Upload
-
View
177
Download
4
Embed Size (px)
Citation preview
-
( )
. . - , , , . :
..-.., .. (. ), ..-.., .. (. ), ..-.., .., ..., .., ..-.., .., ..-.., .., ..-.., ..
:
..-.., .., ..-.., .., ..-.., .., ..-.., .., ..., .., ..-.., .. ( .. -
, . ), ..-.., .. ( , . , ), ..-.., Ph. D., , .. ( , . , ), Ph. D., (Kim Kisik, INHA-, . , ), Ph. D., (Kim Jaewan, KIAS,
. , ), Ph. D., - .. ( , . , )
Copyright & A K-C
South Ural State University
The main purpose of the series Mathematics. Mehanics. Physics is to promote the results of re-search in mathematics, mechanics and physics, as well as their applications in natural, technical and economic sciences.
Editorial Board A.A. Mirzoev, South Ural State University, Chelyabinsk, Russian Federation E.V. Golubev, South Ural State University, Chelyabinsk, Russian Federation V.I. Zalyapin, South Ural State University, Chelyabinsk, Russian Federation A.O. Chernyavskii, South Ural State University, Chelyabinsk, Russian Federation N.D. Kundikova, South Ural State University, Chelyabinsk, Russian Federation Yu.M. Kovalev, South Ural State University, Chelyabinsk, Russian Federation A.V. Keller, South Ural State University, Chelyabinsk, Russian Federation Editorial Counsil L.D. Menikhes, South Ural State University, Chelyabinsk, Russian Federation V.V. Karachik, South Ural State University, Chelyabinsk, Russian Federation D.A. Mirzaev, South Ural State University, Chelyabinsk, Russian Federation V.P. Beskachko, South Ural State University, Chelyabinsk, Russian Federation S.B. Sapozhnikov, South Ural State University, Chelyabinsk, Russian Federation V.I. Zhukovsky, Moscow State University, Moscow, Russian Federation S.I. Pinchuk, Indiana University, Bloomington, United States of America V.A. Strauss, University of Simon Bolivar, Caracas, Venezuela Kishik Kim, INHA-University, Incheon, Korea Jaewan Kim, Korea Institute for Advanced Study KIAS, Seoul, Korea E.S. Puzyrev, Vanderbilt University, Nashville, USA
Copyright & A K-C
3
.. .......... 5 .., .. ...................................................................................
14
.., .. , - , ........
25
.. 3- ............................................. 31
.., .. (111) ................................................................................................
44
.. ........................................................................................................................................
50
.., .., .. -: ab initio ..............................................
56
.. .......................................... 64
.. ...... 70 .. - ..........................................................................................
73
, 2015
Copyright & A K-C
CONTENTS
Mathematics
ALIYEV R.A. Inverse Problem of Determination of Coefficient in the Elliptic Equation ................. 5 DMITRIEV A.V., ERSHOV A.A. The Analysis of Energy Absorption in a Blanket for Contact Electric Resistance...............................................................................................................................
14
ERSHOVA A.A., SIDIKOVA A.I. Uncertainty Estimation of the Method Based on Generalized Residual Principle for the Restore Task of the Spectral Density of Crystals.......................................
25
KARACHIK V.V. Cauchy and Goursat Problems for Differential Equation of Third Order............. 31
Physics
VIATKIN G.P., MOROZOV S.I. Ab Initio Modeling of Segregation of Iron Atoms on the (111) Nickel Surface......................................................................................................................................
44
PROKUDINA L.A. Effects of External Influences on the Development of Disturbances in the Oregonator with Diffusion...................................................................................................................
50
RIDNYI Ya.M., MIRZOEV A.A., MIRZAEV D.A. Carbon Impurities in Paramagnetic FCC Iron: Ab Initio Simulation of Energy Parameters .........................................................................................
56
SOKOLOVA N.M. Energy Intensive Mechanical Dispersion............................................................ 64
Short communications
GULYASHCHIKH I.A. Neumann Problem for Polyharmonic Equation in the Unit Ball ................. 70 KHAYRISLAMOV K.Z. A Steady-State Flow of a Power-Law Fluid between Rotating Coaxial Cylinders..............................................................................................................................................
73
Copyright & A K-C
2015, 7, 2 5
517.946
.. 1
- . . - .
: ; .
[19]. [2, 10] [1112] - . .
2{ ( ), ( , ), ( )}q u u x y y
1 2( , ) ( ) ( , ),u u h x y q u h x y + = + ( , ) ,x y D (1)
1 1 2(0, ) ( ), ( , ) ( ),x xu y y u l y y = = 20 ,y l (2)
1 2 2( ,0) ( ), ( , ) ( ),yu x x u x l x = = 10 ,x l (3)
10 0( , ) ( ),0 ,u d y y d l= < < 20 ,y l (4)
0 ( ),y yu g x= = 10 ,x l (5)
1 1 1 1 2 2 1 2 2 1 2 2(0) (0), ( ) (0), (0) ( ), ( ) ( ),x x x y x yl l l l = = = =
1 2 1(0) (0), (0) ( ),y x y xg g l = = 1 0 2 0 2( ) (0), ( ) ( )yd d l = = . 1{( , ) 0 ,D x y x l= < <
20 }, ( , ),iy l h x y< < ( ),i x 11,2, ( ), ( ), ( )i y y g x = , 3 4 3 3
1 1 2 1 1 2( , ) ( ), 1,2, ( ) [0, ], ( ) [0, ], ( ) [0, ],a a
ih x y C D i x C l x C l y C l + + + + = [ ]4 2( ) 0, ,y C l +
31( ) [0, ],
ag x C l+ 11 2 2(0), ( ),R R l = = 0 < 1 < . . 2{ ( ), ( , ), ( )}q u u x y y (1)(5),
2( ), ( )q u y M 1 2[ , ]R R N 2[0, ],l 10 < 2 3
,2 1 2 1 2 2 2 2 2( ) ( ) 0, ( ) [ , ], ( ) 0, ( ) 0, ( ) [0, ],yq u q u q u C R R y y y C l < > ( , )u x y 4( )C D (1)(5).
0 1 1 1 21 2(0), ( ), ( ).l l = = = 1 2. < 1 2[ , ]z - ( )zs x ( ),zy s x= ( , ( )) ,zu x s x z= ( , ( )) .zx s x D : {( , ) ,zD x y D= ( , ) }.u x y z< 1 2[ , ]z - 0( , ( )) ( )z zu x s d f x= . , 0( ) .zf d z= ( ) 0y > ( )z
( )y , 0( ) ( )zz s d = .
1. 2( , ) ( )ija x y C D ,2 2
2
11 1
, 1, 0ij i j i
ii ja
== > 2( , ) ( )u x y C D
( )C D D
11 12 22 1 2 3( , ) 2 ( , ) ( , ) ( , ) ( , ) ( , )xx xy x yyya x y u a x y u a x y u b x y u b x y u b x y u+ + = + + ,
1 - , , , - , . , . E-mail: [email protected]
Copyright & A K-C
. . . 6
( , ), 1,2,3ib x y i = D . D
11 12 222xx xy yy x ya u a u a u M u u u + + + + , 0,M > ( , ) ,x y D
1 1( , ) 0, ( , ) 0,xu l y u l y= = [ ]20,y l , ( , ) 0u x y D [11, c. 99].
1. (1)(5)
11( , ) 0, ( , ) 0, ( , ) 0, 1,2, ( ) ,i x i y xh x y h x y h x y i x > > > =
2 1 1( ) , ( ) , ( ) , ( ) 0.yx y g x y
2( , ), ( , )x yu x y u x y > 0, (6)
}11 2min{min ( ),min ( ),min ( ),min ( )x y
xx x
g x x x y = ,
1
1 1
1 22
( , ) ( , )min{ ,min ,
1 ( , )Dx xh x y h x y
h x y
+=
1
1 1
1 2( , ) ( , )min }.1 ( , )Dy yh x y h x y
h x y
+
. (1) (3) x ,y , (6) . .
(6) ( )zs x x , , 2( , )u x l z -, 0 1[ , ].x d l
. 1 2 < ( )q z 0 1( , ).z -
2
42( , , ( )) ( )u q y C D M 0 2[ , ] N 2 0[0, ( )]s d (1)
(5), (6) , 4( , ) ( ).u x y C D . 2. 1 2[ , ], ( )c q z 0[ , ]z c . ( , )u x y -
cD .
. (6) 0 02( ) (0, ), ( ( ))c cs d l s d c = 0( )cs d , .
1 10 0(0, ) (0, ( )),G d s d = 0(0, )cG d= 1 0 0( ( ), ( ))cs d s d
1c > 0(0, )G d= 0(0, ( ))cs d . , 1 cG G G= .
11( ( )),( , ) ,
( , )( ( )), ( , ) .c
q x x y GL x y
q y x y G
=
0 0(0, ) (0, ( ))cG d s d= { ( ), ( , )}f x u x y
1 2( , ) ( , ) ( , )u u h x y L x y h x y + = + , ( , ) ,x y G (7)
1 0(0, ) ( ), ( , )) ( ),xu y y u d y y = = 0(0, ( )),cy s d (8) 1 0( ,0) ( ), ( , ( )) ( ),c cu x x u x s d f x= = 0(0, ),x d (9)
0 ( )y yu g x= = , 0(0, ),x d (10)
1 1 1 0 1 0(0) (0), ( ) (0), (0) ( ( )),x cx cd f s d = = = 0 0( ) ( ( )),ccf d s d=
0 1(0) ( ), (0) (0).y y xg d g = = , : 1 1{ ( , ), ( )}cu x y f x 2 2{ ( , ), ( )}.cu x y f x
2 1 2 1( , ) ( , ) ( , ), ( ) ( ) ( ).c c cu x y u x y u x y f x f x f x= = 0u u + = , (11)
(0, ) 0,xu y = 0( , ) 0,u d y = (12)
Copyright & A K-C
..
2015, 7, 2 7
( ,0) 0,u x = 0( , ( )) ( )c cu x s d f x= (13)
0 0y yu = = . (14)
(11)(13) , ,
1/ 2
1
2 1 2 122 20 0
( , ) {sh[( ) 1] }cosn
n nn d d
u x y A y x
=
+ += + .
0( )cy d s d=
2 1/ 22 1
2 0sh[( ) 1]
nn n
d
Ad
+
=+
,
02 2 1
20 00( )cos
c
dn
n d df d += .
1/ 2
1/ 21
2 1 22 2 10
22 1 02 0
sh[( ) 1]( , ) cos
2sh[( ) 1]n
n
d nn dn
d
yu x y x
d
=
+
++
+=
+ .
, (14),
2 1 2 12 1/ 22 12 22 1/ 20 01 2 0
[( ) 1] cos 0sh[( ) 1]
n nnnd d
n d
xd
+ +
+=
+ =+
.
