вестник южно уральского-государственного_университета._серия_математика._механика._физика_№1_2013

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( . ),

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( . ),

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. .- . ., . .,

. . ., . .,

. . ., . .,

. . ., . .

2009 .

77-26455 13 2006 .

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-. -

«Ulrich’s Periodicals Directory».

- «

-, -

-».

29211 -« ».

– 2 .

Copyright ОАО «ЦКБ «БИБКОМ» & ООО «Aгентство Kнига-Cервис»

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.............................26 . . .........34 . ., . . -

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, II........................................................................................55 . ., . .

................................................................................................................................63 . . - -

...................................................................69

. ., . . Ab initio -- .....................................................................................................................76

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............................................................................................................................................88 . ., . ., . ., . ., . .

- -.............................................................................................................................................95

. ., . ., . . -............................................................................................100

- . . mp .....................................................................107

. ., . ., . ., . ., . ., . . (Y1–xNdx)3Al5O12 ...............................110

. ., . . –..................................................................................................114

© , 2013


3

CONTENTS Mathematics AZOV D.G. Estimation of bijective projection area of a surface with negative curvature ................. 4 AKIMOVA A.A. Classification of knots in the thickened torus with minimal diagrams which are not in a circule and have five crossings ............................................................................................... 8 HERREINSTEIN A.V., KHAYRISLAMOV M.Z. Explicit difference scheme for the solution of one-dimensional quasi-linear heat conductivity equation.................................................................... 12 ZIMOVETS A.A. A boundary layer method for the construction of approximate attainability sets of control systems ................................................................................................................................ 18 KAMALTDINOVA T.S. Approximate solution of inverse boundary problem for the heat conduc-tivity equation by nonlinear method of projection regularity .............................................................. 26 KARACHIK V.V. On one generalized mean theorem for harmonic functions .................................. 34 LEBEDEV P.D., USHAKOV V.N. A variant of a metric for unbounded convex sets....................... 40 MENIKHES L.D. On connection between sufficient conditions of regularizability of integral equa-tions...................................................................................................................................................... 50 PASIKOV V.L. Extreme strategies in game-theory problems for linear integral differential Volterra systems, II.............................................................................................................................. 55 PATRUSHEV A.A., PATRUSHEVA E.V. A variant of the solution of Markushevich boundary problem ................................................................................................................................................ 63 YULDASHEV . . Inverse problem for nonlinear integral differential equation with hyperbolic operator of a high degree ..................................................................................................................... 69

Physics VERKHOVYKH A.V., MIRZOEV A.A. Ab initio modeling of the grain boundary formation en-ergy in BCC iron.................................................................................................................................. 76 GROMOV V.E., RAYKOV S.V., SHERSTOBITOV D.A., IVANOV Yu.F., KHAIMZON B.B., KONOVALOV S.V. Analysis of carbon dissolution in titanium under electron beam treatment ...... 82 RECHKALOV V.G., BESKACHKO V.P. Simulation of experiments to measure surface tensionby the shape of a drop surface at the existence of irregularity in its hanger or bearing....................... 88 SOZYKIN S.A., SOKOLOVA E.R., TELNOY K.A., BESKACHKO V.P., VYATKIN G.P. Quan-tum-chemical modeling of deformation processes of chiral carbon nanotubes ................................... 95 SHEVYAKOV I.A., TAMBOVCEV V.I., KUCHURKIN A.A. Radio physical properties of colli-sional plasma in gas discharge............................................................................................................. 100

Short communications

AL-DELFI J.K. Quasi-Sobolev spaces mp .......................................................................................... 107

RYBINA E.N., BRYZGALOV A.N., SVIRSKAYA L.M., VIKTOROV V.V., VOLKOV P.V., ZHIVULIN D.E. The magnetic properties of solid solutions (Y1–xNdx)3Al5O12.................................. 110CHIRKOV P.V., MIRZOEV A.A. Ineratomic potential for iron-carbon system and martencitic phase transition problem...................................................................................................................... 114


. « . . » 4

514.772

. . 1

, . -

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( , )z f x y= (1) ( , )K x y . ,

2( , ) 0,x y α≤ − < (2) (1) .

. . [1]: 0 0a > , 2C - ( , )f x ya (1) (2), 0 /a a α≤ . . [2]

, . . : 0 0r > , 2C - (1) (2) -

r , 0 /r r α< . . , 0 3r e≤ . : 0 0,5r ≥ . 0r 0a . -

[3] . . , - (1). [4–7].

2 2 2 2( , )(1 ) ,xx yy xy x yz z z K x y z z− = + + (3) . . . : -

– (3) 2C - 0 /r r α>

0 /a a> α , ( , )K x y (2).

, (1). 1. 2( , )z z x y C= ∈ ( , ) 0K x y < -

2 2 2.x y R+ ≤ 0C > , ,

2 2 2

, 0 4, 0,( , )

m

x y r

dxdy Cr m rK x y

+ ≤

≤ < < > (4)

2 12 44 3 , 2;

2

3 , 2.

m mm C mR

Ce m

π

π

− −− ⋅ ≠≤

=

(5)

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2013, 5, 1 5

1. . . :

( ) ( )2 2 2

2 22 2 2

0 0

1 ( , ) ( , ) 2 ( ) .xx yy xyx y r

d z d z d z z z dxdydr r

π π

ϕ ρρ ϕ ϕ ρ ϕ ϕ+ ≤

= − − (6)

2( , ) ,z x y C∈ ( , ) ( cos , sin ),z zρ ϕ ρ ϕ ρ ϕ= ,ρ ϕ – .

( )2

22

0 0

1( ) 1 ( , )r

g r d z dπ

ϕρ ρ ρ φ ϕρ

= + .

(0) 0,g = 2( )g r rπ≥ ( ) 0g r′ > 0 .r R< < ( )g r , :

( )2 2 2 2 2 2 2 2 2

222 2 2 2 2( ) (1 ) ( , ) 1 .

( , )x y x yx y r x y r x y r

dxdyg r z z dxdy K x y z z dxdyK x y

+ ≤ + ≤ + ≤

≤ + + ≤ ⋅ + + (7)

(4), (6) (7), 22 ( )( ) .m

g rg rCr

′′ ≥ (8)

(8) (0, )Rρ ∈ , 32 22( ) ( ) .

3

m

g g rC

ρ ρ− −

′ ≥

rρ → r 1R 2R 1 2(0 )R R R< < < 2,m ≠

2 21 12 22 2

1 2 2 12( ) ( ) ( ).

(2 ) 3

m m

g R g R R Rm C

− −− −

− ≥ −−

2( ) ,g r rπ≥2 2

2 22 1

1

1 2 .(2 ) 3

m m

R RR m Cπ

− −

≥ −−

(9)

2 4.m< < (9) 2R R , 22

221

2 32

mm m CR Rπ

−− −≥ + . (10)

(10)

14

13 mCRπ

−= -

22(4 )4 3

2

mmm C

π

−−− .

222(4 )2 4 3

2

mmmm CR

π

−−−−≥ , ,

2 12 44 3 .

2m mm CR

π− −−≤ (11)

0 2m< < (10) : 22

221

2 3 .2

mm m CR Rπ

−− −≤ +


. « . . » 6

143 .mCR

π−

= (11).

2m = (9) 2

11

1 1 ln .3

RRR Cπ

≥

11

3 1ln lnCR RRπ

≤ + 3ln 1 ln .CRπ

≤ +

3 .CR eπ

≤ (12)

(12) (11) 2.m→ 1 .

1. 2( , ) 0,K x y α≤ − <2 2 2

2

2( , )x y r

dxdy rK x y

πα+ ≤

≤ 2m = (5)

. 3 .eRα

≤

2. 1 , (4) 0 ,r r≥ 0r – . 0.r r>

0r , .C 3. (4) m , ,

. , ( )2 2

1( , ) .1

nK x yx y

= −+ +

1n <2 2 2

,( , )

m

x y r

dxdy CrK x y

+ ≤

≤ 0 4.m< < 1

. 1n >4m > , . -

, 2 11 (1 ) ,nA R −= + + 2 10

1 1(1 )

r

nz dtA t −= −− +

, 2 2 ,r x y= + -

R( )2 2

1( , ) .1

nnK x y

x y

−=+ +

–.

2 ( , , , , ).xx yy xy x yz z z F x y z z z− = (13)

( )22 2( , , , , ) ( , ) 1 ,x y x yF x y z z z K x y z z≤ ⋅ + + ( , ) 0K x y < – .

2. . 2. (13) 2C - 2 2 2x y R+ ≤

( , )K x y (4), R (5). 2 1.

1. , . . / . . . – .: . , 1953. – 640 .

2. Heinz, E. Über Flachen mit eineindeutiger Projektion auf eine Ebene, deren Krümmungen durch Ungleichungen eingeschränkt sind / E. Heinz // Math. Ann. – 1955. – V. 129, 5. – P. 451–454.


�� .�. ��

2013, 5, 1 7

3. , . . – / . . // . – 1976. – . 100(142), 3(7). –

. 356–363. 4. , . . – / . . // -

. – 1983. – . 38, 1. – . 153–154. 5. , . . -

/ . . // . – 1985. – . 28. – . 19–21.

6. , . . n- - / . . // . – 1994. – 1(2).

– . 12–17. 7. , . . . n- -

/ . . // . – 1985. – 5. – . 72–74.

ESTIMATION OF BIJECTIVE PROJECTION AREA OF A SURFACE WITH NEGATIVE CURVATURE

D.G. Azov1

The article deals with a surface of negative Gaussian curvature which is bijectively projected onto a circle. It provides sufficient conditions of existence of an estimate for the circle radius.

Keywords: surfaces with negative Gaussian curvature, hyperbolic Monge–Ampere equation, esti-mation of bijective projection area.

References 1. Efimov N.V. Issledovanie polnoj poverkhnosti otricatelnoj krivizny [Research of a complete sur-

face with negative curvature]. Dokl. AN SSSR. 1953. 640 p. (in Russ.). 2. Heinz E. Über Flachen mit eineindeutiger Projektion auf eine Ebene, deren Krümmungen durch

Ungleichungen eingeschränkt sind. Math. Ann. 1955. Vol. 129, no. 5. pp. 451–454. 3. Efimov N.V. Ocenki razmerov oblasti regulyarnosti reshenij nekotorykh uravnenij Monzha–

Ampera [Estimation of the range of domain of regularity for the solution of Monge–Ampere equations]. Matematicheskij sbornik. 1976. Vol. 100(142), no. 3(7). pp. 356–363. (in Russ.).

4. Azov D.G. On a class of hyperbolic Monge–Ampère equations. Russian Mathematical Surveys.1983. Vol. 38. p. 170. DOI:10.1070/RM1983v038n01ABEH003390

5. Brys'ev A.B. Ocenka oblasti regulyarnosti reshenij nekotorykh nelinejnykh differencial'nykh neravenstv [Estimation of the range of domain of regularity for the solution of nonlinear differential ine-quality]. Ukrainskij geometricheskij sbornik. 1985. Issue. 28. p. 19–21. (in Russ.).

