
A3 =
A4 =
l a2 0 0
0 l 0 0
0 0 l 0
0 0 0 l
l 0 a3 0
0 l 0 0
0 0 l 0
0 0 0 l
l 0 0 a4
0 l 0 0
0 0 l 0
0 0 0 l
Here al... a4 are the parameters on which the transformations of the adjoint group
Let's make the matrixes of the internal automorphisms. The formulas of the transformation (48), for example, for the operator El, look like
x. = x1, x2 = [l-(l-n)a1 ]x2, x'3 =(l-a1)x3, x4 = [l-(3-2n)a1 ]x4.
Hence, the matrix corresponding to the operator El will be
l 0 0 0
0 l-(l - n) al 0 0
0 0 l - al 0
0 0 0 l -(3 - 2n) al
depend. Multiplying these matrixes we receive the matrix of the general automorphism
l a2 a3
0 l-(l - n) a 0
0 0 l - al
0 0 0
a
4
0
0
l-(3 - 2n) a
Let's consider the operator of one-parameter group
X = e1X l + e2X 2 + e3X3 + e%, (50)
where el...eA are the coordinates of the vector e. Transforming this vector by means of the transposed matrix Ax we receive the vector el with the coordinates
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e1, a2e1 + [l-(1 -n)a1]в2,
a3 e +(l - ц) e, (51)
a4 e1 + [l-(3 - 2n) a1 ] e4.
We can see from (51) that the coordinate e1 remains invariable. Therefore, it is possible to suppose that e1 = 0 or e1 = 1.
Let us assume that e1 = 0. Considering that any two magnitudes from e2, e3,e4 are equal to zero we receive the vectors:
0, 1, 0, 0
0, 0, 1, 0
0, 0, 0, 1
with the accuracy within a constant multiplier. If one of these magnitudes is equal to zero we have
0, 1, a, 0
0, 1, 0, a
0, 0,1, a .
Let e1^1. From (51) we choose a2 =- [1 -(1 - n) a1 ] e2, a3 =-(1 - a1) e3, a4 =-[1 -(3 - 2n) a1 ] e4
and receive a vector
1, 0, 0, 0 .
4 Invariant-group solutions
Let's find the invariant-group solutions on the one-parameter subgroups of the optimum system (52). These solutions will be essentially various relatively to g. It means that it is impossible to get any of these solutions from another transformation of the group g.
Besides, for the reasons which become obvious from the following, let’s consider an unessentially different solution of the rank 1 in the subgroup generated by the operator
X2 + a 2 X3 + a1 X4, (53)
a1 is the arbitrary constant.
Thus, using the linear transformation the vector e is deduced to one of the following forms:
1, 0, 0, 0 ;
0, 1, a, 0 ;
0, 0, 1, 0 ;
0, 0, 0, 1 .
0, 1, 0, 0 ; 0, 1, 0, a ; 0, 0, 1, a ];
Hence, we have the possibility of transformation (50) by the appropriate automorphism A to one of the operators
Xi,
X2 X2 + aX3 X2 + aX4,
X3 X3 + aX4, (52)
X4
These are the operators of the optimum system of one-parameter subgroups. Further on, the constant a is written as a2 to
distinguish it from a
The method of the determination of the invariant-group solutions is the following. Let’s assume that a subgroup corresponds to any operator from (52), (53). In this subgroup we find a complete set of the functionally
independent invariants Jv, i. e. such a set of invariants for which the rank t of the Jacoby
matrix
J3
dx‘
is equal to t = к + m - R.
where Xare the independent and desired variables, к and m is the number of independent and desired variables, R = 1 is the rank of the matrix of the operator (9). For the existence of the invariant solutions it is necessary that the completeness degree 5 of
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the complete set of the functionally independent invariants should be equal to the number of the desired variables m. s is the
rank of the matrix
dJ'
du
where u1 are the
desired variables. if s = mthe invariant solutions are found as follows below. Let’s set-up the equation for a variety
Фc (J) = 0; c = 1,..., r. (54)
Let's admit that the first r invariants depend on the desired variables (they can simultaneously depend on independent variables too) and the others t - r are the functions only of x, i. e.
J° = Jс (x, u); c = 1,..., r;
Jrtl = Jrtl(x); j = 1,...,t-r.
Then, the equations (54) are set-up this
way
Фс (J ) = Jc -wc (Jr+1,..., Jf ) = 0; c = 1,..., r.
Substituting the values of Jc here we find the connection between wc and uc , i. e.
Фc(J) = 0; c = 1,...,r. (55)
Let's determine the values of
Pi
du
dxi
from the equations
^ ( ^c\ dФc , i dJ
Di (Фc )= — + P
dx * 1 du
i = 1,...,k; c = 1,...,r.
0;
It is clear that p. will depend on
wc, Jr+1,..., /.
c
Substituting the values of uc from (55) and pi into the system (8) we will receive
the equations in which wc will be the desired variables and Jr+1,..., Jt will be the
independent invariants. Having solved these equations relatively to wc and then using
(55), we receive the invariant solutions uc(Jr+1,..., Jt) in the given subgroup.
Let's consider the operators (52) and (53). It’s necessary to cast out the operators X3X4, and X3+ a2 X4 at once as there are no invariant solutions in the corresponding subgroups. For all the other operators we can select the necessary number of the
of the matrix
is equal to the
functionally independent invariants Jv for which the completeness degree, i. e. the rank
dJ у /du1
number of the desired variables. Hence, the invariant solutions potentially exist in the corresponding subgroups. Further for reducing records we shall denote a subgroup generated by the operator Х as H[X].