0n = n , 2 1
2 0{cos }n
dx+ -
, ( ) 0cf x = . . . 0 1 2[ , ] [0, ]d l l ( ( , ), ) .u z y y z =
(1)(4)
2
1 23 3 2
1 1( ) 2 ( ) ( , ) ( , ),y y zy yy
zz z z
q z h y h yzz
+ + = + 1 0 2( ) ( ),0 ( ),d z l y z < < < < (15)
0( , ( ))z z d = . (16) (6) (15) . -
{ }0( ) (0, ( ))c cD d D y c= . 1 0{( , ) : ( ) ,cH z y d z c= < < 0 y< < ( )}z 0( ) \cD d G . 0{ ,x d= 0 y< < ( )}z { ( ),y z= 1 0( ) }.d z c < < -, 1( , )z y 2( , )z y (15)(16). ( , )z y = = 2 1( , ) ( , ).z y z y 0 0( , ), ( , )xu d y u d y (0, ( ))y z ,
( , ( ))z z , ( , ( )).z z z , ( , ( ))z z z 0
1
( , ( ))xu d z= ,
( , ( )) ( , ( )) 0.zz z z z = = , ( )q z cD . ( , )z y cD . , ( , )z y
1 2 3 4 5 62 0,zz zy yy z yk k k k k k + + + + + = 1 0( ) ,0 ( ),d z c y z < < < < (17) ( , ( )) ( , ( )) 0zz z z z = = , (18)
21 2 2 2 33 2
2 2 2
1 1 1( , ) (1 ), ( , ) , ( , ) ,y y
z z z
k z y k z y k z y
= + = = 2 234 1 1 3 31 1
1( , ) [ (1 )(yy y
z z
kk z y k k
= + + +
Copyright & A K-C
. . . 8
121
1) zz
z
+ + 1 1 11
12 ( ) ]y zy
z
k
+ , 5( , )k z y =23 3 1 2 1[ ( )y y zzk k + 12 ],zy 6( , )k z y = 1 2( ) .q z h h
(17)(18)
1 2 32 [ ]zz zy yy z yk k k M + + + + , ( , ) cz y H , (19) ( , ( )) ( , ( )) 0zz z z z = = . (20)
1 , ( , ) 0,( , ) cz y z y H . - ( , )u x y .cD . 2
3. ( , )u x y 1
D . ,
( , )u x y 0{( , ) : ,x y x d < 1(0, ( ))}y s x > 0.
0 > .
1 1{( , ) ( , ), ( ) }G z y z y z = < , 1 1{( , ) ( , ,), ( ) }S z y z y z = + < . 2 , ( , )u x y .G -
, (1)(5). ( , ), ( ), 1,2.i iz y q z i = 2 1 2 1( , ) ( , ) ( , ), ( ) ( ) ( ).z y z y z y q z q z q z = = (14)(15) 3 { ( , ), ( )}z y q z ,
1 2 3 4 5 6 1 12 ( , ) ( ),( , )zz zy yy z yk k k k k k h y q z z y S + + + + + = , (21) ( , ( )) 0z z = , (22)
1 1( , ) ( , ) 0,( , )zy y z y S = = . (23) (21) (23) , ,z z y y = = .
. ( )z 1 1 1( ) ( ), ( ) , 0.z z z K K = + >
{( , ) (0, ), ( ) }.S z y z y z = <
1 2 3 4 5 6 1 12 ( , ) ( ),( , )zz zy yy z yk k k k k k h y q z z y S + + + + + = , (24) ( , ( )) 0z z = , (25)
(0, ) (0, ) 0zy y = = . (26) ( )q z (24)
2 3 1 2 3( ) ( , )( 2 ) ,zz zy yy z yq z Q z y k k a a a = + + + + + (27) (0, ) (0, ) 0zy y = = , (28)
322 3
1 1 1
1( , ) , ( , ) , ( , )
kkQ z y k z y k z y
h k k= = = , 1 4 2 5
1 1
1 1( , ) , ( , ) ,a z y k a z y k
h h= = 23 1
1
( , ) .q
a z y hh
=
(27) y . .
( , ) [ (ln )]P z y Qy y = +
.
2 3 1 2 32 ( , ) ( , ) ( , ) ( , ) ( , ) ( , )zz yz yy z yP k P k P l z y P z y l z y P z y l z y P z y+ + + = + + +
4 5( , ) ( , ) ( , ) ( , )zl z y z y l z y z y + + , ( , )z y S , (29) (0, ) (0, ) 0zP y P y= = , (30)
( , ) /yz y Q Q = , 544 51 1
( , ) , ( , ) ,kk
k z y k z yk k
= = 1 2 4( , ) 2 yl z y k k= ,
Copyright & A K-C
..
2015, 7, 2 9
2 3 5( , ) 2 ,yl z y k k= 3( , )l x y =2
2 3 22 2y
z y
ak k l
Q + ,
14 1 2( , ) 2 ( 2 ) ,
yz
al x y l k
Q = 5( , )l x y = 2 32zz zyk k + +
31 2 .
yz y
al l
Q
2 32 [ ], 0zz yz yy z y yP k P k P M P P P M + + + + + + > , (31)
(0, ) (0, ) 0zP y P y= = . (32) M C , , . 4.
1 2( )
( , ) ( , ) ( , ) ( , ) ,y
z
z y z y z P z d
= 1( , ) ( ), 1,2,i z y C S i = (33)
( ) ( ) ( )
( , ) [ ( , ) ], 0y y y
z z yz z z
z y M P d P z d P d P M
+ + + > . (34)
.
( , )yyQ
P z yQ
+ = , (35)
( , ( )) 0z z = . (36) (35)(36), (33). (33) z ,
( )
( , ( )) ( , ) ( , )y
z
PP z z P z y z d
y =
.
{1 2 1( ) ( )
( , ) ( , ) ( , ) ( , ) ( , ) [ ( , ) ( , ) ] ( )y y
z z yz z
z y z y z P z d z y P z y P z d z
= + + +
2( )
( , ) ( , )y
zz
z P z d
+ + 2( )
( , ) ( , ) }y
zz
z P z d
.
, (34). .
, const 0, = > 21(0, ), (0, )2
. :
2 1( , ) ( ( )) , ( , ) exp(2 ),4
z y z y z z y = + + =
1{( , ) ( , ) , 0}
4H z y z y z = < + > ,H S
1{( , ) ( , ) }.
4A z y H z y = = + (37)
H H . ,
{ ( , ) : 0}.H A U z y H z = = (38) 5. ( , ) ( )H z y C H
2 1 1 2 21 1[ ( , ) ] ( ) (4 ) { exp[2 ( ) ] }4 4( )
yH z d dzdy H dzdy H dzdy
H H Hz
+ + + + . (39)
. : 1/ 2{ ( )}, { ( )}, ( ) ( ) ( )H H y z H H y z d z z z + = > = < = + .
, (38)(39),
2
( )
[ ( , ) ]y
z
H z d dzdyH
+
0
dz [ ]
( ) ( )2
( )
( , ) ( )d z d z
z
H z d y z dy
1 11
( ) (4 )4
+ +
Copyright & A K-C
. . . 10
( ) ( )
2
0 ( )
[ ( , ) ( ) ]d z d z
z
dz H z d dyy
= 1 1 2 21 1( ) (4 ) { exp[2 ( ) ] }
4 4H H
H dzdy H dzdy
+ +
+ + + + .
H , .
6. , , - 2 3,k k , C , , [11, c. 93].
2 3 4 2 2 22 3( 2 )zz yz yyP P C P k P k P P
+ + + + 2 2
2 3( 2 )zz yz yyC P k P k P ++ + + + div ,U ( ), ,z y H 0 0, , (40)
3 3 2 2 2 2 2[ ]z yU C P P P
+ + . (41) ( [11, c. 9399]). (40), (33)
(34),
( )22 3 4 2 2 2 z yP P MC P P P P P + + + +
( )2
2 2
( )
{y
z
C M M P dz
+
+ + +
2 2
( )(
})
y y
yzz
P d P d
+ +
( )222 2 div .CM P P U ++ + +
2 23 4 2 2 2 2 1/ 2 3/ 2 2( ) ( )P P M M C P P + + + +
( )2
2 2
( )
{y
zz
C M M P d
+
+ + +
2 2
( ) ( )
div}y y
yz z
P d P d U
+ + ,
C . 22 1/ 21 ( )M M C P + +
3 4 2 2 2 5 / 2 21 ( )M M C P + 2 2 2
2
( ) ( ) ( )
( ) div ,{ }y y y
z yz z z
M M C P d P d P d U
+ + + +
.C
H (37)(39) (41),
2 1/ 21 ( )M M C + 2 3 4 2 5/ 21 ( )
H
P dzdy M M C
+ +
2 2 2
H
P dzdy
( )22 1 2 2 21( ) ( )4 {H
M M C P P dzdy
+ + + + +
221exp 2 ( ) ( )4
}H
P P dzdy
+ + + + 3 4 2 21 1exp 2 ( ) ( )
4 4C + +
22( )A
P P ds
+ .
22 1/ 2 2 1 111 ( ) ( ) ( )4
H
M M C M M C P dzdy
+ + + + +
3 4 2 5 / 2 1[1 ( ) (4
M M C + + + 1 2 4 1 2 2 21) ( ) ( ) ]4
H
M M C P dzdy
+ + + +
Copyright & A K-C
..
2015, 7, 2 11
1exp 2 ( )
4C +
22( ) }H
P P dzdy
+ +3 41 1exp 2 ( ) (
4 4C + +
22 2 2) ( )A
P P ds
+ .
, 2 1/ 2 2 1 11 1( ) ( ) ( )
4 2M M C M M C ++ + + + ,
2 5 / 2 2 4 11 1( ) ( ) ( )4 2
M M C M M C ++ + + + .
, , 2 3 4 2 2 2 1exp 2 ( )
4H H
P dzdy P dzdy C
+ +
22( ) expH
P P dzdy C
+ +23 4 2 2 21 12 ( ) ( ) ( )
4 4A
P P ds
+ + + .
, 3 4 2 2 2 3 4 2 21 1( ) exp 2 ( )
4 4H
P dzdy C
+ + .
1 (0, ) . (37) ,
111
( , ) ,( , )4
z y z y H < + .
1
3 4 2 2 2 3 4 2 21 1( ) exp 2 ( )4 4
H
P dzdy C
+ + .
1
3 4 2 2 2 3 4 2 21 1
1 1 1 1( ) exp 2 ( ) ( ) exp 2 ( )4 4 4 4
H
P dzdy C
+ + + + .
3 4 2 21 1
1 1( ) exp 2 ( )4 4
+ + .
1
21
1 1exp2 ( ) ( )
4 4H
P dzdy C
+ + .
, ,
1
2 0H
P dzdy
= .
, ( , ) 0P z y = 1
H . , 1 (0, ) ( , ) 0,( , )P z y z y H= . (33) ( , ) 0,z y ( , )z y H , (27)
( ) 0, (0, ).q z z 1 . = + 2 ( , )u x y D - .
2D -
. , .
1. , .. -
/ .. // . .. 1978. 2. . 8085. 2. , .. -
/ .. // . 1984. . 20, 11. . 19471953.
Copyright & A K-C
. . . 12
3. Sylvester, J. A Global uniqueness theorem for an inverse boundary value problem / J. Sylvester, G. Uhlmann // Annals of Mathematics. 1987. Vol. 125. P. 153169.
4. , .. - / .. // . 1988. . 24, 12. . 21252129.
5. , .. / .. // - . 2007. . 47, 8. . 13651377.
6. Yang, R. Inverse coefficient problems for nonlinear elliptic equations / R. Yang, Y. Ou // ANZIAM. 2007. Vol. 49, no. 2. P. 271279.
7. , .. - / .. // . 2010. . 10, 2. . 93105.
8. , .. / .. . .: , 1995. 206 .
9. , .. / .. // . . . . . 2011. . 11. . 1. . 39.
10. , .. - / .. // . 1986. . 27, 5. . 8394.
11. , .. / .. -, .. , .. . .: , 1980. 288 .
12. , . / . . .: , 1965. 379 .
19 2012 .
Copyright & A K-C
..
2015, 7, 2 13
Bulletin of the South Ural State University
Series Mathematics. Mechanics. Physics 2015, vol. 7, no. 2, pp. 513
INVERSE PROBLEM OF DETERMINATION OF COEFFICIENT IN THE ELLIPTIC EQUATION R.A. Aliyev 1
The inverse problem of determination of coefficient in the elliptic equation in a rectangle is consid-ered. Identification problem of unknown denseness of sources and coefficients lead to similar inverse problems. The theorem of uniqueness of the formulated inverse problem is proved using Karlemans evaluation method. Researches are carried out in a class of continuously differentiable functions deriva-tives of which satisfy the Holder condition.