6. Azov D.G. Vestnik Chelyabinskogo gosudarstvennogo universiteta. 1994. no. 1(2). pp. 12–17. (in Russ.).

7. Azov D.G. Vestnik Moskovskogo universiteta. 1985. no. 5. pp. 72–74. �� !" # $%&"'( ) 7 �� 2012 �.

1 Azov Dmitry Georgievich is Associate Professor, Department of General Mathematics, South Ural State University. E-mail: [email protected]


. « . . » 8

515.162.3

,

1

. . 2

T×I, -.

�� : ��, � �� , �� .

F×I, F – -, .

S3 T×I, T = S1 ×S1 – . T×I -

, . -: . -

[1, 2] - [3] . [4].

, [5], . . . -

.

( . . ) . K ⊂ T×I -

G ⊂ T 4, « » , . K -

( ), -, – t ⊂ I. -

, . -,

. 1. K , ( -

) , K. G ⊂ T , -

. , . . , -

. , - T×I,

[6]. , , - T. ,

, [3]. -, , .

. 1. 69 T×I, -

. . 1. 1 -

, [5]. - 5 , – 5 ,

, – . - 1 �� !�� "��#$��! %#��&�' ())� 12-01-00748, %#��&�' *+-1414.2012.1 "� %��,��#�&��- "��#$.� ��,/�0��,1�!0 2.��, � &�.$� %#��&�' 12-3-1-1003/2 45* (6*. 2 6.�'�� 6�7�� 6��#�� – �&,��&, 8$��-9#��:�.�- %��,��#�&��!- ,��#��&�&. E-mail: [email protected]


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2013, 5, 1 9

. , T×I . [7, 8].

, T×I, , -.

1. 6 -, ( . 2).

1 - 2 [5],

n 4 . 2. 34 T×I,

5 , ( . 3).

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. « . . » 10

2 2 [5], n 4 .

1 , 2 ( . . 3), T, -

. n 2n, 32 . , -

[5]. , . , , -. ,

, , .

[7, 8]. : 3 (K) ( ) ( ) 2 2 ( ) ( )(K) ( ) ( )s s s s

sX a a a a xω α β γ δ− − −= − − − ,

(s) (s) – A B s, (s) (s) – -, ,

s. , , ω(K) . . , , . 1,

. 34 , - 2.

1. 5 : 521, 522, 556–558. 2. 13 : 56, 512, 513, 517, 521 , 531, 544, 546,

549, 551, 553, 556, 568. 3. , , 6.

53 57.

��. 3. �� 5 �� T, �� . �� T ��


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2013, 5, 1 11

1. , . . RP3

– / . . // . – 1990. – T. 2, 3. – . 171–191. 2. Drobotukhina, Yu.V. Classification of links in RP3 with at most six crossings

/Yu.V. Drobotukhina // Advances in Soviet Mathematics. – 1994. – V. 18, 1. – P. 87–121. 3. Gabrovshek, B. Knots in the solid torus up to 6 crossings / B. Gabrovshek, I.M. Mroczkowski //

J. Knot Theory Ramifications. – 2012. – V. 21. – 1250106. [43 .] DOI: 10.1142/S02182165125010644. Enumerating the k-tangle projections / A. Bogdanov, V. Meshkov, A. Omelchenko, M. Petrov //

J. Knot Theory Ramifications. – 2012. – V. 21, 7. – 1250069. [17 .] DOI: 10.1142/S0218216512500691

5. , . . / . . , . . // . : ,

, . – 2012. – . 12. – . 3. – C. 10–21.6. Matveev, S.V. Prime decompositions of knots in T×I / S.V. Matveev // Topology and its Applica-

tions. – 2012. – V. 159, 7. – . 1820–1824. DOI: 10.1016/j.topol.2011.04.022 7. Kauffman, L. State models and the Jones polynomial // Topology. – 1987. – V. 26, 3. –

P. 395–407. 8. , . . , , / . . ,

. . . – .: , 1997. – 352 .

CLASSIFICATION OF KNOTS IN THE THICKENED TORUS WITH MINIMAL DIAGRAMS WHICH ARE NOT IN A CIRCULE AND HAVE FIVE CROSSINGS

A.A. Akimova1

We compose the table of knots in the thickened torus T×I with minimal diagrams which are not in a circle and have five crossing intersections.

Keywords: knot, thickened torus, knot table.

References 1. Drobotukhina Yu.V. Analog polinoma Dzhonsa dlya zaceplenij v RP3 i obobshhenie teoremy

Kaufmana–Murasugi [An analogue of the Jones polynomial for links in RP3 and a generalization of the Kauffman-Murasugi theorem]. Leningrad Mathematical Journal. 1991. Vol. 2, Issue 3. pp. 613–630.

2. Drobotukhina Yu.V. Classification of links in RP3 with at most six crossings. Advances in Soviet Mathematics. 1994. Vol. 18, no. 1. pp. 87–121.

3. Gabrovshek B., Mroczkowski I.M. Knots in the solid torus up to 6 crossings. J. Knot Theory Ramifications. 2012. Vol. 21. 1250106. [43 pages] DOI: 10.1142/S0218216512501064.

4. Bogdanov A., Meshkov V., Omelchenko A., Petrov M. Enumerating the k-tangle projections // J. Knot Theory Ramifications. 2012. Vol. 21, no. 7. 1250069. [17 pages] DOI: 10.1142/S0218216512500691

5. Akimova A.A., Matveev S.V. Klassifikaciya uzlov maloj slozhnosti v utolshhennom tore [Classification of Low Complexity Knots in the Thickened Torus]. Vestnik Novosibirskogo gosu-darstvennogo universiteta. Seriya: Matematika, mekhanika, informatika. 2012. Vol. 12. Issue. 3. pp. 10–21. (in Russ.).

6. Matveev S.V. Prime decompositions of knots in T×I. Topology and its Applications. 2012. Vol. 159, no. 7. pp. 1820–1824. DOI: 10.1016/j.topol.2011.04.022

7. Kauffman L. State models and the Jones polynomial. Topology. 1987. Vol. 26, no. 3. pp. 395–407.

8. Prasolov V.V., Sosinskij A.B. Uzly, zacepleniya, kosy i tryokhmernye mnogoobraziya [Knots, linkage, braids and three-dimensional manifold]. Moscow: MCNMO, 1997. 352 p. (in Russ.).

�� 24 �� 2012 �.

1 Akimova Alyona Andreevna is Student, South Ural State University. E-mail: [email protected]


. « . . » 12

519.633

. . 1, . . 2

-, . -

- – .

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, [1, 2], -.

- [3]:

( )

( )

0

( ) ( ) , 0 , 0 ,

( , 0) ( ),

( ) ( (0, )) (0, ) ,

( ) ( ( , )) ( , ) ,

l l lx

r r rx L

u uu q u t T x Lt x x

u x x

uq u u t u t Qx

uq u u L t u L t Qx

ϕ

λ θ

λ θ

=

=

∂ ∂ ∂= < ≤ < <∂ ∂ ∂=

∂− = − +∂

∂ = − +∂

(1)

( , )u u x t= – ( ), 0 , 0x L t T≤ ≤ ≤ ≤ ; L – ; T – ; ( )u – ;

( )q u – ; ( )xϕ – -; ( )l uλ – ;

( )r uλ – ; lθ – ; rθ – ; lQ – -

; rQ – . ( )c u= , ( )q q u= , ( )l l uλ λ= ( )r r uλ λ=

, .

( ) uq ux∂∂

, :

0

( ) ( )u

G u q dξ ξ= .

G2

22( )G Ga u

t x∂ ∂=∂ ∂

, (2)

1 �� !�"#$% &#'%()�*%+ – $,-��, "#�$%$#� .%/%",-0#��0#�%+�'"%1 �#2", "#.�$�# 3�%"(#$�, 0#��0#�%"%, 45�,-6�#()'"% 7,'2$#�'�*��8 2�%*��'%��. 2 9# �%'(#0,* :%1#%( ;%�#�2((#�*%+ – #'3%�#��, "#.�$�# 3�%"(#$�, 0#��0#�%"%, 45�,-6�#()'"% 7,'2$#�'�*��8 .#"2()��. E-mail: [email protected]


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2013, 5, 1 13

( )( )( )

q ua uc u

= – .

( )G u .

( , )x t [2]

1, , 1, 2, ..., ,2h h h ix i h i Nτ τω ω ω ω= × = = − =

{ , 0, 1, ...},jt j jτω τ= = =

LhN

= – x , τ – t . -

. 1( ) ,2iG t G i h t= − – G 1

2x i h= − . -

(2) 1 ,2

i h t− , -

:

2 1 12

( ) ( ) 2 ( ) ( )( )i i i ii

dG t G t G t G ta udt h

− +− += , [0; ]t τ∈ , 2, ..., 1i N= − . (3)

1( )iG t− 1( )iG t+ :

11 1( ) (0) (0)i

i idGG t G t

dt−

− −≈ + , 11 1( ) (0) (0)i

i idGG t G t

dt+

+ +≈ + .

(3) 2 2

1 11 12 2

( ) 2 ( ) ( )( ) (0) (0) (0) (0)i i i i ii i i

dG t a u a u dG dGG t G G tdt dt dth h

− +− ++ = + + + . (4)

2

2Ohτ , 1 (0)idG

dt−

1 (0)idGdt+ .

(4) 2

22 ( )

( ) ( (0) )ia u t

hi iG t G B e At B

−= − + + , [0; ]t τ∈ ,

1 11 (0) (0)2

i idG dGAdt dt− += + ,

21 1

2(0) (0)

2 2 ( )i i

i

G G hB Aa u

− ++= − ⋅ .

�� !" #�$"!%&"!' %()*+

1ix −0 ix 1ix +

0t

h 2h ih Nh L=x

t

1 2 N 1N +

0t τ+

12

iG+

τ

0t τ−

12

iG−

iG1iG − 1iG +


. « . . » 14

G (+1), – (–1), -

– 12

+ , –

12

− ( . ). 1 (0)idGdt−

1 (0)idGdt+ :

1 12 2

1 1 1(0)i i iG GdGdt τ

+ −

− − −−= ,

1 12 2

1 1 1(0)i i iG GdGdt τ

+ −

+ + +−= .

2

22 ( )

( 1) ( )ia u

hi iG G B e A B

ττ

−+ = − + + , (5)

1 1 1 12 2 2 2

1 1 1 112 i i i iA G G G Gτ

+ − + −

− − + += − + − , 2

1 122 2 ( )

i i

i

G G hB Aa u

− ++= − ⋅ .

G t τ= , -:

2 2

2 22 ( ) 2 ( )

1 1( ) 12

i ia u a ui ih h

i iG GG G e e

τ ττ

− −− ++

= + − , (6)

2 2

2 21 ( ) ( )2 1 11

2

i ia u a ui ih h

i iG GG G e e

τ τ+ − ⋅ − ⋅− ++

= + − . (7)

(5)–(7) iG iu -

, 0

( )iu

iG q dξ ξ= . ( )G u

( ).

0 1N + ( . -): , -

. (1) :

( )0

( (0, )) (0, )l l lx

G u t u t Qx

λ θ=

∂− = − +∂

. (8)

1G− , G , 0x

Gx =

∂∂

1 0

0x

G GGx h=

−∂ =∂

, (0, )G t – 0 1(0, )2

G GG t += . -

(8) 1 1 10 1 0 1 0 112 2

2 2 2l l lG G G G G GG G G G G Q

h hλ θ− − −+ + +

− + ⋅ = − + . (9)


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2013, 5, 1 15

1 0 1

2G Gz G− +

= , (9) z :

( ) ( )( ) 122l l l

GG z z z Qh h

λ θ⋅ − − = + . (10)

, ( )q u , ( )c u , ( )l uλ ( )r uλ.