1. The subgroup H [X1]. We have the following invariants
1 1
J1 = (x2)n-1 u1, J2 = (x2)n-1 u2,
2-n
1
J3 = (x2)n1 u3, J4 = —j = 1.
x
For these invariants the 4-th order determinant corresponds to the Jacoby matrix
dJ / where Xare the independent and
dx1
rank of the matrix
is equal to 3, i.
desired variables. Hence the specified intervals are functionally independent. The
dJ у
/du1
e. s = = m. Therefore the invariant solutions potentially exist.
The Н-solution satisfies the equation л 7\ d2w n - 3 . dw 1 1 ’dl2 n -1 dl
2n - 3 (1 - n )2
w-5e
0
n
s
f dw Y
v d 1 у
0,(56)
n-2
n-1
u3 = (x2) w.
2. The subgroup H [X2]. The set of the invariants is the following
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т1 _ 1 т2 _ 2 т3 ___ 3 т4 ____ 2
J — u , J — и , J — и , J — x
andthe Я-solution looks like
U3 — gi X2 + g2
u
de0 ( 2 )
( X 2 )
2-n
(1 - n)(3 - n)
+
X
-1
(59)
(57)
+ a 2x + —2 + g2 x2.
x x
Here and further g1 и g1 are the arbitrary constants. 5. The subgroup H [ X2 +a2 X3 +a1X4 ]
3. The subgroup H [X2 +a 2X3 ]. The We have
invariants of this subgroup are
2
t1 1 T2 _ x 2, 3
J — U , J —---------2 U + и ,
2
т3 1 и 4 2
J — x ----------, J — x ,
2 a 2
r1 — 1 t2 — x 2 1 U , 3
J — U , J —-----2U ---------~+ U ,
2a2 x2
2
2a 2
and the H-solution looks like
T3 1 U 4 2
J — x ----------, J — x
2a
M
U
— de0 (sa 2)'
( x 2 )
n+2
U
x
f (x 2)
+
2 | a1
x +-----
Л
2
a 2 x1 +
■ +
(1 + n)(3 + n) (58)
+ a 2 x1x2 + g 2 x2.
4. The subgroup H [ X2 + a2 X4 ]. We have
т 1 _ 1 T2 _ 2
J — U , J — U ,
1 3
3 _ x U t4 _ 2 ’
J —~2-------, J — x
+ \ +g 2
X
/
a 2 x J
\
x a
2
f (x)—1
-(x2 )2 j
2 + a1 x +-------
v a 2 x
2
( x2 )2 +
2
2 x J
n
(60)
a
n-2
(x2)
(x2 )2 +
a
(x2)n
-dx2 -
-dx2,
a1
a — —L, M a
de0 ( sa 2 ) 2
5 The invariant solutions of the equation of induction and the boundary problems of an electromagnetic field
n
n
Let's consider the problem of the possibility of using the received results for solving boundary problems of an
electromagnetic field.
As (56)-(60) are valid for any interval Ax1, and the parameters a1 and a2 are arbitrary (but constant within the Ax1) the estimation of using any of these solutions is to find out the possibility to coordinate it with a boundary problem when there is an
appropriate approximation of the boundary conditions.
Further we shall consider a piecewise-linear approximation. From this point of view the invariant solutions (58)-(60) are of interest.
We can formulate the boundary conditions for the solution (58) as follows:
Ul
P1 — 0
— 0 3
Ul f-,2
P 2
— b2 x1.
From here we receive
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3
и
&Q (Ф2У
(1 + n )(3 + n)
(УГ
р2
-р2 х2
+
Р2 1 2 —X X •
Р2
+
(61)
So, the solution (61) corresponds to the case of movement of liquid through a cylindrical channel of the radius p2.
For the invariant solution (59) the additional conditions should look like
и3 ~P1x1, и3 = P2x1
p1 1 Р 2 2
(the radiation conditions).
From here we find
3
и
M
(pi2 -р2 )
2t 2 X
{Pi2 [ f ( X 2 )-f (Р2 )_
-p2 [f (x2)-f (p1)]+
+ (x2) [f (p2 )-f (p1 )]} +
u 3 - b1 X1, p1 1
U|
P 2 -
Hence, at n >2 it follows that
v 2-n
u
8в0 (sP )n
2
(1 - n )(3 - n)
+ b1p1
+1
— 0.
¥
that a1 —
3 "
n Р. +
Pi- 2 a2
X
a^
2
X У
1
X ,
(62)
X
X
P-р2 lplp2,
— p1b1 - p2 P2 2 2 ’
Pi -P2
Thus, we have the case of movement of liquid in space the internal border of which is specified by the cylinder of the radius p1.
Hence, the solution (63) corresponds to the case of movement of liquid on the channel formed by the coaxial cylinders of the radii p1 and p 2.
2
2
The boundary conditions for the invariant solution (60) should be written down in the form of
Conclusion
The research of the group properties of equations of an electromagnetic field for an isotropic medium with the nonlinear
conductivity of a type s — c| j|1, having a
cylindrical symmetry, shows the existence of invariant-group solutions on one-parameter subgroups.
The operators, corresponding to the optimum system of subgroups for the
electromagnetic field equations, are expansion operators, i. e. the invariant solutions, corresponding to these subgroups, are self-similar.
The invariant solutions admit their coordination with the boundary conditions presented like the form of the boundary values of a vector potential if for them we use the piecewise-linear approximation.
Reference
1. Theconductivity of gases at the elevated electron temperature / J. Kerrebrok // Engineering questions of magnetic hydrodynamics. - М., 1965. - 326 p.
2. Групповые свойства дифференциальных уравнений / Л. В. Овсянников. -
Новосибирск :Изд-во Сиб. отд. АН СССР, 1962. - 239 c.
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