Keywords: inverse problem; elliptic equation.
References 1. Iskenderov A.D. Izvestiya AN Az.SSR. 1978. no. 2. pp. 8085. (in Russ.). 2. Klibanov M.V. Differentsial'nye uravneniya. 1984. Vol. 20, no. 11. pp. 19471953. (in Russ.). 3. Sylvester J., Uhlmann G. A global uniqueness theorem for an inverse boundary value problem //
Annals of Mathematics. 1987. Vol. 125. pp. 153169. 4. Vabishchevich P.N. Differentsial'nye uravneniya. 1988. Vol. 24, no. 12. pp. 21252129. (in
Russ.). 5. Solov'ev V.V. Zhurnal vychislitel'noy matematiki i matematicheskoy fiziki. 2007. Vol. 47, no. 8.
pp. 13651377. (in Russ.). 6. Runsheng Yang and Yunhua Ou Inverse coefficient problems for nonlinear elliptic equations.
ANZIAM. 2007. Vol. 49, no. 2. pp. 271279. http://dx.doi.org/10.1017/S1446181100012839 7. Vakhitov I.S. Dal'nevostochnyy matematicheskiy zhurnal. 2010. Vol. 10, no. 2. pp. 93105. (in
Russ.). 8. Denisov A.M. Vvedenie v teoriyu obratnykh zadach (Introduction into inverse problems theory).
Moscow, Nauka Publ., 1995. 206 p. (in Russ.). 9. Aliev R.A. An Inverse Problem for Quasilinear Elliptic Equations. Izvestiya of Saratov Univer-
sity. New Series. Series: Mathematics. Mechanics. Informatics. 2011. Vol. 11. Issue 1. pp. 39. 10. Klibanov M.V. Sibirskiy matematicheskiy zhurnal. 1986. Vol. 27, no. 5. pp. 8394. 11. Lavrent'ev M.M., Romanov V.G., Shishatskiy S.P. Nekorrektnye zadachi matematicheskoy fiziki
i analiza (Illposed problems of mathematical physics and analysis). Moscow, Nauka Publ., 1980. 286 c. (in Russ.).
12. Khermander L. Lineynye differentsial'nye operatory s chastnymi proizvodnymi (Linear differen-tial operators with partial derivatives). Moscow, Mir Publ., 1965. 379 p. (in Russ.). [Hrmander L. Lin-ear partial differential operators. Academic Press and Springer-Verlag, New York, 1963.]
Received 19 December 2012
1 Aliyev Ramiz Atash oqli is Cand. Sc. (Physics and Mathematics), Associate Professor, Information and Information System Department, Azerbaijan University of Cooperation, Baku, Azerbaijan. E-mail: [email protected]
Copyright & A K-C
. . . 14
517.955.8
1 .. 2, .. 3
, . xp , p , - , .
: ; ; - ; ; ; - ; ; ; .
[1] -. , . x0,5 , 50 % -, . , x0,5, . . x0,5, . , . , . - [1] x0,5 [2, . 3, . 45], , . xp p- 2, -. , , - . 1. ( )
- (. 1).
, . , ( 1x y = = ). [3], :
1 12-01-00259-, 02.740.11.0612, 1.731.2011 . 2 , , , - . E-mail: [email protected] 3 - , , , - . E-mail: [email protected]
. 1
Copyright & A K-C
.., ..
2015, 7, 2 15
/ 2, / 2
0,
2 20,
0 {0 , / 2 / 2},
0,
0, ( / 2, ) ( , / 2),
1 1, ( , ).
y b b
x a
x a
u D x a b y b
u
y
uy b b
x
uy
x y
=
=
=
= = < < <
. . . 16
, .p
p p
xW W
b= , .
px , p- ,
, .. px
., . 2
p
WW p= . (1)
, , ( )px 0 .
0
1
21 2 2 2
( , ) ch ch ( ) cos2
shn
nJ
x n n nbu x y x a x y
nb b b bn a
b
=
= +
,
1
0 20
2 cos( )( )
1
xyJ x dy
y=
.
,
/ 2 02
210 / 2
21 4 2
( , ) sh 2sh2
sh
a b
nb
nJ
n n abW u x y dydx a a
n b b bn a
b
=
= = +
,
2/ 2 0
2
210 / 2
21
( , )2
sh
px b
pnb
nJ
bW u x y dydx
nn a
b
=
= =
4 4 4sh sh ( ) sh
2 2sh ( 2 ) sh .
2 2 2
p pp
p
n n nx a x a xn nb b b
a x ab b b
+ + + +
W [3],
1
2 21 ln ( )
ch
an
b
n
a b eW I U O
abn n
b
=
= = + +
,
0
1
22
( ,0) (0,0) ( ) th ( )n
nJ
a nabU u a u O O
b n b
=
= + = + + =
1
2 2ln ( ).
ch
an
b
n
a b eO
abn n
b
=
= + +
,
2/ 2 0
2
10 / 2
2
( , ) tha b
nb
nJ
nbu x y dydx I U a
n b
=
=
0
1
2
th ( )n
nJ
nba O
n b
=
=
,
Copyright & A K-C
.., ..
2015, 7, 2 17
, ( , )u x y ( )O . .
,
.1
2ln ( )
ch
an
b
n
b eW O
an n
b
=
= +
,
20
., .
21
.1 2 3
2 4 4sh sh ( )
1 2sh ( 2 )
22 2 2sh
1( ).
2
p p
p p
n n nJ x a x
W nb b bW a x
n bn a
b
WS S S
=
= + + +
= + + +
1S , 2S 3S . , ,
~px , 0 1< < .
20
121
2 4sh
( )2 2sh
p
pn
n nJ x
b bS O x
nn a
b
=
= =
,
20
221 1
22 1
sh ( 2 ) ( )2 2
sh shp p
n n
nJ
nbS a x O x
n nbn a n a
b b
= =
= = +
,
4 42 20 0
32 21 1
2 4 2sh ( )
2 22 4sh sh
pn n
a xp b b
n n
n n nJ x a J
e eb b bS
n nn a n a
b b
= =
= = +
4 4 22 20 0
21 1 1
2 2
( ) ( ).2 24 4sh sh
pn n n
a x ab b b
p pn n n
n nJ J
e e eb bO x O x
n nnn a n a
b b
= = =
+ + = +
240
1
2p
nx
b
nJ
bS e
n
=
= .
0( ) 1 ( )J x O x= + 20 ( ) 1 ( )J x O x= + . , 0x > ,
.. 0M > , 20 ( ) 1J x Mx + 0x > . 4
4 4
1 1 1
(1 ( ))p
p p
nxn nbx x
b b
O n eS e O e
n n
= = =
+ = = + =
4 4 4
41 1
1
p p p
p
n nx x x
b b b
x p b
e e eO O
n n xe
= =
= + = +
.
Copyright & A K-C
. . . 18
4
1
pn
xb
e
n
= [6, . 2]:
4 4 4 41 1
1 1 11
1 1 1p p p p
n y n yx x x xn nb b b b
n n
e e e e dydy dy
n y n y n y
+ +
= = =
= + + =
4
1
,p
yx
bedy C R
y
= + +
1
1
1 n
n
dyC
n y
+
=
=
, 4 4
1
1
1 1.
p pn y
x xnb b
n
e eR dy
n y
+
=
=
,
4
11
4 4( ) ln ( )
py
xb
p p pe y
dy E x C x O xy b b
= = +
0px ,
111
( )( ) [ ] ln
!
xy z n
nx
e e xE x dy xy z dz C x
y z n n
=
= = = = =
[7].
R ,
4
1( )
py
xbe
F yy
= .
4 4 42 2
2 2 3
16 4 1( )
p p py y y
x x xb b bp px xe e eF yy bb y y
= + + .
.
4 4 4 41 1/ 2
1 1 1/ 2
1 1 1 1| | ( )
p p p pn y n y
x x x xn nb b b b
p n n
e e e eR dy dy O x
n y n y
+ +
= =
= = +
4 4 41/ 2
2 2 21 2 32 3
1 1 1 11/ 2
1 1( ) ( ) ( )
2
p p pn n n
x x xn b b b
n p p pn n n nn
e e eF n y n dy O x M x M x M
n n n
+
= = = =
+ + + + +
4
2 21 2 32 3
1 1 1
41( ) ( ) ( ).
pn
xb p
p p p p pn n n
nxeO x M x M x M O x O x
n n bn
= = =+ + + + =
,
., .
1
1 1ln ( )
2 4h
na
b
p pp pn
W b eW O x O
nx xn a
b
=
= + + + =
41 1 1ln ln ( ) ln ( ) .
4p
p pp p p
xb bO x O O x O
x x x
= + + = + +
, .p W (1), px :
Copyright & A K-C
.., ..
2015, 7, 2 19
.1 1ln ln ( ) ( )4 2
p p
p p p
Wb bO x O p O x O
x x x
+ + = + +
.
, .W 0 , ,
1
ln (1 ) ln ( ) ( ).4
ch
na
b
pp pn
b b ep p O x O
nx xn a
b
== + + +
px , :
1
11(1 ( ) ( ))
4
ppQ p p
pb
x e O Ob
= + +
0 , 1
1 ch
na
b
n
eQ
nn a
b
==
.
, a
b 1pQe
. , ,
2 2
2 20y
x
u u
x y
+ =
,
:
11
1
1exp (1 ( ) ( ))
4ch( )
y
x
na
p bp px
py n y
x
b ex p O O
b nn a
b
=
= + +
0 .
2. ( )
, , . 2.
.
0,
2 2
0
2 2
0 {0 ,0 },
0,
0,
2 1, (0, ),
0 ( , ),
2 1, ( , ).
y b
x a
x
u D x a y b
u
y
u
x
yy
uy b
xy b b
y
=
=
=
= = < < <
. . . 20
2, .0 0
( , )px b
p W u x y dydx= , px .
px , p - , ..
px
., . 2
p
WW p= .
( )px 0 .
0
0
(2 1)( )4 (2 1) (2 1)
( , ) ch ( ) cos(2 1)
(2 1)shm
mJ m mbu x y a x y
m b bm a
b
=
++ + = + +
.
,
20
22
00 0
(2 1)( )8 (2 1) 4 2
( , ) cth ln ( ),(2 1)
a b
m
mJ m bbW u x y dydx a Q O
m b
=
+ + = = = + + +
20
(2 1)2exp( )
(2 1)(2 1) shm
ma
bQm
m ab
=
+=
+ +
.
, b a>> , , , . , :
4 4ln ( )
b b aW O O
a b
= + + +
.
, , :
20
2, .
200 0
(2 1)4 2 (2 1)
( , ) sh ( )(2 1)
(2 1)sh
px b
p pm
mJ
mbW u x y dydx W a x
m bm a
b
=
+ + = = = + +
24 4 4 4
ln ( ) ln ln ( ) .p pp p p p
b b bW Q O x O O x O
x x x x
= + + + = + +
,
, .4 4 4
ln ln ( )p pp p
b bW O x O p W
x x
= + + =
.
,
24
ln (1 ) ln ( ) ( )pp p
b bp pQ O x O
x x
= + + ,
, p -
2
111 4 (1 ( ) ( ))
4 4
ppQ p p
pb
x e O Ob
= + +
, 2
0
(2 1)2exp
(2 1)(2 1)shm
ma
bQ
mm a
b
=
+ =
+ +
.
Copyright & A K-C
.., ..
2015, 7, 2 21
3. ( ) (. 3)
.