1 2{ 0, , ..., }mz z z= . -- . (10) 1[ ; ]i iz z + ,

1, 1i m= − .

2 0Az Bz C+ + = , (11)

1 1

1 1

( ) ( ) ( ) ( )( )

i i l i l i

i i i i

q z q z z zAz z h z z

λ λ+ +

+ +

− −= +

− −,

1 1 1 1 1

1 1 1

( ) ( ) ( ) ( ) ( ) ( )2 i i i i l i l i l i i l i il

i i i i i i

q z z q z z z z z z z zBh z z z z z z

λ λ λ λθ+ + + + +

+ + +

− − −= ⋅ − +

− − −,

21 1 1 1

1 10

( ) ( ) ( ) ( ) 22 ( ) ( )2

izi i i l i i l i i

i i l li i i i

q z q z z z z z z GC q d q z z Qh z z z z h

λ λξ ξ θ+ + +

+ +

− −= ⋅ − + ⋅ − − −

− −.

z∗ – (11), 1[ ; ]i iz z + , 0G 0

0 12 ( )G G z G∗= − . .

:

( )

( )

0

0

( ) ( ) , 0 , 0 ,

( , 0) ,

( ) ( (0, )) (0, ) ,

( ) ( ( , )) ( , ) ,

l l lx

r r rx L

u uu q u t T x Lt x x

u x

uq u u t u t Qx

uq u u L t u L t Qx

ϕ

λ θ

λ θ

=

=

∂ ∂ ∂= < ≤ < <∂ ∂ ∂=

∂− = − +∂

∂ = − +∂

(12)

T = 100 , L = 1 , 0ϕ = 22 ° , lθ = 1400 ° , rθ = 1400 ° , 5 210 /( )lQ = , 20 /( )rQ = , ( )u , ( )q u , ( )l uλ ( )r uλ . 1.

�� !"� 1 #$�%&$!' ()*+$,) -�.�/&0.*( 1�+�%!, '( '23!)4' 56$7"!'/! 0&/-&.�06.,

, °C

0 100 200 300 400 500 600 700 800 1000 6 3( ), 10 /( )u ⋅ ° 3,414 3,568 4,040 4,347 4,812 5,272 5,886 7,286 7,218 7,218

3( ), /( )q u ⋅ ° 22,5 23,4 24,8 26,7 27,2 27,7 28,1 28,6 27 27 2( ), /( )l uλ ⋅ ⋅ ° 100 100 110 120 130 140 150 160 170 170 2( ), /( )r uλ ⋅ ⋅ ° 100 120 130 140 150 150 150 150 150 150


. « . . » 16

. 2. , (12) , ,

, , , - [3, 5].

�� 2 ��

τ , N 0,01 0,05 0,1 0,5

40 44,7 10−⋅ 44,5 10−⋅ 31,52 10−⋅ 31,72 10−⋅60 31,12 10−⋅ 31,12 10−⋅ 31,19 10−⋅ 32,18 10−⋅80 49,9 10−⋅ 31,0 10−⋅ 31,06 10−⋅ 32,15 10−⋅

100 47,5 10−⋅ 47,7 10−⋅ 48,7 10−⋅ 32,25 10−⋅150 44,2 10−⋅ 44,7 10−⋅ 45,9 10−⋅ 32,43 10−⋅200 42,7 10−⋅ 43,4 10−⋅ 45,0 10−⋅ 32,67 10−⋅

1. , . . / . . , . // . – 2008. – . 15, 5. – . 870–871.

2. , . . / . . , . . , . // . «

». – 2008. – . 1. – 15(115). – . 9–11. 3. , . . / . . . – .: , 1989. – 616 . 4. , . . / . . , B.C. . – .: , 1977. –

440 . 5. , . . / . . ; . . . . . – .: -

, 1978. – 512 . 6. , . / . . – .: , 1982. – 235 . 7. , . .

« 4.0» / . . , . , . . // - 9776, 20.02.2008. – .:

, 2008. 8. , . -

« 5.0» / . , . . , . . // -2008612210, 30.04.2008,

.


�� .., �� .�. ��

2013, 5, 1 17

EXPLICIT DIFFERENCE SCHEME FOR THE SOLUTION OF ONE-DIMENSIONAL QUASI-LINEAR HEAT CONDUCTIVITY EQUATION

A.V. Herreinstein1, M.Z. Khayrislamov2

Numerical method for the solution of the third mixed boundary value problem for one-dimensional quasi-linear heat conductivity equation of a parabolic type based on the use of explicit difference scheme is given. Dependence of coefficients on temperature is overcome by the introduction of the new required function that is a primitive integral of conductivity.

Keywords: thermal conductivity, quasi-linear heat conductivity equation, explicit difference schemes, approximation.

References 1. Herreinstein A.V., Mashrabov N. Nagrevanie kruga dvizhushhimsya teploistochnikom [Circle

heating by moving heat source]. Obozrenie prikladnoj i promyshlennoj matematiki. 2008. Vol. 15, no. 5. pp. 870–871. (in Russ.).

2. Herreinstein A.W., Herreinstein E.A., Mashrabov N. Ustojchivye yavnye skhemy dlya urav-neniya teploprovodnosti [Steady Obvious Schemes for Equation of Heat Conductivity]. Vestnik YuUrGU. Seriya «Matematicheskoe modelirovanie i programmirovanie». 2008. Issue 1. no. 15(115). pp. 9–11. (in Russ.).

3. Samarskij A.A. Teoriya raznostnykh skhem [Theory of difference schemes]. Moscow: Nauka, 1989. 616 p. (in Russ.).

4. Godunov S.K., Ryaben'kij B.C. Raznostnye skhemy [Difference schemes]. M.: Nauka, 1977. 440 p. (in Russ.).

5. Kalitkin N.N. Chislennye metody [Numerical methods]. M.: Nauka, 1978. 512 p. (in Russ.). 6. Shup T. Reshenie inzhenernykh zadach na EVM [The solution of engineering problems with a

computer]. Moscow: Mir, 1982. 235 p. (in Russ.). 7. Herreinstein A.V., Mashrabov N., Herreinstein E.A. Raschet temperaturnykh polej v cilindre pri

dejstvii poverxnostnykh teplovykh istochnikov «Teplo 4.0» [Calculation of temperature patterns in a cylinder at surface heat sources “Teplo 4.0” effect]. Gosudarstvennaya registraciya v Otraslevom fonde algoritmov i programm 9776. 20.02.2008. Moscow: FGNU GKCIT, 2008. (in Russ.).

8. Mashrabov N., Herreinstein A.V., Herreinstein E.A. Raschet temperaturnykh polej v cilindre pri dejstvii poverkhnostnykh teplovykh istochnikov «Teplo 5.0» [Calculation of temperature patterns in a cylinder at surface heat sources “Teplo 5.0” effect]. Svidetel'stvo o gosudarstvennoj registracii pro-gramm dlya EVM 2008612210. 30.04.2008. ROSPATENT [Certificate of state registration of com-puter program No. 2008612210. 30.04.2008. ROSPATENT]. (in Russ.).

�� 27 �� 2012 �.

1 Herreinstein Arcady Vasilevich is Cand. Sc. (Physics and Mathematics), Associate Professor, Applied Mathematics Department, South Ural State University. 2 Khayrislamov Mikhail Zinatullaevich is Post-Graduate student, Applied Mathematics Department, South Ural State University. E-mail: [email protected]


. « . . » 18

517.977.58

1

. . 2

, -

n- -. -

, -,

. -.

�� : �� , �� , �� , �� .

. , , ,

, . . , , -

. , -, , ,

. , -, .

, , . , -

. -,

, [1–6].

, -

( , , )x f t x u= , 0[ , ]t t ϑ∈ , u P∈ , (1) x – n - nR , u – -

, P – pR . , (1) .

. - ( , , )f t x u ( , , )t x u

0[ , ] nt R Pϑ × × , 0[ , ] nD t Rϑ⊂ × -( ) (0, )L L D= ∈ ∞ ,

* ** *( , , ) ( , , )f t x u f t x u L x x− ≤ −

( , , )t x u D P∈ × . B. (0, )μ∈ ∞ ,

( )( , , ) 1f t x u xμ≤ +

1 �� !�"#$#� � %�&'�( !%�)%�&& *%$+,-,.&� �/0 «1,#�&,2$3',$ 3,3�$& , �$�%,4 .!%��"$#,4» !%, 5,#�#3��6 !�--$%7'$8%9 �/0 (!%�$'� 12-*-1-1002), � ��'7$ !%, !�--$%7'$ )%�#�� *%$+,-$#�� 33,63'�6 :$-$%�;,, !� !�--$%7'$ �$-.<,( #�.2# (='�" > 0?-5927.2012.1. 2 @,&��$; /%�$& /#��"A$�,2 – 3��%=,6 !%$!�-��$"A, '�5$-%� ,#5�%&�;,�## ( �$(#�"�),6, B#3�,�.� %�-,�C"$'�%�#,', , ,#-5�%&�;,�## ( �$(#�"�),6 – �D:, 8%�"A3',6 5$-$%�"A# 6 .#,�$%3,�$�. E-mail: [email protected]


�� .�. ��

2013, 5, 1 19

0( , , ) [ , ] nt x u t R Pϑ∈ × × . (1) 0[ , ] nt RϑΦ ⊂ × ,

0[ , ]t t ϑ∈ ( )tΦ . C.

( )* *

* 0 *

**

, [ , ]:sup ( ), ( ) ( )

t t �� t t td t t

ϑ ρχ ρ

− ≤Φ Φ ≤ ,

( )χ ρ – 0ρ ↓ , -

0lim ( ) 0ρ

χ ρ↓

= . Φ ( ( )tΦ ) -

t . (1) -0X ≠∅ : 0 0( )X t⊂Φ .

1. ** *( , , )Y t t x *t (1),

Φ , * *( )x t x= -* nx R∈ , ( )x t (1), -

( )u t , , * *( )x t x= , * *( )x t x= ( ) ( )x t t∈Φ*

*[ , ]t t t∈ . ( )u t

( )u t , ( )u t P∈ , 0[ , ]t t ϑ∈ .

* *

* ** * * *( , , ) ( , , )

x XY t t X Y t t x

∈= , *

nX R∈ .

1. 0 0( , , )Y t Xϑ (1), -Φ .

(1) ( . .) ( , )x F t x∈ , 0[ , ]t t ϑ∈ , (2)

{ }( , ) ( , , ) :F t x co f t x u u P= ∈ – { }( , , ) :f t x u u P∈ nR .

2. ** *( , , )X t t x *t . . (2),

Φ , * *( )x t x=* nx R∈ , ( )x t . . (2), ,

* *( )x t x= , * *( )x t x= ( ) ( )x t t∈Φ **[ , ]t t t∈ .

* *

* ** * * *( , , ) ( , , )

x XX t t X X t t x

∈= , *

nX R∈ .