0,
2 2
0
2 2
0 {0 ,0 },
0,
2 1, (0, ),
0, ( , ),
0, (0, ),
2 1, ( , ).
y b
x
x a
u x a y b
u
y
yuy
xy b
y bu
y b bxy
=
=
=
= < < <
. . . 22
, .2 2 2
ln ln ( ) ( ).2p pp p
b bW O O x
x x
= + +
px , p - ,
, px
., . 2
p
WW p= .
,
( )31
11 2 1 ( ) ( )4 2
ppQ p p
pb
x e O Ob
= + +
, 3
1
( 1)
sh( )
na nb
n
eQ
nn a
b
=
=
.
, p - , , ,
11 2
4 2
px
py
bx
b
=
0 . (2)
. , , . 50, 90 95 % . ,
0,5x , 0,9x 0,95x ,
. . . . , , . . [1] 50, 90 95 % , 0,04 , 0,2 0,26 . [1] . . 56 , , . 0,1 0,05 . 56 [1]. , 0,001 , 1 . .
1 , , , 1 100100 2. . ,
Copyright & A K-C
.., ..
2015, 7, 2 23
(3) x x , y . (2), 0,5x 1,995 0,02
. [1] 0,5x 0,04 ,
. 0,5 0,02x = 0,335 62 .
[1] . 90
95 %. 0,4
0,9
0,5
5,3652
x
x b
=
, 0,45
0,95
0,5
6,612
x
x b
=
.
0,9x 0,95x 0,107 0,132 . 0,335
320400 . . , , . [1] . . . , [5], 23, 0,1 ~4 . , 0,9x 0,95x
.
1. , ..
/ .. , .. // . 2011. 6. . 4754.
2. , .. / .. , .. . .: , 1973. 212 .
3. , .. / .. // . 2011. . 51, 6. . 10641080.
4. , . / . . .: , 1961. 464 .
5. , .. / .. . : , 2005. 197 .
6. , .. / .. , .. . .: , 2008. 248 .
7. , . / . , . . .: , 1974. 295 .
11 2014 .
Copyright & A K-C
. . . 24
Bulletin of the South Ural State University Series Mathematics. Mechanics. Physics
2015, vol. 7, no. 2, pp. 1424
THE ANALYSIS OF ENERGY ABSORPTION IN A BLANKET FOR CONTACT ELECTRIC RESISTANCE A.V. Dmitriev 1, A.A. Ershov 2
The authors investigate piculiarities of the dissipation of electric energy at the point contacts between the graphite flakes in a composition based on it. The analysis reveals the surface layer on the surface of the graphite flake and carries out the calculation of the proportion of the output electrical energy. The calculations are performed in the two-dimensional approximation at the location of contacts with the current flowing across and along one of the flat faces of the plate. The sizes of plate conductors and electrical crosssection of the electrical spots of contacts between them have been used as parameters. The authors have discovered the analytical dependences of the layer thickness of xp with a predetermined proportion of the energy dissipation of p of the crosssection contacts of 2 tending to zero. The authors have used boundary value problems as an elliptic function for the normal derivative. The general solution is obtained using the Fourier expansion of the Bessel functions, special cases are performed at the location of contact on the opposite side of the base and in the end sections. The method of asymptotic expansions is used as the solution. For all three cases, the thickness of xp is expressed through the multiplication of the power function of the proportion of crosssection of contact in the face area and exponentially dependentant on the p factor. The value of p linearly enters the index grade-dependence. The asynptotical analysis of the thickness of the layer with the release of 50, 90 and 95 % of the electrical energy for the particular case is carried out. The compensation for the calculation of the conditions under which the grid method is not applicable is given. The value of x0,5 and the thickness of the surface layer in which 50 % of the energy is dissipated by the order of magnitude thinner than the thickness of the natural graphite flakes are shown. When the defect layer is formed it determines the properties of the electrical contacts in the composition of natural flake graphite with non-conductive binder. It characterizes the contact electrical resistance at the pressing point contacts.
Keywords: boundary value problem; lamellar crystal of graphite; asymptotic decomposition; small parameter; thickness of a defective layer; composite material; network of resistance; chain model; prevalence of contact electric resistance.
References 1. Dmitriev A.V., Ershov A.A. Khimiya tverdogo topliva. 2011. no. 6. pp. 4754. (in Russ.). 2. Lavrent'ev M.A., Shabat B.V. Metody teorii funktsiy kompleksnogo peremennogo (Methods of
the theory of functions of a complex variable). Moscow, Nauka Publ., 1973. 212 p. (in Russ.). 3. Ershov A.A. Zhurnal vychislitel'noy matematiki i uravneniy matematicheskoy fiziki. 2011.
Vol. 51, no. 6. pp. 10641080. (in Russ.). 4. Khol'm R. Elektricheskie kontakty (Electrical contacts). Moscow, Inostrannaya literatura Publ.,
1961. 464 p. (in Russ.). 5. Dmitriev A.V. Nauchnye osnovy razrabotki sposobov snizheniya udel'nogo elektricheskogo
soprotivleniya grafitirovannykh elektrodov (The scientific basis for the development of ways to reduce electrical resistivity of graphite electrodes). Chelyabinsk, ChGPU Publ., 2005. 197 p. (in Russ.).
6. Il'in A.M., Danilin A.R. Asimptoticheskie metody v analize (Asymptotic methods in the analysis). Moscow, Fizmatlit Publ., 2008. 248 p. (in Russ.).
7. Beytman G., Erdeyn A. Vysshie transtsendentnye funktsii (Higher transcendental functions). Moscow, Nauka Publ., 1974. 295 p. (in Russ.).
Received 11 December 2014
1 Dmitriev Anton Vladimirovich is Cand. Sc. (Engineering), Lecturer, Chemistry of a Firm Body and Nanoprocesses Departament, Chelyabinsk State University. E-mail: [email protected] 2 Ershov Aleksandr Anatol'evich is Cand. Sc. (Physics and Mathmatics), Associate Professor, Calculus Mathematics Departament, Chelyabinsk State University. E-mail: [email protected]
Copyright & A K-C
2015, 7, 2 25
517.948
, , .. 1, .. 2
-. .. , -.
: ; ; -; ; .
, , [1], . - , , . -.
[2] . . - -, . [3].
1.
- , ,
( )( ) ( , ) ( ) ;0 ,
b
a
f tSn s K s t n s ds t
t= = < (1)
2
3 2( , ) ,
2 sh2
sK s t
st
t
=
2 2( )
( ) [ , ], (0, )f t
n s L a b Lt
, ( )n s ,
( )f t , .
, 0( ) ( )f t f t= 0( )n s (1), M ,
{ }' 2( ) : ( ), ( ) [ , ], ( ) 0 ,M n s n s n s L a b n a= = (2) '( )n s s .
0( )f t , 2( ) (0, ), 0f t L > ,
2
0( ) ( ) .L
f t f t
t t <
( ),f t M ( )n t - 0( )n t 2[ , ].L a b , (1) [4].
1 , , . E-mail: [email protected] 2 - , , , - . E-mail: [email protected]
Copyright & A K-C
. . . 26
B, 2[ , ]L a b 2[ , ]L a b ,
2 2( ) ( ) ( ) ; ( ) [ , ], ( ) [ , ],s
a
n s Bu s u d u s L a b Bu s L a b = = (3)
C
2 2( ) ( ); ( ) [ , ], ( ) (0, ).Cu s ABu s u s L a b Cu s L= (4) (3) (4) ,
( ) ( , ) ( ) ,b
a
Cu s P s t u s ds= (5)
( )( , ) , .s
b
P s t K t d = (6)
, (1) C - nC , nh , -
.n n C h nh (1)
2
3 2( , ) , ,0
2 sh2
sK s t a s b t
st
t
= < <
( )N t
2 2
3 2 3 2max ( ).
2 sh 2 sh2 2
a s b
s bN t
s at t
t t
=
(7)
, 2( ) (0, ).N t L ( , )K s t ( )N t . ,
2
42
(0, )6 40
1( ) .
4 sh2
L
bN t dt
at
t
=
,t 4
22
2 1( ) ~
bN t
a t
, 0, ( ) 0t N t . , 2( ) (0, )N t L .
nC [ , ]a b n
( ), ( , )i nP t P s t
( ) ( , ),i iP t P s t= (8)
1 1( 1)( ) ( )
, , , 0,1... 1,2
i ii i i
s s i b a i b as s a s a i n
n n+
++ + = = + = + =
a
1( , ) ( ); , (0, ); 0,1... 1.in i iP s t P t s s s t i n+= < = (9) (9), nC
( ) ( , ) ( ) ; (0, ),b
n na
C u s P s t u s ds t= (10)
2 2: [ , ] (0, ).nC L a b L
Copyright & A K-C
.., , , ..
2015, 7, 2 27
(5)(10) ,
( )2
32
( ) .n nLb a
C C N t hn
= (11)
2. (1) ..
2
22
( )inf ( ) ( ) : ( ) [ , ] , 0.
b
na
f tC u s u s ds u s L a b
t
+ >
(12)
[5] ( )nh
u s (12).
, ( )nf t , 2(0, )L ,
( ),( )
( ) ; ,nnf t
f t pr R Ct
=
(13)
2(0, )L ( )f t
t -
nC .
( ), ( ), ,n nC f t h = (12) - [4]
,( ) ( ) ( ) .n nn h h nnC u s f t u t h = + (14)
, '0, ( ) ( ) nnf t n s h > +
( ), ( ), ,n nC f t h (14). ( , ( ), , )( )n n
n
C f t hhu s
(12), (14) ( )nhu s ,
( )nh
n s (1)
( ) ( ).n nh h
n s Bu s = (15)
(8)(10) , 11
0
( ) ( ) ( ) .i
i
sn
ini s
C u s P t u s ds+
== (16)
1
( ) ,i
i
s
is
b au s ds u
n
+ = (17)
( )1
( ) ,i
i
s
is
nu u s ds
b a
+
=
(18)
(16) (17) , 1
0
( ) ( )n
in ii
b aC u s P t u
n
=
= . (19)
2[ , ]nX L a b - ( )s { }1( ) : , 0,1... 1 .i i is c s s s i n += < = (20)
Copyright & A K-C
. . . 28
, nX , { },ie 0,1... 1i n=
1
1
1;( )
0; [ , ), 0,1... 1.i i
ii i
s s se s
s s s i n+
+
.., , , ..
2015, 7, 2 29
3. (1) , [6] [7]
{ }2 2 ,inf ( ) : ( ) [ , ], ( ) ( ) ( )n nnu s u s L a b C u s f t u s h + (28)
{ }2 2 ,inf ( ) : ( ) [ , ], ( ) ( ) ,n nnu s u s L a b C u s f t h + (29)
, ( )0
n
n
f t
h
.
, [7], , , ( ) nnf t h > + -
( )nh
u s (29).
( ) .nh
u s = (31)
3. '0, ( ) ( ) nnf t n s h > + , (28) - (30) (31). [8].
4. '0, ( ) ( ) nnf t n s h > + , (28) -
(12) , (14). [8]. 2[ , ]L a b .
( , ), , 0r r >
{ }2( , ) sup ( ) : ( ) , ( ) ,L rn
r n s n s M Sn s =
r rM BS= , S (1).
5. 0( )n s M , ( )nhn s (15) '0, ( ) ( ) nnf t n s h > + .
0r > , ,
20 [ , ]
( ) ( ) 2 ( 2 , ).nh nL a b
n s n s rh r +
[3]. [2] ,
1
221( , ) 1 ln .4
rr r
+
(32)
( )nh
u t , (15),
( )2
1
22
0 [ , ]
1( ) ( ) 2 1 ln
4 2nh L a b n
rn s n s r
rh
+ + ,
( )nh
n s (1), (26).
1. , .. - / .. // . 1954. . 26, 5. . 551556.
2. , .. / .. , .. // - . 2014. . 17, 2(58). . 125136.
Copyright & A K-C
. . . 30
3. , .. , - , / .. , .. , .. // . : , , -. 2014. . 14, 4. . 5964.
4. , .. / .. , .. // . . 1987. 892 87.
5. , .. / .. // . . 1963. . 151, 3. . 501504.
6. , .. / .. , .. , .. - // . 1973. . 13, 2. . 294302.
7. , .. - - I / .. // . . 1975. . 224, 15. . 10251029.
8. , .. / .. . .: , 1981. 156 .
11 2014 .
Bulletin of the South Ural State University Series Mathematics. Mechanics. Physics
2015, vol. 7, no. 2, pp. 2530 UNCERTAINTY ESTIMATION OF THE METHOD BASED ON GENERALIZED RESIDUAL PRINCIPLE FOR THE RESTORE TASK OF THE SPECTRAL DENSITY OF CRYSTALS A.A. Ershova 1, A.I. Sidikova 2
The article studies the task of identification of phonon spectrum of a crystal according to its hit ca-pacity. The authors reveal the evaluation of accuracy of Tikhonov regularization method with regulari-zation parameter which was chosen form generalized residual principle.
Keywords: regularization; module of continuity; uncertainty estimation; ill-conditioned task; gen-eralized residual principle.
References 1. Lifshits I.M. Zhurnal eksperimental'noy i teoreticheskoy fiziki. 1954. Vol. 26, Issue 5. pp. 551
556. (in Russ.). 2. Tanana V.P., Erygina A.A. Sibirskiy zhurnal industrial'noy matematiki. 2014. Vol. 17, no. 2(58).
pp. 125136. (in Russ.). 3. Tanana V.P., Sidikova A.I., Vishnyakov E.Yu. Ob otsenke pogreshnosti regulyarizuyushchego
algoritma, osnovannogo na obobshchennom metode nevyazki, pri reshenii integral'nykh uravneniy (On Error Estimates for Regularizing Algorithm Based on Generalized Residual Method when Solving Inte-gral Equations). Bulletin of South Ural State University. Series of Computer Technologies, Automatic Control & Radioelectronics. 2014. Vol. 14, no. 4. pp. 5964. (in Russ.).
4. Tanana V.P., Boyarshinov V.V. Dep. v VINITI. 1987. 892.V87. (in Russ.). 5. Tikhonov A.N. Dokl. AN SSSR. 1963. Vol. 151, no. 3. pp. 501504. (in Russ.). 6. Goncharskiy A.V., Leonov A.S., Yagola A.G. Zhurnal vychislitel'noy matematiki i mate-
maticheskoy fiziki. 1973. Vol. 13, no. 2. pp. 294302. (in Russ.). 7. Tanana V.P. Dokl. AN SSSR. 1975. Vol. 224, no. 15. pp. 10251029. (in Russ.). 8. Tanana V.P. Metody reshenija operatornyh uravnenij (Methods of solution of operator equa-
tions). Moscow, Nauka Publ., 1981. 156 p. (in Russ.). Received 11 December 2014
1 Ershova Anna Aleksandrovna is Post-graduate Student, Department of Theory of Management and Optimization, Chelyabinsk State Univer-sity. E-mail: [email protected] 2 Sidikova Anna Ivanovna is Cand. Sc. (Physics and Mathematics), Associate Professor, Calculating Mathematics Department, South Ural State University. E-mail: [email protected]
Copyright & A K-C
2015, 7, 2 31
517.956.32
3- .. 1
- 3- . .
: ; ; 3- ; .
3
(3)
1
( ) ( ) ( ) ( ) ( ), ,x iii
uD u x D u a x c x u f x x G
x=
+ + =
L (1)
1 2 3( , , )x x x x= , (3) 3
1 2 3/xD x x x= , ( )f C G , - G . , (1) - constix = . . ( ) 0x = - 3 4C 0 3x . , ( ) 0x =
0i ix x= , 1,2,3i = , G , G iS .
i iS = . , x i - , - ( )ix . (1) .
. ( )u x , , 2( )u C G (3) ( )xD u C G , - (1)
|( ), , 0,2,
k
kk
us s k
l = =
(2)
3l , || || 1l = , 3( ) ( )l x C .
{ }3 0 1: , 1,3i i ix x x x i= < < =D . { }( )k ki i iS x x x= =D , 0,1k = . 0( )i iS S x= ,
31
1
( )ii
S x=
= .
. ( )u x , , 1( )u C D ( )
(2) ( )ix
D u C D , (3) ( )xD u C D ,
(1)
0 ( )| ( )( ) ( ), 1,2,3.
ii iS x
u x x i= = (3)
( )u x , , -
. , ( )nxD ,
. 0ia = , 1,3i = , ( ) constc x = , , , [1]. [2, 3]. [2] , [3] . 2- , [4, 5], . 1.
* ( )DL , ( )DL :
1 - , , , - . E-mail: [email protected]
Copyright & A K-C
. . . 32
( )3
* (3)
1
( ) ( ) ( ) ( ) ( ) ( ) ( ).x iii
D v x D v x a x v x c x v xx=
+
L (4)
, * ( )DL ( )v x , : 1( )v C D ,
( )
(2) ( )ix
D v C D (3) ( )xD v C D . kx jx
( )ix .
. (1) ,
1
*
( )|
( ) ( ) 0, ,
( ) ( ), 1,3,i i
i ix x
D v x x
v x w x i=
=
= =
L D
(5)
( )( )i iw x
1( )
1
(2) 1( ) ( ) ( )|
0 1( ) |
( ) ( ) ( ) 0, ( ),
( ) 1, , , .
i i i
s s
i i i i i i ix x x
i i s s sx x
D w x a x w x x S x
w x s k j x x x
=
=
+ =
= = < < (6)
(1) R . , 1R R( , )x x= .
1. , , / ( )i ic a a x C D . (1) .
. (3)(4) 1R( , )x x . , -
* 1( )R( , ) 0D x x =L 1
.x
xd , xD ,
( )( )
1 1
1 1( )
1( )
31 1 1 1 1
| |1
31 1 1
| ( )|1
R( , ) R( , ) R( , ) R( , )
( )R( , ) ( )R( , ) ( )R( , ) 0.
i i i i
i
i ii ii
x x x xi
x x
i i x ixx xi
x x x x x x x x
a x a x d c x d
= ==
===
+
+ =
1 1|
( )R( , )xa y = - 1 2 3 1 2 3 1 2 3( , , )R( , , ; , , )a x x y y y . (6), (5),
( )1 1( )
( )
3 31 1 1
( ) | ( ) | ( )1 1
R( , ) ( ) 2 ( ) ( ) ( )R( , ) ( )R( , ) .ii i i ii
x x
i i i x i i i x ix xi i
x x w x a w x a x d c x d = == =
= +
, 1( )
( )( ) | ( ) ( )( ) ( ) ( ) 1, , 1,3.
i
i ii
x
i i i x i i i ixw x a w x d x S i =+ = =
1 1( )
( )
31 1 1
| ( )1
R( , ) 1 ( )R( , ) ( )R( , ) , .ii ii
x x
i x ix xi
x x a x d c x d x ==
= D (7)
1R( , )x x ( )v x , (7) 1 1( )
( )
3
| ( )1
( ) 1 ( ) ( ) ( ) ( ) .ii ii
x x
i x ix xi
v x a v d c v d ==
= (8)
(8) . -
, ( )v C D ( )v x
( )1 01
( ) ( ) ( ) ( ),n nn
v x v x v x v x
=
= + (9)
0( ) 1v x = , 1n
Copyright & A K-C
.. 3-
2015, 7, 2 33
1 1( )
( )
3
1 | ( ) 11
( ) 1 ( ) ( ) ( ) ( ) ,ii ii
x x
n i n x i nx xi
v x a v d c v d = =
=
(9) D . - [6] (5)(6). (9).
| |,| |ic a M , D 1 0max { ,1}i i iK x x= .
( )| | ,| |i j j k k i i j j k kx y x y x y x y x y x y x y = + = + + , , ,i j k
, 1, 2, 3. xD 1 1 1 1 1 1 2 1
( )( )( ) / 2 | | , ( )( )( ) /3 | |,i i j j k i i j j k kx x x x K x x x x x x x x K x x (10)
xD
()
1 1 1 11 0 1 1 2 2 2 2 3 3
1 1 1 1 1 2 11 1 3 3 1 1 2 2 3 3
| ( ) ( ) | ( )( ) ( )( )
( )( ) ( )( )( ) 4 /3 | | .
v x v x M x x x x x x x x
x x x x x x x x x x MK x x
+ +
+ +
, xD 1n 1 2 1,!
1| ( ) ( ) | / 3(4 ) | | .n n
n nv x v x K KM x x
+ (11) (11) n 1n + ,
( )( )
1 2 1,! 1 2 2,!1
1 2 1,! 1
| ( ) ( ) | / 3(4 ) 3 | | | |
/ 3(4 ) | | 3 | | /(2 2) .
n n nn n
n n
v x v x K KM M x x x x
K KM M x x x x n
+ ++
+
+
+ +
3/(2 2) 1n + < , 1n , 1 1 2 1,!
1| ( ) ( ) | / 3(4 ) | |n n
n nv x v x K KM x x+ +
+ ,
.. (11) 1n xD . (11) - (9), , ( )v x , D -
( )3 1| ( ) | 1 2 / 3 sh 4 | | .v x MK MK x x + (8). . - ( )w x . , ( )w x
1 1( )
( )
3
| ( )1
( ) ( ) ( ) ( ) ( ) .ii ii
x x
i x ix xi
w x c w d a w d ==
=
, 1 1
( )
( )
3
| ( )1
| ( ) | | ( ) | | ( ) | .ii ii
x x
x ix xi
w x M w d w d ==
+
sup | ( ) |x
W w x
=D
2 1| ( ) | 4 /3 | |w x MWK x x .
, ( )w x , (11), 1 2 1,!| ( ) | / 3(4 ) | | , .n nw x W KM K x x x D
, n , 0, - n , ( ) 0w x = xD . .
, 1R( , )x x , (7).
, (8) : 1( )v C D ,
( )
(2) ( )ix
D v C D , (3) ( )xD v C D , (5)(6), (1)
.
Copyright & A K-C
. . . 34
, ( )v x D
. (9), 1x . ,
( )nv x D
( ) ( )12
2 221 3 1 2|
1
( ) ( ) ( ) ( ) ( )x
n n n n xxv x v x a x v x v x d
x + =
=
( ) ( )1 1 13 2 3
2 23 33 2 3
3 3
2 1 3 1 1 3 2|1
( ) ( ) ( ) ( ) ( ) ( ) ( ) |x x x
xn n n nxx x xx
a x v x v x d c x a x v x v x d dx
= =
=
+
( )1 12 3
2 22 3
3 3
1 1 3 21
( ) ( ) ( ) |x x
xn nx x xa x v x v x d d
x
==
(12)
1 sup( , / )i ix
M M a x
= D
, , (10), ,
( )1 1 11 0 1 2 2 3 3 (1)1
( ( ) ( )) 2( )( ) | | ,v x v x M x x x x x xx
+
11 0 1 (1)
1
( ( ) ( )) ( 1) | | .v x v x M K x xx
+
(13)
11
( ( ) ( ))n nv x v xx +
. (10),
1 2 ,!1 1 1
1
( ( ) ( )) 2 /3 ( 1) (4 ) | |n nn nv x v x K K M KM x xx + + +
( )1 12 3
2 31 1 3 2
1
x x
n nx xM v v d d
x
+
. (14)
( )1 12 3
2 3
1 2 ,!1 1 1
1
2 2,!2 1 1 11 1 (3) 3 3 3 2
( ( ) ( )) 2 /3 ( 1) (4 ) | |
2 / 3 ( 1) (4 ) | |
n nn n
nx xn
x x
v x v x K K M KM x xx
K K M KM x x x d d
+
+
+ +
+
+ +
1 1 1 12 3 2 3
2 3 2 3
1 1 1 12 3 2 3
2 3 2 3
2 1 2 ,!1 1 2 3 2 3 2 1 1
1
21 1 2 3 2 3 2
1
( ) 4 /3 ( 1) (4 ) | |
( ) .
x x x x n nn nx x
x x x x
n nx x
M v v d d d d K K M KM x xx
M v v d d d dx
+ + +
+
2n (13),
1 2 ,! 1 1 2 1,!1 1 1 1 (1)
1
( ( ) ( )) 2 /3 ( 1) (4 ) | | ( 1) | | .n n n nn nv x v x nK K M KM x x K M x xx+ +
+ + + +
, (9) 1x ,
D . , (9) - 1x D . -
2x 3x . , 1( )v C D . , , 1( )v C D .