, *0 0( , , )Y t t x *

0 0( , , )X t t x (1) . . (2)

* *0 0 0 0( , , ) ( , , )X t t x clY t t x= , 0[ , ]t t ϑ∈ ,

*0 0( , , )clY t t x – *

0 0( , , )Y t t x nR .

-, -

, , -. , nR

« » hΛ h , ,

0[ , ]t ϑ { }0 1, , , Nt t t ϑΓ = = , N < ∞ -


, 2013 20

0, 1max i

i N= −Δ = Δ , 1i i it t+Δ = − , 0, 1i N= − , ,

. . (2), hiX , 0,i N= hΛ , h

Δ 0 0( , , )iX t t X . , Γ -, . . iΔ = Δ , 0, 1i N= − .

( , )nnX δΛ

x nX R⊂nδΛ

nR nδ ,

),( pp P δΛ , u pRP ⊂

pδΛpR pδ . X P

-.

0XP nΛ pΛ 0 0( , )h nX X h= Λ

( , )p ppP Pδ δ= Λ , ( )p pδ δ= Δ , ( )pδ Δ – 0Δ > ,

0Δ→ .

{ }( , ) ( , , ) :p pF t x co f t x u u Pδ δ= ∈ , 0( , ) [ , ] nt x t Rϑ∈ × . (3)

0( , ) [ , ] nt x t Rϑ∈ ×

( )ˆ ( , ) ( , ),pnnF t x F t xδ δ= Λ , (4)

n hδ = Δ . (3)–(4)

( ) ( ) ( )ˆ ˆ( , ), ( , ) ( , ), ( , ) ( , ), ( , )p pd F t x F t x d F t x F t x d F t x F t xδ δ≤ + , (5)

( )ˆ( , ), ( , )2

pn

nd F t x F t xδ δ≤ .

(5) ( , , )f t x u( , , )t x u D ,

*( )ϕ δ ( , )n pδ δ δ= ( *( ) 0ϕ δ ↓ 0δ ↓ ), ( ) *ˆ( , ), ( , ) ( )d F t x F t x ϕ δ≤

( , )t x D∈ .

, { }1 2, ,i

h i i ii NX x x x= . -

1hiX +

hiX .

1. i hj ix X∈ , 1, ij N=

1ˆ ( , , )i

i i jX t t x+ :

1ˆ ˆ( , , ) ( , )i i i

i i j j i jX t t x x F t x+ = + Δ . (6)

2. 1ˆ ( , , )i

i i jX t t x+

1 1ˆ ˆ( , , ) ( , , )

i hj i

h ii i i i i j

x XX t t X X t t x+ +

∈= .

3. 1ˆ ( , , )h

i i iX t t X+ 1( )it +Φ -Φ :

1 1 1ˆ ( , , ) ( )h h

i i i i iX X t t X t+ + += ∩Φ .


�� .�. ��

2013, 5, 1 21

« », 1hiX + -

ijx h

iX . , n hδ = Δ

1ˆ ( , , )h

i i iX t t X+ hΛ . [3] , ( )h h= Δ , *( ) ( )h hΔ = Δ Δ , *( )h Δ – -

0Δ > , 0Δ→ , hiX , 0,i N=

0 0( , , )iX t t X . . (2) 0Δ→ .

« » , , -

h Δ . -, [5, 6], , -

1hiX + , -

1 0 0( , , )iX t t X+ , i hj ix X∈ .

. 3. Ξ hΛ ,

hΛ , n - h , -.

*y – hΛ . *( )yΞ

hΛ , *y . 4. hΛ ,

hΛ . hΛ -, .

hhX ⊂ Λ – , hΛ .

5. h hx X∈ hX , hX .

6. h hx X∈ hX , , hX .

7. hXhX hX∂ .

, [5, 6], -

1hiX +

hiX -

, . ,

1ijx + (6).

-,

, (6), . -

, ( , [5]) . , -

. . , , -, .

1hiX +

hiX , -


, 2013 22

. -, , -

-, , hiX .

hiXε∂ – h

iX ε , ( )h h h

i i iX X Xε ε∂ = ∩ , ( )h

iX ε – hΛ , ε -hiX . 1

hiX +

hiX .

1. *i

hiXε∂ ii εε 2* = h

iX , iε

(1).

2. ( )* *1( , , ),i i

n hi i iX X t t X hε ε

Ξ Ξ+= Λ ∂ ,

**

1 1( , , ) ( , , )h hi

ii

h hi i i i i

x XX t t X X t t x

εε

Ξ Ξ+ +

∈∂∂ = , (7)

*1 1( )

ˆ( , , ) ( , , )h i

h hi i i i i

xX t t x coX t t XεΞ

+ +Ξ∈Ξ

= Ξ∩∂ . (8)

3. *i

XεΞ∂ *

iXεΞ .

4. ( )* * *( ) \ ,i i i

h nhX X X hε ε ε

Ξ Ξ Ξ∂ = Λ ∂ h -

*i

XεΞ .

5. *1i

iS XεΞ

+ ⊂ ∂ , -

**i

hx XεΞ∈∂ ,

( )1 *ˆ ( , , ) \

ih h

i i i iZ t t x X Xε+ ∩ ∂ =∅ ,

1 * * *ˆ ˆ( , , ) ( , )i i iZ t t x x F t x+ = − Δ . 6. 1

hiX + , 1iS + ,

, 1( )it +Φ Φ . . -

1hiX + .

1 hiX ,

*iε . iε , -

1iS + 5.

2 *i

XεΞ , -

, *i

hiXε∂ -

. *i

XεΞ -

(8). 3 , 2. ,

*i

XεΞ∂ , 1

hiX + .


�� .�. ��

2013, 5, 1 23

4 *i

XεΞ , -

*i

XεΞ , *

iXεΞ .

5 1iS + ,

1hiX + . 1iS + ,

*i

XεΞ∂

1hiX + . ,

, -, h

iX ,

iε . 6 ,

1iS + , 1hiX + .

, 5 -

1hiX + , , .

ˆ ( , ),max

hi i

if F t x x X

f K∈ ∈

≤ .

( ) Δ≤+ ihh

ii KxxttXd ),,,(ˆ1 , (9)

h hix X∈ . (7)–(8), (9)

( ) 2 2 2 21( , , ), ( )h h

i i id X t t X X K h nΞ+ ≤ Δ + Δ Δ . (10)

2 2( )i iK K h nΞ = + Δ . (10)

( )( )1( , , ), ,2

n h hi i i i i

nd X t t X h X K hΞ Ξ+Λ ≤ Δ + . (11)

(11) , * 2i iK h nε Ξ> Δ + -

( )*int 1( , , \ ),i

n h hi i i iX X t t X X hε

Ξ+= Λ ∂

( )1( , , ),n hedge i i iX X t t X hΞ

+= Λ ∂ , , *i

XεΞ

1hiX + , .

*1( )

2i i inK K h hε Ξ

+> + Δ + + , (12)

*1iK +

1

*1ˆ ( , ),

maxh

i ii

f F t x x X hBf K

++

∈ ∈ +≤ ,

B – . 1 *ˆ ( , , )i iZ t t x+

*x 1iS + * 1\i

iX SεΞ

+∂ .

, , (12) 5 -, 1

hiX + .


, 2013 24

1 2

2 1 2sin

x xg kx x x ul m

=

= − − + (13)

9,8g = , 3l = , 1,2k = 1m = . 21 2( , )x x x R= ∈ –

(13), u uu r≤ , 3ur = . , x -[ 15,15] [ 5,5]x∈Φ = − × − . , (13) -

A–C. , 0 0( , , )X t Xϑ (13)

5ϑ = { }0(0) ( ,0)x X π∈ = Φ .

0[ , ]t ϑ { }0 1, , , Nt t t ϑΓ = = -

0tN

ϑ −Δ = . hΛ

32h = Δ .

( , , )f t x u , , hiX , 0Δ→ 0 0( , , )iX t t X , 1,i N= , -

3* 253 5ε = Δ + Δ .

0 0( , , )X t Xϑ (13) 5ϑ = 0,05Δ = , 0,01h = . -

1 10 .

1. , . . / . . . – .: , 1968. – 476 .

2. , . . / . . , . . // -

. – 1994. – . 34, 7. – . 965–977.

�� (13) � �� 5ϑ =


�� .�. ��

2013, 5, 1 25

3. , . . / . . , . . , . . // . – 1998. – . 62, 2. – . 179–187.

4. , . . - / . . , . . // -

. – 2001. – . 41. – . 6. – . 895–908. 5. , . .

: . … . .- . / . . . – , 2005. – 160 . 6. , . .

: . … . .- . / . . . – , 2009. – 206 .

A BOUNDARY LAYER METHOD FOR THE CONSTRUCTION OF APPROXIMATE ATTAINABILITY SETS OF CONTROL SYSTEMS

A.A. Zimovets1

A boundary layer method for the construction of approximate attainability sets of dynamical sys-tems in the n-dimensional Euclidean space with phase constraints is given. This method belongs to the group of grid methods and it is based on the approach when only the points of a boundary layer are used in an iterative process but not the points of a constructed set. This method allows us to speed up the cal-culations by reducing the number of calculating points in traditional grid methods.

Keywords: control system, differential inclusion, attainability set, grid-based method.

References 1. Krasovskij N.N. Teoriya upravleniya dvizheniem [Theory of motion control]. Moscow: Nauka,

1968. 476 p. (in Russ.). 2. Ushakov V.N., Khripunov A.P. Computational Mathematics and Mathematical Physics. 1994.

Vol. 34, no. 7. pp. 833–842. 3. Gusejnov X.G., Moiseev A.N., Ushakov V.N. Prikladnaya matematika i mekhanika [Journal of

Applied Mathematics and Mechanics]. 1998. Vol. 62, no. 2. pp. 179–187. 4. Neznakhin A.A., Ushakov V.N. Computational Mathematics and Mathematical Physics. 2001.

Vol. 41, no. 6. pp. 846–859. 5. Pakhotinskikh V.Yu. Postroenie reshenij v zadachax upravleniya na konechnom promezhutke

vremeni: dis. kand. fiz.-mat. nauk [The solution of control problems at finite time period: Cand. phys. math. sci. diss.]. Chelyabinsk, 2005. 160 p. (in Russ.).

6. Mikhalev D.K. Postroenie reshenij v differencial'nyx igrax na konechnom promezhutke vremeni i vizualizaciya reshenij: dis. … kand. fiz.-mat. nauk [The solution of the problems in differential games at finite time period and visualization of the solutions: Cand. phys. math. sci. diss.]. Chelyabinsk, 2009. 206 p. (in Russ.).

�� !" # $%&"'( ) 4 �'�*+$* 2012 ,.

1 Zimovets Artem Anatolievich is Senior Teacher, Information Technology Department, Ural Federal University. E-mail: [email protected]


. « . . » 26

517.948.00

. . 1

, -- -

. .

�� : �� , ��, � �� - �, �� , �� .

[1] , , ( )h t [ )2 0,C ∞ , .

.

[2] , ( )( )

( )3 12

20

,h t Wε

ε

−

>∈ −∞ ∞ -

. , [2],

.

( ) ( )2

2, ,

; 0 1, 0u x t u x t

x tt x

∂ ∂= < < >

∂ ∂, (1)

( , 0) 0; 0 1,u x x= ≤ ≤ (2) (0, ) ( ); 0,u t h t t= ≥ (3)

( )h t – - ,

( )h t′ , (0) 0h = (4)

0 0t > , 0t t≥( ) 0;h t = (5)

(1, ) 0; 0.u t t= ≥ (6) ( , )u x t (1)–(3), (6), . . ,

[ )( ) ( ) ( )( )2,1( , ) [0,1] 0, 0,1 0, .u x t C C∈ × ∞ × ∞, [3, . 424]

( ),u x t . -t .

1. ( ) [ )0,t CΦ ∈ ∞ . -:

1 �� !"#$� %��&'"� ()*+))$"� – ,��*-!. /*)/# �$��)�&, 0�1) *� $23!,�!�)�&"#. ��)��!0!, 45"#-6*��&,0!. +#,7 �*,�$)"-"2. 7"!$)*,!�)�. E-mail: [email protected]


�� .�. ��

2013, 5, 1 27

( ) ( ) ( ) ( )0 0

, ,xu x t t dt u x t t dtx

∞ ∞∂′ Φ = Φ∂

( ) ( ) ( ) ( )2

20 0

, , .xxu x t t dt u x t t dtx

∞ ∞∂′′ Φ = Φ∂

2. ( ),u x t – (1)–(3), (6).

( ) ( ) ( )0 1

0 0

lim , lim , 0x x

u x t h t dt u x t dt∞ ∞

→ →− = = .

, (1)–(6) - (3), ( )h t , , ( )1 0,1x ∈ , ( )f t , -

( ) ( )1, ; 0u x t f t t= ≥ . (7)

(1)–(2), (6)–(7)

[ )2 0,Z L∈ ∞ . ( )h t Z∈ ,

( ) ( ) ( )1

,n

jj

h t t tϕ ψ=

= +

( ) ( )2 2 ;,2

0; ,

jj j j j j j

j

j j j j

at c b c b t c b

tt c b t c b

ϕ− − − ≤ ≤ +

=< − > +

, , 0,j j j j ja b c c b> ≥ j kc c≠ j k≠ , ( ) [ )3 22 0,t Wψ ∈ ∞ .

, ( )0( )f t f t= , (7),

0 ( )h t , Z , ( )0f t , -

( ) [ ) [ )2 10, 0,f t L Lδ ∈ ∞ ∞ 0δ > ,

0f fδ δ− ≤ . (8) , , ,fδ δ Z , ( )h tδ (1)–(2),

(6)–(8) 20 Lh hδ − ( )h tδ 0 ( )h t .

[ ) [ )2 20, 0,H L iL= ∞ + ∞ – , F – ,

[ ) [ )2 10, 0,L L∞ ∞ H

( ) ( )0

1 ; 0i tF h t h t e dtτ τπ

∞−= ≥ . (9)

1. F , (9), . . [1].

1 2 - (1)–(2), (6)–(8).

, : ( ) ( ) ( )

2

2

ˆ ,ˆ , ; 0,1 , 0,

u xi u x x

xτ

τ τ τ∂

= ∈ ≥∂

(10)


. « . . » 28

( ) ( )ˆ , , .u x F u x tτ = (6)–(7) ,

( )ˆ 1, 0; 0u τ τ= ≥ (11)

( ) ( )1ˆˆ , ; 0,u x fτ τ τ= ≥ (12)

( ) ( )ˆ .f F f tτ =

2 , ( )ˆ ,u x τ (10)–(12)

[ ] [ )0,1 0,× ∞ . (10)

( ) ( ) ( )0 0ˆ , x xu x t A e B eμ τ μ ττ τ −= + , (13)

( )01 12

iμ = + , ( )A τ ( )B τ – .

(11)–(13) ,

( ) ( )0

0 1

shˆ ˆ , 0.sh (1 )

h fx

μ ττ τ τμ τ

= ≥−

(14)

, 112

x ≤ , [ ]0,2τ ∈ .

( ) ( )

20

0 1 1 1

sh ch 2 cos 2sh (1 ) ch 1 2 cos 1 2x x x

μ τ τ τμ τ τ τ

−=− − − −

, (15)

(15) ,

( ) ( )

20

1 10 1

sh ch cosch 1 cos 1sh (1 ) x xx

μ τ λ λλ λμ τ−=

− − −−, (16)

2λ τ= . , (16) ,

( )20

0 1

2 1sh 0 22sh (1 )

e

exμ τ τ

μ τ

+≤ ≤ ≤

−−. (17)

0

0 1

shsh (1 )x

μ τμ τ−

2τ ≥ .

( ) ( ) ( ) ( ) ( )2 21 1

20

0 1 1 11 1 1

chsh ch 2 cos 2 22 ,sh (1 ) ch 1 2 cos 1 2 sh 1

2x x

ex x x x e e

τ τ

ττμ τ τ τ

μ τ ττ τ − − −

−= ≤ ≤− − − − − −

, , 112

x ≤ , 2τ ≥ , ( )12 1

22x

e eτ−

≤ ≤ ,

10 2

0 1

sh4 2

sh (1 )

xe

x

τμ τ τμ τ

≤ ≥−

. (18)

( )22 14

2

e

e

+>

−,

(17) (18)


�� .�. ��

2013, 5, 1 29

10 2

0 1

sh, 0

sh (1 )

xae

x

τμ τ τμ τ

≤ ≥−

, (19)

( )22 1

2

ea

e

+=

−. (20)

(14)

, (1)–(2), (6)–(7) -T ,

[ )2 0,L ∞ [ )2 0,L ∞

( ) ( )0

1 0

shˆ ˆ , 0,sh(1 )

Tf fxμ ττ τ τμ τ

= ≥−

(21)

( )01 12

iμ = + , ˆ ˆ,f Tf H∈ .

(14) : ( ) ( )ˆ ˆ ; 0h Tfτ τ τ= ≥ . (22)

Z [ )2 0,L ∞ ,

[ ]Z F Z= , (23) F (9), (22) -

T . , 0

ˆ ˆ( ) ( )f fτ τ= ( )0ˆh Zτ ∈ ,

0 ( )f τ , ( )f Hδ τ ∈ 0δ > ,

( ) ( )0ˆ ˆf fδ τ τ δ− ≤ . (24)

, ( )fδ τ δ , -

( )hδ τ , ( )0h τ Z ,

( ) ( )0ˆ ˆh hδ τ τ− .

{ }: 0Tα α ≥ , :

( ) ( )ˆ ;ˆ , 0,0;

TfT fατ τ ατ ατ α≤= >>

(25)

T (21). , ( )hαδ τ T

( )ˆ ˆh T fαδ α δ τ= , (26)

Tα (25). ( )ˆ ,fδα α δ= (26)

( ) ( )21 2ˆ ˆ 9T h fα

δ δτ τ δ− − = . (27)

, 1 [4], ,

( ) ( ) [ )21 ˆ ˆ 0,T h f Cα

δ δτ τ− − ∈ ∞ α , 2

fδ α →∞


. « . . » 30

0α → . , 2 2ˆ 9fδ δ> (27). ,

, (27) [ ]1 2,α α

[ ]1 2,α α α∈ 1ˆ ˆh hααδ δ= . .

T (27). , ( )hδ τ (22), (24)

( ) ( ) ( )ˆ ,

0ˆ ˆ ; ,

fh pr h Hδα δδ δτ τ= (28)

[ )0 2 0,H F L= ∞ .

( ) ( )0ˆ ˆh hδ τ τ− hδ (22), (24)

( ) ( )0ˆ ˆh hδ τ τ− ( )hδ τ

( )0h τ . ( )0ˆh Zτ ∈ , (23) 0c > ,

0τ ≥

( )0 4ˆ

1

ch ττ

≤+

. (29)

(29) 0d > , 10,2

ε ∈

( ) ( )23 1

00

ˆ1 dh dετ τ τε

∞−+ ≤ . (30)

( )0 0ˆ ,h Hτ ∈ (31)

, [2], (28), (31) ,

( ) ( ) ( )0ˆ ˆ 6 , ,dh h Gδ ετ τ α δ ε

ε− ≤ (32)

( )Gε σ , (14), (19), (20) (30) :

( ) ( )11

3 1 22 , 1 ; 0,x

ae Gτ

εεσ τ τ τ

−−= = + ≥ (33)

( ),α δ ε( ) .

Gd ε α δε α

= (34)

(32) 10,2

ε ∈ , ( )ε δ , -

,

( ) ( ) ( )( )(

( )10, 2

6 , min 6 ,d dG Gεε δε

α δ ε δ α δ εε δ ε∈

= . (35)

(32)–(35) ( )hδ τ

( ) ( ) ( ) ( ) ( )( )0ˆ ˆ 6 ,dh h Gδ ε δτ τ α δ ε δ

ε δ− ≤ . (36)


�� .�. ��

2013, 5, 1 31

(36), . , 10,2

ε ∈

( )( )

( )

13 1 2

6 121

21 lnGax

εε

εσσ

−−−= + . (37)

(37) , aσ > 10,2

ε ∈

( )( )

( )3 12 2 3 11 ln

2xG

a

εε

εσσ

−− −≤ . (38)

(37) (38) , aσ > 10,2

ε ∈

( )( )

( )3 12 2

3 11

lim 1

ln2

G

xa

εεσ

ε

σ

σ−→∞

− −

= , (39)

(37) (39) 1 aσ > , 1σ σ≥ 10,2

ε ∈

( )( )

( )3 1

3 11 ln2xG

a

εε

εσσ

−− −≥ . (40)

, ( )( )

1

3 13 11

lim 0

ln2x

a

εσε

σ

σ

−

−→∞− −

= 2 1σ σ> ,

2σ σ> ( )

( )3 1

3 11 ln2x

a

εε σσ

−− −< ,

(40), 2σ σ>

( )21 1 Gε σσσ< . (41)

( ),α δ ε (34), ( )ˆ ,α δ ε

2dα δε

− = . (42)

(42) ,

( ) 42

ˆ , dα δ εεδ

= . (43)

(34), (41) (42) , ( ) ( )ˆ , ,α δ ε α δ ε≤ . (44)

, (32), (38) (44) 0 0δ > , -

( )00,δ δ∈ 10,2

ε ∈


. « . . » 32

( ) ( )( )

( ) ( )3 12 3 11

0ˆ ,ˆ ˆ 36 ln

2xdh h

a

εε

δα δ ε

τ τε

−− −− ≤ . (45)

(44) (45) , ( )00,δ δ∈ 10,2

ε ∈

( ) ( ) ( ) ( )2 3 1 3 1 40 1 21ˆ ˆ 36 lnd dh h xa

ε εδ τ τ

ε εδ− − −− ≤ . (46)

( ) 11ln ln

ε δδ

= ,

(46) ,

( ) ( )( )

( )( )

322 3 440 4 2 4 4

1ˆ ˆ 36 ln ln ln lnd dh h da a

ε δδ

δτ τδ ε δ δ ε δ δ

−− ≤ . (47)

, 1 0δ δ> 2

14

1

1ln ln 1daδ

δ< .