( )
(2)ix
D (3)xD ( )v x D .
, (8) ( )
(2)ix
D
Copyright & A K-C
.. 3-
2015, 7, 2 35
1,3i = , (3)xD ,
D . ( )
(2) (3), ( )i xx
D v D v C D . ,
1
( )
(2) ( ) ,iji i
x
i j x ix xj i
D v a v a v cv d=/
+ =
(15)
1( )
(2)|
0,i i i
i ix x xD v a v x S
=+ = .
1| 1i ix xv
== , 1,3i j = ( i j=/ )
( )
(2) ( )ix
D v C D ,
1( ) |( ) ( )
i ii i x x
w x v x=
= (6). -
(15) ix
D . * ( ) ( ) 0D v x =L . , ( )v x , - (8), (5)(6). .
1. 1, ( )ic a C D , 2 2/ ( )i ia x C D , -
1R ( , ) ( )i jx x
x x C D .
0( ) R( , )U x x x= . 2. (1) ,
1. U(x) : 1( )U C D , ( )
(2) ( )ix
D U C D , (3) ( )xD U C D ,
(1) (6)
0( )
0
(2) 0( ) ( ) ( )|
0 1( ) |
( ) ( ) ( ) 0, ( ),
( ) 1, , , ,
i i i
s s
i i i i i i ix x x
i i s s sx x
D w x a x w x x S x
w x s k j x x x
=
=
+ =
= = < < (16)
0( ) |( ) ( ) i ii i x xw x U x
== .
. 0x x= 1x x= (7). - (9),
0
0
( ) ( , ), ,nn
U x U x x x
== D (17)
00( , ) 1U x x = , 1n 1
( )00
( )
30
1 1 ( )|1
( , ) ( ) ( , ) ( ) ( , ) .ii ii
x x
n n i n ixx xi
U x x c U x d a U x d ==
= (18)
0( , )nU x x , 2 ( )U x . -
(18) , 01( , )nU x x+ . ,
0( , )nU x x 0
1( , )nU x x+ . (17). (11) ,
( )30 36| ( , ) | ,9(2 1)!
n
n
MkU x x
n
0| |i ix x k , | |,| |ia c M . ix nD U , ( )(2)
i nxD U , (3)x nD U .
, (17) , ( )U x .
(18) 0i ix x= . , 00
( )|( ) ( , )
i in n ix x
U x w x x=
=
( )00
( )
0( ) 1 ( ) ( )|
( , ) ( ) ( , ) .ii ii
x
n i i n i ixxw x x a w x d == (19)
Copyright & A K-C
. . . 36
0j jx x= , j i=/ . 00
( ) |( , ) 0
j jn i x x
w x x=
= 1n . , 0( ) |( ) 1j ji i x xw x
== . -
(19) jx
D j i=/ . 1n ,
( )0 00 0
( )
0( ) 1 ( ) 1 ( ) ( )|
( , ) ( ) ( , ) ( ) ( , ) ,|
k i
j ji i i ik i
j j
x x
x n i i n i k i x n i ix xx xx
D w x x a w x d a D w x d
= ==
=
2n ( )
00( )
0( ) 1 ( ) ( )|
( , ) ( ) ( , ) .ij j i ii
x
x n i i x n i ixxD w x x a D w x d ==
kx
D ,
( )00( ) ( )( )
(2) (2)0( ) 1 ( ) ( )|
( , ) ( ) ( , ) .ii i i ii
x
n i i n i ix x xxD w x x a D w x d == (20)
1n = , 0( )
(2) 01 ( ) |( , ) ( )
i i ii ix x x
D w x x a x=
=
0( )
(2) 0 01 ( ) 0 ( )|( , ) ( ) ( , )
i i ii i ix x x
D w x x a x w x x=
= .
, (19) (20),
0( )
(2) 0 0( ) 1 ( )|
( , ) ( ) ( , )i i i
n i i n ix x xD w x x a x w x x== . (21)
n , 0( ) ( )lim ( , ) ( )n i i in
w x x w x
= ,
(16).
0( , )nU x x . , , ,i j k , 1, 2, 3
( )0 00 0 0 ( )( )
( )0 0( )
(3) 0
3(2)
( ) 1 || |1
(3) (3)1 | ( ) 1 1
( , )
( ) ( ) ( ) ( , )
( ) ( , ) ( ) ( , ) ( ) (
k j i
i iij j k kk j i
i
i ii
x n
x x x
j k k j i n xxx xx x xi
x x
i x n x i x n nx x
D U x x
a d a d c d D U x
a D U x d c D U x d c x U x
== ==
=
=
= + +
+
+
0, )x
(22)
, ( , ) 0nU x x = , 1n . 1n = (22). , 0
0( , ) 1U x x = , -
3(3) 0 0 0
11
( , ) ( ) ( , ) ( ) ( , )ix n i x n n
i
D U x x a x D U x x c x U x x+=
= (23)
0n = . (21) , i ix =
( )
(2)1( , ) ( ) ( , )i n i nxD U x a x U x = . (24)
, (23) 1n m= . n m= . , (23) (24), n m= , (22),
1n m= + . ( )
0 00 0 0( )
3(3) 0
1 ( ) 1 || |1
( , ) ( ) ( ) ( ) ( ) ( , )k j ii ij j k kk j i
x x x
x m i j k k j i m xx xx x xi
D U x x a x a d a d c d U x + == ==
= + + +
( )
( )0( ) 0
0
3
1 1 ( )1 |
3
1 11
( ) ( ) ( , ) ( ) ( , )
( ) ( ) ( , ) ( ) ( ) ( , ) .
i
ji
i i
i
x
i j x m m ixj x
x
i x m mxi
a a x D U x c x U x d
a x c D U x c x c U x d
= =
=
+ + +
+ +
(25)
( )ia x ( )ib x . (18),
Copyright & A K-C
.. 3-
2015, 7, 2 37
( )0 00 0 0
( )
( )00 0
( )
( ) 1 || |
30
1 ( ) 1|1
( ) ( ) ( ) ( ) ( , )
( ) ( , ) ( ) ( , ) ( , ).
k j i
i ij j k kk j i
j
i i ij jj
x x x
i j k k j i m xx xx x x
x x
j x m j x m x mxx xj
b x a d a d c d U x
a D U x d c D U x D U x x
== =
==
= + + +
+ + =
, ( )c x ( )d x
( )00 0
( )
30
1 ( ) 1|1
( ) ( ) ( , ) ( ) ( , ) ( , )ii ii
x x
i m i m mxx xi
d x a U x d c U x d U x x ==
= + = .
( )ib x ( )d x (25) (23)
n m= . , (24) n . (23) n . ( ) ( ) 0D U x =L . - .
2. 1, ( )ic a C D , 2 2/ ( )i ia x C D ,
0( , ) ( )i ix x
R x x C D .
2. . 3, ( )u v C G
( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )
2 3 3 1 1 231 2
2 3 1 3 1 231 2
2 3 1 3 1 231 2
*
1 2 3
1 2 3
1 2 3
( ) ( )
,
,
.
x x x x x x xx x
x x x x x x xx x
x x x x x x xx x
v D u u D v
u v a uv u v a uv uv a uv
uv a uv u v a uv u v a uv
u v a uv uv a uv u v a uv
+ =
= + + + + +
= + + + + +
= + + + + +
L L
,
( ) ( )( ) ( )
2 3 2 3 2 31
1 3 3 1 1 3 1 2 1 2 1 232
*1
2 3
3 ( ) ( ) 3
3 3 .
x x x x x x x
x x x x x x x x x x x x xx
v D u u D v u v u v uv a uv
u v u v uv a uv u v u v uv a uv
+ = + + +
+ + + + + +
L L
(26)
, , , G 0x , ( ) 0x = . 0( )G G x= . -
(26) x , ( )v R( , )x ( )G x . 1J , 2J 3J .
( ) 0x = i
n , 1,2,3i = . 1J .
( )( )
2 3 2 3 3 2 2 31
2 3 2 3 3 2 2 3 1
1 1( )
1
R 1/ 2 R R R 3 R
R 1/ 2 R R R 3 R .
G x
S
J u u u u a u d
u u u u a u n d
= + + + =
= + + +
(6),
( )( ) ( ) ( )
2 3 2 3 3 2 2 3 1
2 3 2 3 2 31 3 2
1 1( )
1( )
R 1/ 2 R R R 3 R
( R) 3 / 2 R R 3 R R .
x
S x
J u u u u a u n d
u u u a u ds
= + + +
+ + +
( )x ( ) 0x = , ( )G x . ( ) ( ) ( )i j kP x x x = . 1( )S x 1l .
(6),
( ) ( )2 2 3 31 3 21 ( )1
( R) 3 R ( R) 3 R2 S x
l u u u u ds = + =
Copyright & A K-C
. . . 38
( ) ( ){ }2 2 3 3 3 21( )1
( R) 3 R ( R) 3 R2 x
u u n u u n ds = +
( ) ( ){ }
2 3
2 32 3
2 2 3 3 3 21
( ) ( )
2 3
( )
2 2 3 3
1 1( R) ( R)
2 21
( R) 3 R ( R) 3 R2
1/ 2 ( )R( , ) 1/ 2 ( )R( , ) ( ),
P x P x
x x
x
u d u d
u u n u u n ds
u P P x u P P x u x
=
= +
+
i
n 1( )x 1 1x = . , 1J
( )2 3 2 3 3 2 2 3 11 1( ) R 1/ 2 R R R 3 RxJ u u u u a u n d = + + + ( ) ( ){ }2 2 3 3 3 21( )
2 2 3 3
1( R) 3 R ( R) 3 R
21/ 2 ( )R( , ) 1 / 2 ( )R( , ) ( ).
xu u n u u n ds
u P P x u P P x u x
+ +
+ +
2J 3J . 1J , 2J 3J ,
( ) ( )1 1 2 2 3 3
3
( )1
1 1 1( ) ( )R( , ) ( )R( , ) ( )R( , )
3 3 3
1R 2 R R 2 R
2 j j k k k ji ixi
u x u P P x u P P x u P P x
u u n u u n d
=
= + +
+ +
( )3( ) ( )1
1R R R R 3 R R( , ) ( ) .
2j k j k k j j k iix G xiu u u u a u n d x f d
=
+ + + + (27)
, , -
kxu ,
k ix xu ( ) 0x = . , ,
( )l x ( ) 0x = . , , [5].
( ) 0x = 1 2 3( , )x x x= 4C . , 2 3( , )k k x x = . , ( ) 0x =
0 02 3 1 2 3 1
1 2 2 1 3 3 1 2 3
, ,u u u u u u u
l l lx x x x x x x x x
+ = + = + + =
, (28)
/i ix = . |/ iu x , ( , ) 0l n =/ ( ) 0x = . (28) 2 |/ i ju x x .
( )1 1 1 2 1 3 2 2 2 3 3 3 |, , , , ,x x x x x x x x x x x xX u u u u u u = , A ,X b= (29)
2 3 3 2
22 2
23 3
1 2 1 2 2 3 2 2 3
1 3 2 3 1 3 3 2 3
2 2 21 1 2 1 3 2 2 3 3
0 1 0
2 0 1 0 0
0 2 0 0 1A ,
0
0
2 2 2
l l l l l l
l l l l l l
l l l l l l l l l
= +
+
Copyright & A K-C
.. 3-
2015, 7, 2 39
1 1 1|
2 2 20 0 0
23 22 332 2| |2 3 2 3
, , ,x x xb u u ux x x x
=
3 3 31 1
22 2 3 31 1 , 1| | |
, , .ji i ii i i ji i i j
ll lu u ul
x x x x x x x x
= = =
A 4det A= . .
3. (1)(2) , 1, ( )ic a C G , 2 2/ ( )i ia x C G ,
4 ( )kk C ( )l x 4C -
. (27), |/ iu x (28),
2 |/ i ju x x (29). . (1)(2) (27).
(1)
1 1 1|2
| 1 1 |( ) ( )x x xv u u x u x u = ,
1|xu 1 1|x xu i (28) (29). , (1) 1( )D v f=L ,
(2) . 1 2 ( )i ix x
u C G . ,
01 ( ( )f C G x . ,
1( )( ) R( , ) ( )
G xv x x f d =
(1)(2), . ,
1 | ( ) 1( ) ( )
2 ( )
1 1 | ( )( )
RR( , ) ( ) ( , ) ( ) ,
RR( , ) ( ) ( , ) ( )|
i ii
j
i i i ij ik k
x iS x G xi i
P x
x j x ix S xi k kx
vx f d x f d
x x
vx f d x f d
x x x
=
= ==
=
= + +
.
2
1 | ( ) 1( ) ( )
R R( , ) ( ) ( , ) ( ) ,
k kkx kS x G x
i i k
x f d x f dx x x
= +
3 3 ( )1 1
1
2 33
1 | ( ) 1( ) ( )1
RR( , ) ( ) ( , ) ( )
R R( , ) ( ) ( , ) ( ) ,
|i
k kij j
i ii
P x
x ixi j k ii x
x iS x G xi k i j ki
vx x f x x f d
x x x x
x f d x f dx x x x x
== =
==
= +
+
3 ( )
1 11
23
1 | ( ) 1( ) ( )1
R( ) ( ) ( , ) ( )
R( , ) ( )R( , ) ( ) ( )R( , ) ( ) .
|i
k kij j
i ii
P x
x ixii x
i x i xS x G xk ji
D v f x x f dx
x a x x f d D x f dx x
== =
==
= +
+ +
L
L
1( )D v f=L . , | | | 0i i kx x xv v v = = = -, , 1,2,3i k = , . .
3. -
. u(x) v(x) , -
, (3) (3), ( )x xD u D v C D . ,
Copyright & A K-C
. . . 40
( )( ) ( )3
(2) (2)(3) (3) (3)
1
( ) .i ii ix x x x xx x
i
D uv vD u uD v v D u u D v=
= + + + (30)
( )3
* (3) (3)
1
( ) ( ) ,i
x x i xi
v D u u D v vD u uD v a uv=
+ = + +L L
D ,
( ) ( )( ) ( )3 3
(2) (2)(3) *
1 1
( ) ( ) ( ) .i ii ii
x i x xx xxi i
D uv a uv v D u u D v v D u u D v= =
+ = + + + L L (31)
(30)
( ) ( ) ( )
(2) (2) (2)( ) .k j j ki i i x x x xx x x
D uv vD u uD v u v u v= + + +
ix
v v= i .
( ) ( ) ( )
3 3 3(2) (2) (2)(3)
1 1 1
( ) 3 2 .i i ii i ix x x xx x x
i i i
D uv v D u uD v u D v= = =
= + +
(31). , -
, D
( )( ) ( )3 3
(2) (2)(3) *
1 1
( ) ( ) ( ) ( ) .ii i
ix x ix x
xi i
D uv D uv v D u u D v u D v a v= =
= + + L L (32)
( )v x (1), .. 1( ) R( , )v x x x= .
1 (3) ( )xD v C D . * ( ) 0D v =L . x = -
( 1,2,3)i i = 0ix ix .
( )0 030 0 0 0| |1
( )R( , ) ( )R( , ) ( ) ( , ) ( )R( , )i i i ix x x x
i
u x x x u x x x u x R x x u x x x= =
=
0 00
3 30 0
| |1 1,
( )R ( , ) ( )R ( , ) ( )R ( , )|
i
i i i i ij j i ii
i i
x
x ix x xxi i j i
x
u x x x u x x x u x x x d
= = == = =/
=
+ =
( )( )( )0 0 ( )( )3
(2)( )
1
R( , ) ( ) ( ) ( ) R( , ) ( )R( , ) .|
i
ii i i
x x
i ix x xi
x D u d u D x a x d
==
= + L (33)
,
0 0( ) ( )| |
0
( ) ( ) ( );
(0) (0) (0) ,
j j i ii i j j k kx x x x
i j k
x x x
= =
= =
= = = (34)
, ,i j k , 1, 2, 3. , 1R( , )x x (5)(6),
( )0 0
0 00
30 0
( ) 0| |1
3 30
( )|1 1,
( ) ( )R( , ) ( )R( , ) R( , )
( )R ( , ) ( )R ( , )|
i i i i
i
i ii i j ji
i i
i i i ix x x xi
x
i i j j ix x xxi j j i
x
u x x x x x x x x x
x x x x x d
= ==
= == = =/
=
= +
+
( )( )0 00( )( )3
(2)( ) ( )
|1
( ) R( , ) ( )R( , ) ( )R( , ) .iii i i
x x
i i i ix xxi
D x a x d f x d
==+ + + (35)
, .
Copyright & A K-C
.. 3-
2015, 7, 2 41
4. , , , / ( )i i if c a a x C D , ( 1,3)i = , ( )( )i ix , -
(34), ( )
(2) ( )i ix
D C D ,
, (35).
. , (3) ( )xD u C D , (35). . - , (35) , . , ( )u x , (35), (1) - (3). . (5)(6) - (34), , (35)
0i ix x= , , , ( )( )i ix . -
, (3) .
00( ) ( )R( , ) .x
xu x f x d = (36)
, 0( ) ( ) ( )D u x f x=L . ,
( )0 0( )( )
3(2)(3) (3)
0 | ( )1
( ) ( ) ( ) R( , ) ( ) R( , ) ,ii iii
x x
x x i xxx xi
D u x f x f D x d f D x d ==
= + +
( )0 0( )
0 | ( )( ) ( )R( , ) ( ) R( , ) ,i
i i i ii
x x
x x i xx xD u x f x d f D x d == +
, (16),
00( ) ( ) ( ) ( ) ( )R( , ) .x
xxD u x f x f D x d = + L L
2, . 1 0( ) ( ) ( )u x u x u x= . , 1( ) ( ) 0D u x =L . , 1( )u x (35), ( ) 0f x = .
0 0
0
( ) |( ) ( ) , ( ) ( ) ,
|i i k kj j
i i i ix x x x
x x
x u x x u x = =
=
= =
, ,i j k 1, 2, 3, (3) (33)
0 ( ) ( ) ( ) .x
xf D u d L
, , (33),
( ) ( ) ( )0 ( ) ( )3 3
(3) (2) (2)1 1 1
1 1
( )R( , ) ( ) ( , ) ( ) R( , ) ( )R( , ) 0.( )ii i
i
x
ixi i
D u x x D u x R x u x D x a x d
= = + + =
1x x= , 0x x= , 1R( , ) ( )x x v x= -
(3)xD .
( ) ( )( )( ) ( )3 3
(2) (2)(3)1 1 1
1 1
( ) ( ) 0.ii i
ix x ix x
xi i
D u v D u v u D v a x v
= = + + =
(32) 1( ) R( , )v x x x= , 1( ) 0v D u =L . ( ) 0v x / , 1( ) 0D u =L . . 3. 0( )u x (36)
. . ( ) 0ia x = 1,2,3i = ( )c x =
0(3)
( )| ( )( ), ( ) ( ), 1,2,3.
ix i iS x
D u u f x u x x i + = = =
(7).
Copyright & A K-C
. . . 42
,! ,! ,!1 1 2 2 3 3
0
R ( , ) ( ) ( ) ( ) ( ) .n n n n
n
x x x x
==
(35), [6] ( )U D ( )V D [6, 4.13] C
(1,1,1) = ,
( ) ( )0 03 3
( ) 0 ( ) 01 1
( ) ( ) ( ) ( ) ( ) R ( , ) ( )R ( , ) ,x x
i i i i i i i ix xi i
u x x x x d f x d = =
= + + +
( )i ix (34). - [1]. [6], [7, 8] [9], [10].
1. , .. -
- : . - .-. / .. . , 2001. 213 c.
2. , . / . . .: , 1934. 330 c. 3. , .. / .. // -
. : - . , 1990. C. 9498. 4. , .. / .. . .: , 1976.
296 c. 5. , .. / .. . .: ,
1976. 436 c. 6. , .. / .. . : -
, 2014. 452 . 7. Karachik, V.V. On some special polynomials / V.V. Karachik // Proceedings of the American
Mathematical Society. 2004. Vol. 132, no. 4. P. 10491058. 8. , .. -
I / .. // . -. . . 2011. . 4. 10(227). . 417.
9. , .. / .. , .. // . 2013. . 49, 2. C. 250254.
10. , .. / .. // . 2012. . 52, 2. . 237252.
24 2014 .
Copyright & A K-C
.. 3-
2015, 7, 2 43
Bulletin of the South Ural State University
Series Mathematics. Mechanics. Physics 2015, vol. 7, no. 2, pp. 3143
CAUCHY AND GOURSAT PROBLEMS FOR DIFFERENTIAL EQUATION OF THIRD ORDER V.V. Karachik 1
Cauchy and Goursat problems for the hyperbolic equation of third order are considered. The theo-rem of the existence of the Riemann function is proved and on the basis of the aforementioned function solutions for the Cauchy and Goursat problems are introduced.
Keywords: Cauchy and Goursat problems; hyperbolic equation of third order; Riemann function.
References 1. Karachik V.V. Razrabotka teorii normirovannyh sistem funkcij i ih primeneniya k resheniyu
nachal'no-kraevyh zadach dlya uravnenij v chastnyh proizvodnyh. Diss. dokt. fiz.-mat. nauk (The devel-opment of the theory of normed systems of functions and their application to the solution of the initial-boundary value problems for equations with differential derivatives. Dr. phys. and math. sci. diss.). Tashkent, 2001. 213 p.
2. Munz G. Integralnye uravneniya (Integral equations). oscow, GTTI Publ., 1934. 330 p. 3. Zhegalov V.I. Trechmerniy analog zadachi Gursa (Three-dimensional analogue of the Goursat
problem). Neklasicheskie uravneniya i uravneniya smeshannogo tipa (Non-classical equations and the equations of mixed type). Novosibirsk, Institut matematiki SO AN SSSR Publ., 1990. pp. 9498.
4. Bitsadze A.V. Uravneniya matematicheskoy fiziki (The equations of mathematical physics). Mos-cow, Nauka Publ., 1976. 296 p. (in Russ.).
5. Vladimirov V.S. Uravneniya matematicheskoy fiziki (The equations of mathematical physics). Moscow, Nauka Publ., 1981. 512 p. (in Russ.).