( )10,δ δ∈ (47)

( ) ( ) ( )32 3 4

0 4 2

1ln ln1 1ˆ ˆ 36 ln ln ln lnd

h h da

ε δδ

δτ τδ δδ

−− ≤ . (48)

( )10,δ δ∈( ) 1 1 1ln ln ln ln 11ln ln

ε δ

δ δδ

= = , (49)

(49) ,

( )3

31ln eε δ

δ= , (50)

(48) (50),

( ) ( )2 3 3 4

0 4 2

1ln ln1ˆ ˆ 36 ln ln lnd

h h d eaδ

δτ τδ δ

−− ≤ . (51)

, 2 1δ δ<4

2

1ln ln adδ

> , ( )20,δ δ∈

4 2 2

1ln ln 1d

aδ

δ δ> . (52)

, (51) (52) , ( )20,δ δ∈

( ) ( )2 3 3

01 1ˆ ˆ 36 ln ln lnh h d eδ τ τδ δ

−− ≤ . (53)

(53) , ( )20,δ δ∈

( ) ( )3

3 20

1 1ˆ ˆ 17 ln ln lnh h d eδ τ τδ δ

−− ≤ .


�� .�. ��

2013, 5, 1 33

1. , . . - / . . , . . //

. – 2010. – . 16, 2. – . 238–252. 2. , . .

- / . . , . . , . . // . – 2012. – . 18, 1. – . 281–288.

3. , . . / . . . – .: , 1968. – 576 . 4. , . . - /

. . , . . // . – 2006. – . 9, 4. – . 353–368.

APPROXIMATE SOLUTION OF INVERSE BOUNDARY PROBLEM FOR THE HEAT CONDUCTIVITY EQUATION BY NONLINEAR METHOD OF PROJECTION REGULARITY

T.S. Kamaltdinova1

Inverse boundary problem is solved in the hypothesis that the required solution is a piecewise smooth function, estimates of above approximate solution are given. The estimates are considerably su-perior to the known estimates by the accuracy.

Keywords: operator equation, regularity, optimal method, error estimation, ill-posed problem.

References 1. Tanana V.P., Sidikova A.I. O garantirovannoj ocenke tochnosti priblizhennogo resheniya odnoj

obratnoj granichnoj zadachi teplovoj diagnostiki [On the guaranteed accuracy estimate of an approxi-mate solution of one inverse problem of thermal diagnostics]. Trudy Instituta Matematiki I MekhanikiURO RAN. 2010. Vol. 16, no. 2. pp. 238–252. (in Russ.).

2. Tanana V.P., Bredikhina A.B., Kamaltdinova T.S. Trudy Instituta Matematiki I Mekhaniki URO RAN. 2012. Vol. 18, no. 1. pp. 281–288. (in Russ.).

3. Mikhlin S.G. Kurs matematicheskoj fiziki [Course of mathematical physics]. Moscow: Nauka, 1968. 576 p. (in Russ.).

4. Tanana V.P., Yaparova N.M. Ob optimal'nom po poryadku metode resheniya uslovno-korrektnykh zadach [The optimum in order method of solving conditionally-correct problems]. Sib. Zh. Vychisl. Mat. 2006. Vol. 9, no. 4. pp.353–368. (in Russ.).

�� 29 �� 2012 �.

1 Kamaltdinova Tatyana Sergeevna is a Senior Lecturer, Numerical Mathematics Department, South Ural State University. E-mail: [email protected]


. « . . » 34

517.586

. . 1

-.

�� : �� , �� , �� -�� .

1. , [1–3]. [2],

n

( ){ ( )}G xν , 2 ( )L S∂ , S∂ – n . , ( 3). 5

( ) | 0( ) ( ) xG D u x =ν , ( )u x – S

S . 6

| | 1( )m xx

Q x ds=

, ( )mQ x – .

2.

2 2 ,![ / 2]( 1)

( )0

| |( ) ( 1)

(2,2) ( 1 2 ,2)

i k ikn ns i

k ni ii

x xG x

n s

−−

== −

− +,

,! / !k kx x k= , ( 1) 1 ( 1)( , , )n nx x x− −= ( , ) ( ) ( )ka b a a b a kb b= + + − –

0( , ) 1a b = . 0 {0}= ∪ � [( ) / 2]

2 ,! 2 ,!(2) 1 2

0( ) ( 1) , 0,1

k ss i i s k i sk

iH x x x s

−+ − −

== − = .

1. [2] 2n ≥ 1 n≥ ≥ν ν , 0,1n =ν

11 1

2

( ) ( ) ( 1) (2)1

( ) ( ) ( ),i ni i n

n

n n ii

G x G x H x++ −

−

− +−=

=∏ ν νν ν ν ν

1( , , )n=ν ν ν , 0n∈ν 1ν .

2. [2] { }( )G ν 0n∈ν , -

1 2 n≥ ≥ ≥ν ν ν , 0,1n =ν , .

( ) ( )G xν { ( ) }P x p x= = αα

α-

0( ), ( ) ( ) ( ) xP x Q x P D Q x == � [1]. ( )P x ∈ -

| |DP . , 2 2 21 | 0| | | | 2 .n D xx x x n=+ + = Δ = ( ) ( )G xν

. 3. ( ) ( )G xν 0

n∈ν ,

1 ( 0,1)n n≥ ≥ =ν ν ν .

1 �� ! �� "#�"$%�� – &$�#$� '�(��$-)�# )�#�� *��+ "�,�, -�$' **$�, ��' &�� )�# )�#�� *�$.$ �"��(�, /0"$-1��2*��! .$*,&��*#% ""3! ,"�% �*�# #. E-mail: [email protected]


�� .�. � ��

2013, 5, 1 35

. : ( ) ( ), 0G G= =/ ν μν μ . n . 1 1=ν μ -

1 1=/ν μ , ( ) ( )G xν ( ) ( )G xμ , . – [4]

| | 1 | | 10 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ,( )p m m p x p m xx x

H x H x H x H x ds m p H x H x ds= =

∂ ∂= − = −∂ ∂ν ν

( )pH x ( )mH x p m≠ . 2n = . ( ) ( )G xν ( ) ( )G xμ

1x : 2 0=ν ( ) ( )G xν – , 2 1=ν , . 2> . 2 2<ν μ . G - 0k∈

0skG ≡ 2 2/ nxΔ = Δ − ∂ ∂ . ,

( ) ( ) ( ) ( ) ( ) ( ), ( ) ( ),n

G x G x G x G xx − =

∂ = Δ = −∂ μ μ μ μ

1 2( 1, , , )n− = −μ μ μ μ ( )= − −=μ μ . , 2 21 2 1 2( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )( )G D G x G D G D G x G x− −= ν μ

ν μ ν μν ν μ μ . (1)

k m≥ ( )sH x2 2 2| | ( ) | | ( ) ,

(2,2) ( 2 ,2) (2,2) ( 2 ,2)

k k mm s s

k k k m k m

x H x x H xn s n s

−

− −Δ =

+ +

2 221 2

1 21 2

2

[ / 2] [ / 2]2 2 ,!2

2 1 20 0

( ) ( ) ( )

( 1) ( 1) | |( )

(2,2) ( 1 2 ,2) ( 2 )! (2,2) ( 1 2 ,2)

k k

k kk i k ji i j jn n

i i j ji j

G D G x H x

D x xH x

n k i n

− −

= =

=

− Δ −= =

− + − − +

ν μμ

μν μ1 2 1 2 1

2

2 1 2 1

2 2

[ / 2] [( ) / 2] 2 2 ,!2 2

2 1 20

[( ) / 2] 2 ,!2

1 2 1 220

( 1) | |1 ( )(2,2) ( 1 2 ,2) ( 2 )! (2,2) ( 1 2 ,2)

( 1) | |( , ) ( ) ( , )(2,2) ( 1 2 ,2)

k k k i k k i jj i j in

i i j i j ii j i

k k k k jj jn

kj jj

x x H xn k i n

x xC k H x C k Gn

− + − + −− −

− −= =

− − −

=

−= =

− + − − +

−= =

− +

μ

μ

ν μ

ν νμ

221

( ) ( ),k x H x−μ

μ

1[ / 2]

1 2 2 10

( , ) 1/((2,2) ( 1 2 ,2) ( 2 )!)k

i ii

C k n k i=

= − + −ν ν . (1) 1 1 2k = −ν ν 1 1 2k = −μ μ

21 2 1 2( ) ( ) 1 2 2 ( ) ( )( ) ( )( ) ( ) ( , ) ( ) ( ) ( )[ ]G D G x C G D G x G x− − −= − μ

ν μ ν μμ μ ν νν ν ν . (2)

1 2 1 2 2 2 0− − + = − <μ μ ν ν ν μ , , ( ) ( )G xν ( ) ( )G xμ -. 2 2>ν μ .

2 2=ν μ . (2) , 0 1sG =

( ) ( ) 1 2 2 ( ) ( ), ( , ) ,G G C G G= −ν μ ν μν ν ν . 1 1=ν μ =/ν μ =/ν μ . , , ( ) ( )G xν ( ) ( )G xμ . .

[5], k -32 2

32kk nk nh

nn+ −+ −=−−

. ( ) ( )G xν 1 k=ν

, k .

{ }( ) ( ) : 1, ,ikkH x i h= , ( ) 2( ( ))i

x nkSH x ds

∂=ω .


. « . . » 36

4. ( )u x – S S

( ) ( )

0 1( ) ( ),

khi i

k kk i

u x h H x∞

= == (3)

( ) ( )1/ ( ) ( )i ink kS

h H u d∂

= ω ξ ξ ξ .

. 1 2( , ) ( 2) | | nE x n x− −= − −ξ ξ ( 2n > ), 1 1/ /n nx x x xΛ = ∂ ∂ + + ∂ ∂x S= ∈∂/ ξ

2

1

( ) ( ) 1 | |( , ) ( , ) .| | | |

ni i i i i i

x n ni

x x x xE x E xx x=

− − − −Λ −Λ = − =− −ξ

ξ ξ ξξ ξξ ξ

11 [7]. | | | |x < ξ(2 2)

( ) ( )

0 1

| |( , ) ( ) ( )2 2

khk ni i

k kk i

E x H x Hk n

− + −∞

= ==

+ −ξξ ξ .

| | | | 1x < =ξ2

( ) ( ) ( ) ( )

0 1 0 1

1 | | ( , ) ( , )| |

(2 2) ( ) ( ) ( ) ( ).2 2

k k

xn

h hi i i i

k k k kk i k i

x E x E xx

k k k n H x H H x Hk n

ξξ ξξ

ξ ξ∞ ∞

= = = =

− = Λ − Λ =−

− + + −= =+ −

(4)

, (4) S∈∂ξ x | | 1x ≤ <α .

S∈∂ξ (3) 2

( ) ( )

0 1

( ) ( ) ( ) ( )

0 1 0 1

1 1 | | 1( ) ( ) ( ) ( ) ( )| |

1 ( ) ( ) ( ) ( ).

k

k k

hi i

k knS Sn n k i

h hi i i i

k k k kSnk i k i

xu x u ds H u ds H xx

H u ds H x h H x

∞

∂ ∂= =

∞ ∞

∂= = = =

−= = =−

= =

ξ ξ

ξ

ξ ξ ξω ωξ

ξ ξω

5. ( )u x – S S ,

( ) ( ) | 0| | 1( ) ( ) ( ) ( ) ,xG u ds g G D u x ==

=ν ξ ν νξξ ξ (5)

22 2

( ) ( )| | / | |L Dg G G=ν ν ν . . (3) 4. i k -

ν , 1 nk= ≥ ≥ν ν ( 0,1n =ν )

2

( )( )

( ) ( )

( )( ) ,

| |i

nkL S

G xH x

G ∂= ν

νω

1 2

( )( )2 | | 1

0 ( ) ( )

( )( ) ( ) ( ) .

| |L S

G xu x G u ds

G

∞

== ∂

= νν ξξ

ν νξ ξ

( ) ( )G Dμ ( -| | 1x ≤ <α ) 0x =

1 1 2

( ) ( )( ) | 0 ( ) ( )2 | | 1 | | 1

( ) ( )

, 1( ) ( ) ( ) ( ) ( ) ( ) .| |x

L S

G GG D u x G u ds G u ds

gG= = == ∂

= =μ νμ ν ξ ν ξξ ξνν μ ν

ξ ξ ξ ξ


�� .�. � ��

2013, 5, 1 37

( ) ( )G xμ ( 3). (5). .

5

| | 1( )m xx

Q x ds=

, ( )mQ x – m .

6.

/ 2| | 1

0, 2 1( ) ( ) , 2

!! ( 2)

mm x mx

n

mQ x ds Q x m

m n n m=

∈ −= Δ

∈+ −

ω. (6)

. 4 [7]. 1. 2 ( )m kR x− ( )mQ x

2 22 2( ) ( ) | | ( ) | | ( )l

m m m m lQ x R x x R x x R x− −= + + +

2

210

( 1) | | ( )2 4 2( ) .(2,2) (2,2) (2 4 2 2,2)

s s s km

m kk s s ks

x Q xm k nR xm k s n

+∞

−+ +=

− Δ− + −=− − + −

,

2 2| | 1 | | 1 0

0, 2( ) ( ) ( ) ( ) .

, 2( )m x m m m l xx x n

l mQ x ds R x R x R x ds

R l m− −= =

<= + + + =

=ωm 2 0m l− > 2 (0) 0m lR − = .

m , 1 ( 0s = , / 2k m= ) / 2 / 2

| | 1/ 2 / 2 1

( ) ( )2( ) .(2,2) ( 2,2) !! ( 2)

m mm m

m x n nxm m

Q x Q xnQ x dsn m n n m=

+

Δ Δ−= =− + −

ω ω

. [6] .

. ( )u x S S , -

1

1

2( )

( ) ( ) | 0| | 11

( )1 ( ) ( ) ( ) ( ) ,(2 )!!( ,2) x

n

G xG u ds G D u x

n ==

Δ=

νν

ν ξ νξ νξ ξ

ω ν (7)

( ) ( ) ( )( ) ( ) / | |DG x G x G=ν ν ν . . , 5 6

( 12m = ν ) 1

2 21

2( )2 2 2

( ) ( ) ( )1

( )| | / | | | |

(2 )!!( ,2)L D L nG x

g G G Gn

Δ= = =

νν

ν ν ν ννω

ν. (8)

gν (5)

( )| |DG ν nω (7). .

. ( )u x – S S ,

| | 1 | | 1

2| | 1

1 1 1 1( ) (0), ( ) (0),( 2)

1 1 1( ) (0) (0).( 2)

i i j

i i

i x i j x xn n

i x xn

u ds u u ds un n n

u ds u un n n

= =

=

= =+

= ++

ξ ξξ ξ

ξξ

ξ ξ ξ ξ ξω ω

ξ ξω

I. (5). 1 1n ie e − += + +ν ,

1, ,( , , )i i n ie = δ δ ,


. « . . » 38

3 121 2 2 3 2 1 1( ) ( ) ( 1) (3) (2)

1 1 0 0 00 ( ) 0 ( 1) 1 ( ) 0 (3) 0 (2)

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) .

n nn n nn n

n n i i

G x G x G x G x H x

G x G x G x G x H x x

−− − −−− − −

−

= =

= =

ν ν ννν ν ν ν ν ν ν ν

22 2 2 2

1| | 1 | | 1

1| | ,( ) ni L i x n xx x

x x ds x x dsn n= =

= = + + =ω

2| | 1ii D x ix D x= = ,

22 2| | / | | /i L i D ng x x n= =ν ω

| 0| | 1( ) ( ) .

in

i x xu ds D u xn ==

=ξξ

ωξ ξ

II. 1 1 2 12 2 n j n j n ie e e e− + − + − += + + + + +ν i j> . 3 12

1 2 2 3 2 1 1( ) ( ) ( 1) (3) (2)

2 2 1 1 1 0 0 00 ( ) 0 ( 1) 1 ( ) 0 ( 1) 0 ( 1) 1 ( ) 0 (3) 0 (2)

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) .

n nn n nn n

n n i i j j i j

G x G x G x G x H x

G x G x G x G x G x G x G x H x x x

−− − −−− − −

− − +

= =

= =

ν ν ννν ν ν ν ν ν ν ν

22 2 2

| | 1| |i j L i j xxx x x x ds

== (6).

2 2( )m i jQ x x x= , 4m =/ 2

2 2 2 2 2

2

( ) 1 1 12 2 .!! ( 2) 4!!( ,2) 8 ( 2) ( 2)

( ) ( )m

mi j i j

Q x x x x xm n n m n n n n nΔ

= Δ = Δ + =+ − + +

2 2| | 1

.( 2)

ni j xx

x x dsn n=

=+ω 2| | 1

i ji j D x x i jx x D D x x= = ,

22 2| | / | | / ( 2)i L i D ng x x n n= = +ν ω

| 0| | 1( ) ( ) .

( 2) i jn

i j x x xu ds D D u xn n ==

=+ξξ

ωξ ξ ξ

III. 12e=ν2 2 2

0 0 0 1 1( ) 2 ( ) 0 ( 1) 0 (2)( ) ( ) ( ) ( ) .

2 2( 1)n n

n nx x xG x G x G x H x

n−

−+ +

= = −−ν

, 2 2 2 2 2

1 1( )

| |( ) ,2 2( 1) 2( 1) 2( 1)n n nx x x nx xG x

n n n−+ +

= − = −− − −ν

2 2( )

1 12 ( ) | | ,nnx G x x

n n−= +ν , (5)

2( )| | 1 | | 1 | | 1

( ) | 0 ( ) | 0

2| 0

1 1( ) 2 ( ) ( ) ( )

1 1 12 ( ) ( ) (0) 2 ( ) ( ) (0)

( ) (0).

( )

n

n

n nx x

nx x

nu ds G u ds u dsn n

n ng G D u x u g G D u x un n n n n

g D u x un

= = =

= =

=

−= + =

− −= + = + Δ + =

= +

ξ ν ξ ξξ ξ ξ

ν ν ν ν

ν

ξ ξ ξ ξ ξ

ω ω

ω

gν . , 2( ) ( ) ( )| | ( ) ( )

2 2DnG G D G x

n= =

−ν ν ν , -

(8) ( 1 2=ν ) 2 2 2 2 2 2 4 2 2 4

2 2( ) 2 2

22

2 2

| | 2 | | | |2 2 ( )8 ( ,2) 16 ( 1)( 2) 16 ( 1)( 2)

24 16 ( 2) 8 ( 2) 16 ( 1) 1 .( 2)16 ( 1)( 2) 16 ( 1)( 2)

( ) ( )n n n

n

nx x n x nx x xg n G xn n n n n n n n

n n n n n n nn nn n n n n n

Δ − Δ − +−= Δ = = =− + − +

− + + + −= = =+− + − +

ννω


�� .�. � ��

2013, 5, 1 39

gν nωi n= . 1, ,i n= .

1. , . . - I / . . // . « -

. . ». – 2011. – . 4. – 10(227). – . 27–38. 2. Karachik, V.V. On one set of orthogonal harmonic polynomials / V.V. Karachik // Proceedings of

American Mathematical Society. – 1998. – V. 126, 12. – P. 3513–3519. 3. Karachik, V.V. On some special polynomials / V.V. Karachik // Proceedings of American

Mathematical Society. – 2004. – V. 132, 4. – P. 1049–1058. 4. , . . / . . . – .: , 1976. –

296 c. 5. , . / . ,

. . – .: , 1974. – 332 c. 6. , . . /

. . // . – 2007. – . 10, 2. – . 142–162. 7. , . . -

/ . . // . – 2011. – . 51, 9. – C. 1674–1694.

ON ONE GENERALIZED MEAN THEOREM FOR HARMONIC FUNCTIONS

V.V. Karachik1

Mean value on the unit sphere of a product of a harmonic function in the unit ball and a homogene-ous polynomial is investigated.

Keywords: harmonic functions, mean theorem, harmonic polynomials.

References 1. 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]. Vestnik YuUrGU. Seriya «Matematika. Mekhanika. Fizika». 2011. Issue 4. no. 10(227). pp. 27–38. (in Russ.).

2. Karachik V.V. On one set of orthogonal harmonic polynomials. Proceedings of American Mathe-matical Society. 1998. Vol. 126, no. 12. pp. 3513–3519.

3. Karachik V.V. On some special polynomials. Proceedings of American Mathematical Society,2004. Vol. 132, no. 4. pp. 1049–1058.

4. Bicadze A.V. Uravneniya matematicheskoj fiziki [Equations of mathematical physics]. Moscow: Nauka, 1976. 296 c.

5. Stejn I., Vejs G. Vvedenie v garmonicheskij analiz na evklidovykh prostranstvakh [Introduction to Fourier Analysis on Euclidean Spaces]. Moscow: Mir, 1974. 332 p. (in Russ.). [Stein E.M., Weiss G.L. Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, 1971. 297 p.]

6. Karachik V.V. Matematicheskie trudy. 2007. Vol. 10, no. 2. pp. 142–162. 7. Karachik V.V. Zhurnal vychislitel'noj matematiki i matematicheskoj fiziki. 2011. Vol. 51, no. 9.

pp. 1674–1694. (in Russ.). �� 30 �� 2013 �.

1 Karachik Valeriy Valentinovich is Dr. Sc. (Physics and Mathematics), Professor, Mathematical Analysis Department, South Ural State Uni-versity. E-mail: [email protected]


. « . . » 40

514.17

1

. . 2, . . 3

-, , -

. , -. -

. , .

�� : �� , ��, �� , �� .

[1], [2] -

Rn [3]. , ,

[4]. comp(Rn) Rn.

1. x∈Rn A -

(x, A) = min{||x − y||: y∈A}. 2. [5, 6] A∈comp(Rn) B∈comp(Rn)

h(A, B) = max{ (x, B): x∈A}. (1)

3. [5, 6] A∈comp(Rn)B∈comp(Rn)

d(A, B) = max{h(A, B), h(B, A)}. (2) [5]

, -.