6. Karachik V.V. Metod normirovannyh sistem funkcij (Method of normed systems of functions). Chelyabinsk: Izdatel'skij centr YuUrGU, 2014. 452 p.
7. Karachik V.V. On some special polynomials. Proceedings of the American Mathematical Soci-ety. 2004. Vol. 132, no. 4. pp. 10491058.
8. Karachik V.V. Polinomial'nye resheniya differencialnykh uravnenij v chastnykh proizvodnykh s postoyannymi koefficientami I [Polynomial solutions to partial differential equations with constant coef-ficients I]. Bulletin of South Ural State University. Series of Mathematics. Mechanics. Physics. 2011. Issue 4. no. 10(227). pp. 417.
9. Karachik V.V., Antropova N.A. Polynomial solutions of the Dirichlet problem for the biharmonic equation in the ball. Differential Equations. 2013. Vol. 49, no. 2. pp. 251256.
10. Karachik V.V. Method for constructing solutions of linear ordinary differential equations with constant coefficients. Computational Mathematics and Mathematical Physics. 2012. Vol. 52, no. 2. pp. 219234.
Received 24 December 2014
1 Karachik Valeriy Valentinovich is Dr. Sc. (Physics and Mathematics), Professor, Mathematical and Functional Analysis Department, South Ural State University. E-mail: [email protected]
Copyright & A K-C
. . . 44
669.24.056.9:004+539.211
(111) .. 1, .. 2
- Ni . - Ni(111), : , , . Fe .
: ; ; ; ; ; ; .
, , . - , , -, . , . , , - () . - . [1] , , - - Ni, Fe, Co. () ( , ). . , - , - . [2], - Ni Fe , [3].
- Ni20xFex, , - .
ab-initio - VASP (Vienna ab initio simulation program), [4, 5].
- (DFT) PAW, - PBE (). 500 eV.
1 - , , , - . 2 , - , , - . E-mail: [email protected]
Copyright & A K-C
.., .. (111)
2015, 7, 2 45
(. 1), 2 2 5 (5 4 ). , - , . 10 , - . K- 9 9 1.
- 105 - , , 103 /. - .
Fe Ni . - VESTA [6].
) ; )
. 1. Ni (111)
Ni(111), 9 20 . . 1.
1 Ni11Fe
Fe
Fe
, ,
13 107,35 0,0014 107,35 0,0015 107,35 0,0016 107,35 0,009 107,27 0,08
10 107,27 0,0811 107,27 0,0812 107,27 0,0817 107,16 0,1918 107,16 0,1919 107,16 0,1920 107,16 0,19
13, 14, 15, 16. ( 912). ( 1720) . I 13 (. 2, ).
Copyright & A K-C
. . . 46
I 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20 - . , , . 2.
Fe ( 11), 13. II . () 9, 10, 12 18. - ( 17, 19, 20).
) x = 1; ) x = 2; ) x = 3. . 2. Ni12xFex
Ni , Fe
2
Ni10Fe2 Fe
( Fe 13)
Fe
, ,
11 110,48 0,00 18 110,32 0,16 9 110,31 0,17
10 110,31 0,17 12 110,31 0,17 14 110,29 0,19 15 110,29 0,19 16 110,29 0,19 17 110,14 0,34 19 110,14 0,34 20 110,14 0,34
-
(. 3), 11 , . - , II ( Fe 11 13) - .
Copyright & A K-C
.., .. (111)
2015, 7, 2 47
3 Ni10Fe2
Fe ( Fe 11)
Fe
, ,
13 110,48 0,00 14 110,31 0,17 15 110,31 0,17 16 110,31 0,17 17 110,21 0,27 19 110,21 0,27 20 110,21 0,27 18 110,19 0,29 9 110,11 0,37
10 110,11 0,37 12 110,11 0,37
Fe - II. . 4.
, III (. 2, ), 18 ( ), - . 9, 10, 12.
4 Ni9 Fe3
Fe
Fe
, ,
18 113,42 0,00
14 113,30 0,12
15 113,30 0,12
16 113,30 0,12
17 113,25 0,17
19 113,25 0,17
20 113,25 0,17
9 113,20 0,22
10 113,20 0,22
12 113,20 0,22 I, II III -
(. 5): , , (- ), .
5
- , %
23 34 45
,
,
,
EF,
Ni(111) 0,32 0,22 1,45 8,09 0,64 4,69 3,02
Ni11Fe 0,03 0,73 0,57 8,10 0,58 4,65 2,97
Ni10Fe2 0,24 0,14 1,08 8,12 0,51 4,63 2,90
Ni9Fe3 0,08 0,86 0,90 8,13 0,47 4,70 2,83
Copyright & A K-C
. . . 48
. , I, II, III, , . 4 5. , - 8,09 8,13 .
0,64 ( Ni(111)) 0,47 ( III). . - Ni Fe . 4,69 ( ), 4,63 ( II), - 4,70 ( III).
I, II III (. 3). - . - Fe 1,8; 0,8; 6,9 - 1,4; 0,1; 0,8; 7,0 7,7 .
) ; ) . . 3.
Fe: Ni (111), Fe
- Ni (111) Fe, - . Fe . , - I II III, , .
III, Fe, .
.. , .. - .
1. , -
/ .. , .. , .. , .. // -. , . 2004. 3. . 7073.
2. , .. (001) (111) / .. , .. , .. // . 2011. 53. . 3. . 558563.
3. Mueller, J.E. Structures, Energetics and Reaction Barriers for CHx Bound to the Nickel (111) Surface / J.E. Mueller, A.C.T. van Duin, W.A. Goddard, III // J. Phys. Chem. C. 2009. Vol. 113, 47. P. 2029020306.
Copyright & A K-C
.., .. (111)
2015, 7, 2 49
4. Kresse, G. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set / G. Kresse, J. Furthmller // Phys. Rev. B. 1996. Vol. 54. Issue 16. P. 11169.
5. Kresse, G. From ultrasoft pseudopotentials to the projector augmented-wave method / G. Kresse, D. Joubert // Phys. Rev. B 1999. Vol. 59. Issue 3. P. 1758.
6. Momma K. VESTA 3 for three-dimensional visualization of crystal, volumetric and morphology data / K. Momma, F. Izumi // Journal of Applied Crystallography. 2011. Vol. 44. Issue 6. P. 12721276.
11 2015 .
Bulletin of the South Ural State University Series Mathematics. Mechanics. Physics
2015, vol. 7, no. 2, pp. 4449 AB INITIO MODELING OF SEGREGATION OF IRON ATOMS ON THE (111) NICKEL SURFACE G.P. Viatkin 1, S.I. Morozov 2
The paper is devoted to computer modeling of surface alloys Ni-based ab initio methods. Obtained surface model alloys Ni (111), calculated their physical properties: relaxation, surface energy, work function. Effect of location of the Fe atoms on these characteristics is studied.
Keywords: computer modeling; ab initio methods; surface; nickel; iron; segregation; relaxation.
References 1. Novikova A.A., Kiseleva T.Yu., Tarasov B.P., Muradyan V.E. Issledovanie mikrostruktury
uglerodnogo nanomateriala, poluchennogo na zhelezo-nikelevom katalizatore (A Study of Carbon Nanomaterial Microstructure, Produced on Iron-Nickel Catalyst). Poverkhnost'. Rentgenovskie, sinkhro-tronnye i neytronnye issledovaniya. (A Study of Carbon Nanomaterial Microstructure, Produced on Iron-Nickel Catalyst). 2004. no. 3. pp. 7073. (in Russ.).
2. Mutigullin I.V., Bazhanov D.I., Ilyushin A.S. Effect of coverage by carbon on the possibility of forming an interstitial solid solution in Fe(001) and Fe(111) subsurface layers. Physics of the Solid State. 2011. Vol. 53, no. 3. pp. 599605.
3. Mueller J.E., van Duin A.C.T., Goddard W.A., III Structures, Energetics and Reaction Barriers for CHx Bound to the Nickel (111) Surface. J. Phys. Chem. C. 2009. Vol. 113, no. 47. pp. 2029020306. DOI: 10.1021/jp810555y.
4. Kresse G., Furthmller J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B. 1996. Vol. 54. Issue 16. p. 11169. http://dx.doi.org/10.1103/PhysRevB.54.11169.
5. Kresse G., Joubert D. From ultrasoft pseudopotentials to the projector augmented-wave method. Phys. Rev. B. 1999. Vol. 59. Issue 3. p. 1758 . http://dx.doi.org/10.1103/PhysRevB.59.1758.
6. Momma K., Izumi F. VESTA 3 for three-dimensional visualization of crystal, volumetric and morphology data. Journal of Applied Crystallography. 2011. Vol. 44. Issue 6. pp. 12721276. DOI: 10.1107/S0021889811038970.
Received 11 March 2015
1 Viatkin German Platonovich is a Corresponding Member of Russian Academy Science, Dr. Sc. (Chemistry), Professor, General and Theoreti-cal Physics Department, South Ural State University. 2 Morozov Sergey Ivanovich is Cand. Sc. (Physics and Mathematics), Associate Professor, General and Theoretical Physics Department, South Ural State University. E-mail: [email protected]
Copyright & A K-C
. . . 50
519.87:577.3+544.1
.. 1
, , . . - , .
: ; ; ; ; - ; .
60- XX - [1]. ( ) . - - , . [1] -, , - . , , - [2]. - , - , [3, 4]. - [5].
. - , -. - , .
[6] - ( P) :
22
1 2 3 4 22 ,X
X Xk AY k XY k BX k X D
t = + +
2
1 2 5 2,Y
Y Yk AY k XY fk Z D P
t = + + +
(1)
2
3 5 2,Z
Z Zk BX k Z D
t = +
X 2HBrO , Y Br , Z -
IV, DX, DY, DZ , f -.
. (1) - :
21 0 2 0 0 3 0 4 02 0,k AY k X Y k BX k X + = 1 0 2 0 0 5 0 0,k AY k X Y fk Z P + + = 3 0 5 0 0,k BX k Z = (2)
X0, Y0, Z0 , k1, k2, k3, k4, k5 . [7]: A = B = 6102, k1 = 810
9, k2 = 2109, k3 = 2,1, k4 = 410
7, k5 = 104
1 , - , , - . E-mail: [email protected]
Copyright & A K-C
..
2015, 7, 2 51
(. 1). (2) - Matlab , .
: 1. , -
, (1). 2.
X0, Y0, Z0. 3. -
(. 2). 1
f X0, 108 Y0, 10
17 Z0, 1015
0,15 0,25 0,35 0,45 0,55 0,65 0,75 0,85 0,95 1,05 1,15 1,25 1,35
0,1811 0,1969 0,2126 0,2284 0,2441 0,2599 0,2756 0,2914 0,3071 0,3229 0,3386 0,3544 0,3701
0,0071 0,0129 0,0195 0,0270 0,0352 0,0443 0,0543 0,0650 0,0766 0,0890 0,1022 0,1163 0,1312
0,2282 0,2481 0,2679 0,2878 0,3076 0,3274 0,3473 0,3671 0,3870 0,4068 0,4267 0,4465 0,4664
2
P = 0,8
f X0, 104 Y0, 10
8 Z0, 1010
0,15 0,25 0,35 0,45 0,55 0,65 0,75 0,80 0,85 0,95 1,05 1,15 1,25 1,35
0,99959264 0,99959343 0,99959421 0,99959500 0,99959579 0,99959657 0,99959736 0,99959775 0,99959815 0,99959893 0,99959972 0,99960051 0,99960129 0,99960208
0,16659767 0,16659793 0,16659819 0,16659845 0,16659872 0,16659898 0,16659924 0,16659937 0,16659950 0,16659977 0,16660003 0,16660029 0,16660055 0,16660081
0,12594867 0,12594877 0,12594887 0,1259