( - clos(Rn)). A, B ,

(1) . -

h*(A, B) = sup{ (x, B): x∈A}. (3) -

d*(A, B) = max{h*(A, B), h*(B, A)}. (4) (3) (4) A B + . -

- 1 �� !�"#$#� !%& '&#�#(��) !�**$%+,$ �--. (!%�$,� 11-01-00427-� «/"0�%&�1 & *&#�1&2$(,&$ !%�3$*4% %$5$#&6 �*&''$%$#3&�"7# 8 &0%�8 & 9�*�2�8 4!%��"$#&6», 11-01-12088-�'&-1-2011 «:$��* !�9&3&�## 8 *&''$%$#3&�"7# 8 &0% � 9�*�2�8�$8#&,&, ;,�#�1&,& & ;,�"�0&&» & !%�$,�� 12-01-31247 1�"_� «<!%��"$#&$ & (&#04"6%#�(�& � *&''$%$#3&�"7# 8 &0%�8»), & !%&!�**$%+,$ !%�0%�11 '4#*�1$#��"7# 8 &(("$*��#&) =%$9&*&41� �/> 12-=-1-1002 «<!%��"$#&$ � 4("��&68 ,�#'"&,�� & #$�!%$*$-"$##�(�&. =�9&3&�## $ (�%��$0&& & 0�1&"7��#�� ,�#(�%4,3&& � 9�*�2�8 4!%��"$#&6» & 12-?-1-1017/3 «@�2$(��$##�6 �$�%&6 & 2&(-"$## $ 1$��* *"6 9�*�2 *&#�1&,&, 4!%��"$#&6 & �!�&1&9�3&&» !%& '&#�#(��) !�**$%+,$ <%A �/>. 2 B$�$*$� =��$" CDEFGEHIEJ – ,�#*&*�� KELEMN-DOFHDOFEJHPMEQ #�4,, ��*$" *&#�1&2$(,&8 (&(�$1 .#(�&�4� DOFHDOFEME & 1$-8�#&,& <%A �/>. E-mail: [email protected] <5�,�� R"�*&1&% >&,�"�$�&2 – *�,��% KELEMN-DOFHDOFEJHPMEQ #�4,, ��*$" *&#�1&2$(,&8 (&(�$1 .#(�&�4� DOFHDOFEME &1$8�#&,& <%A �/>. E-mail: [email protected]


�� .�., �� .�. ��

2013, 5, 1 41

. , [7] -

. -– [8–10]. -

. , 2 2

0 0 0 04 40, ,

2+ + + + +a b a b

a0 b0 – A B , ||a|| – . – clos(Rn) [9,

c. 187]. . -. , -

. ,

– . -.

, -, -

, -[12]

1 exp( )x x= − − . (5)

[0, + ) [0, 1). x + x (5) 1x→ . 4. K- A B

( )* *

*

1 exp ( , ) , ( , ) (0, )( , )

1, ( , ) .

h A B h A Bh A B

h A B

− − ∈ +∞=

= +∞. (6)

5. K- A B

( , ) max{ ( , ), ( , )}d A B h A B h B A= . (7) 3 , (7)

( )* *

*

1 exp ( , ) , ( , ) (0, )( , )

1, ( , ) .

d A B h A Bd A B

d A B

− − ∈ +∞=

= +∞. (8)

1. (8) clos(Rn). . , , (8),

[13]. -. , -

. , A, B, C∈clos(Rn) ( , ) ( , ) ( , )d A B d A C d B C≤ + . (9)

, = d*(A, B), = d*(A, C), = d*(B, C).

(9) 1 exp( ) 1 exp( ) 1 exp( )α β γ− − ≤ − − + − − . (10)

, (10) = + . -

α β γ≤ + .


. « . . » 42

1 exp( ) 1 exp( ) 1 exp( )β γ β γ− − − − + − − + − =exp( )(1 exp( )) (exp( ) 1)β γ γ= − − − + − − =

(1 exp( ))(1 exp( )) 0γ β= − − − − − ≤ . , (10) . -

= + , ∈[0, + ]. , - (10) . d .

(8) [0, 1]. d (A, B) =1 , d*(A, B)=+∞. -

. , d, .

clcv(Rn) Rn. 2. A, B∈clcv(Rn).

( )( )( )*( , ) lim 1 exp ( ), ( )d A B d A Bε

ε ε→+∞

= − − , (11)

M( ) = M U(0, ), U(x, ) – Rn x , 0 – Rn.

(11) , , A( ) B( ) -.

. , h*(A, B)≥h*(B, A) , -, d*(A, B)=h*(A, B). ,

( )*( , ) lim ( ), ( )h A B h A Bε

ε ε→+∞

= , (12)

(11). , (12) .

h*(A, B). B – , (12) ,

0, > 0 B( ) = B. , h*(A, B) – , -1, > 1 h(A( ), B( )) = h*(A, B).

h*(A, B) = + , h(A( ), B( )) ≥ − 0 > 0., A , B , , -

, h*(A, B) ≥ h*(B, A). , (12) A

B .

1{ } ,i i iε ε∞= → +∞ >0,

1) ∀i∈N: h(A( i), B( i)) < h*(A, B) − ,

2) ∀i∈N: h(A( i ), B( i)) > h*(A, B) + . , 1). -

1{a }i i A∞= ⊂ ,

( ) *lim , ( , )ε

ρ→+∞

=i B h A Ba . (13)

B – , ai

bi∈B, ||ai − bi|| = (ai, B). ai, bi ,

∀ > max{||ai ||, ||bi ||} h(A U(0, ), B U(0, )) ≥ ||ai −bi|| ≥ (ai, B). (13) , 1 .

, 2). h*(A, B) = r* -, , + .

1 1{ } { ( )}i i i iA A ε∞ ∞= == A 1 1{ } { ( )}i i i iB B ε∞ ∞

= == . ∀i ∈ N

h(Ai, Bi) > r* + . Ai – , ai,


�� .�., �� .�. ��

2013, 5, 1 43

(ai, Bi) > r* + . (14) bi∈B\Bi,

||ai − bi|| ≤ r*. ai r* U(0, i) U(0, i) -

ai∈U(0, i), bi ∉U(0, i) ( . 1), ||ai||∈[ i − r*, i].

, i + ||ai|| + . b0, B .

bi , ai bi , r* -, . , ∠(ai, bi – b0) -

ai bi – b0 : 0lim ( , ) 0

→∞∠ − =i ii

a b b . (15)

i [bi, b0] U(0, ||ai||) U(0, ||ai||) bi

* ( b0∈U(0, i), bi∉U(0, i)). U(0, ||ai||)

Γi = {g∈Rn : g – ai, ai = 0},ai - ( . 2), U(ai, r* + ) -

r*. , . U(0, ||ai||) Γi -ai U(0, ||ai||) + . -

( )* *lim ( , ), ( , ) ( , ) 0γ γ→∞

Γ + ∂ + =i i i iid U r U U ra 0 a a . (16)

(15) (16) , bi* i bi. -

,−= − i i i

i ii

b a ap b

abi Γi. , [bi, b0], -

ai bi – b0 . bi* si -

U(0, ||ai||) , bi ai.

C , U(0, ||ai||) -ai Γi, si pi . -

*lim 0i ii→∞− =b p . (17)

��. 1. �� A � B

��. 2. �� ai, bi, bi

*� pi


. « . . » 44

pi bi , ai,

*i i i i r− ≤ − ≤p a p b .

, 1{ }i i∞=p *

1{ ( , )}i iU r ∞=a

pi ∈ U(ai, r*). (18) (17) (18) , bi

* U(ai,r*) :

( )* *lim , ( , ) 0i iiU rρ

→∞=b a . (19)

bi* ( ) Bi ,

*i i r γ− > +b a .

(19). . -, (12) .

, 1) 2) (12).

1. A B . (11) . , , -

. A = {(x, y): x∈[0, + ), y = 0}, A = {(x, y): x∈N {0}, y=0}. i, i∈N, -

d*(A U(0, i), B U(0, i)) = 0,5. i + 0,9, i ∈ N,

d*(A U(0, i + 0,9), B U(0, i+ 0,9)) = 0,5., (11) .

A B K-. h*(A, B) = + h*(B, A) = + , d (A, B) = 1. ,

. - , .

, , A B -. ,

-, , [14].

6. [14, c. 77] A 0+A = {x ∈ Rn: ∀y ∈ A, ∀ > 0 (y + x∈ A)},

x, A, A. A -

. -A∈clcv(Rn), 0+A = {0} , A – [14, c. 81].

7. - A B clcv(Rn)(A, B) = d(0+A U(0, 1), 0+B U(0, 1)). (20)

(20) [0, 1]. , 0, U(0, 1) , -

1. clcv(Rn). -, ,

. , . G = epi f

f(x) = ax2 + bx + c, a ∈ (0, + ), b, c ∈(− , + ) c , , .


�� .�., �� .�. ��

2013, 5, 1 45

, -.

3. A, B ∈ clcv(Rn). ( ( ), ( ))( , ) lim d A BA B

ε

ε εχε→+∞

= . (21)

. , -

( ( ),0 ( ))lim 0d A Aε

ε εε

+

→+∞= . (22)

0+A A. x ∈ A. 7 ,

{x} + 0+A ⊆ A.A , -

x. h*(0+A, A) ≤ h*(0+A, 0+A + {x}) ≤ ||x||.

, h*(0+A, A) = d1 ∈ [0, ||x||] – . (12)

1(0 ( ), ( ))lim lim 0dh A Aε ε

ε εε ε

+

→+∞ →+∞= = . (23)

, ( ( ),0 ( ))lim 0h A A

ε

ε εε

+

→+∞= . (24)

, (24) . -d2 1{ }i iε ∞

= , ∀i ∈ N: h(A( i), 0+ A( i)) > d2 i, i + .

1{ }i i∞=a

∀i∈N: ai∈ A( i) (25)

∀i ∈ N: (ai , 0+A( i)) > d2 i. (26)

K* = {∀x: (∃y ∈ 0+A : ∠(x, y) ≤ arctg d2)}. (26)

∀i ∈ N: ai∉ (A( i) + U(0, d2)) ⊇ K*( i). (25)

∀i∈N: ai ∉ K*. (27)

{ } 1|/i i i

∞=

a a . ,

. . -,

{ } *1|

/i i i∞=→a a a .

(27) , /i ia a K*. , :

( )* * *\n K K∈ ∂a R .

0+A K* ( ). , a*∈U(0, 1) 0+A. [14, c. 79] -

, { } 1|i i iλ ∞=a { } 1|i i A∞

= ⊂a , i>0, i 0 (

) 0+A.


. « . . » 46

{ } 1|/i i i

∞=

a a , i = ||ai||−1.

, a*∈0+A.

. , (23) (24) , (22).

d(A( ), B( )). d(A( ), B( )) ≤ d(A( ), 0+A( )) + d(0+A( ), 0+B( )) + d(B( ), 0+B( )). (28)

d(0+A( ), 0+B( )) ≤ d(A( ), 0+A( ))+ d(A( ), B( )) + d(B( ), 0+B( )).

d(A( ), B( )) ≥ d(A( ), 0+A( ))− d(0+A( ), 0+B( )) − d(B( ), 0+B( )). (29)

( ( ), ( )) (0 ( ),0 ( )) ( ( ),0 ( )) ( ( ),0 ( ))lim lim limd A B d A B d A A d B Bε ε ε

ε ε ε ε ε ε ε εε ε ε

+ + + +

→+∞ →+∞ →+∞

+≤ + ≤

(0 ( ),0 ( ))lim 0 0d A Bε

ε εε

+ +

→+∞≤ + + (30)

(29) ( ( ), ( )) (0 ( ),0 ( ))lim limd A B d A B

ε ε

ε ε ε εε ε

+ +

→+∞ →+∞≥ . (31)

(30) (31) ( ( ), ( )) (0 ( ),0 ( ))lim limd A B d A B

ε ε

ε ε ε εε ε

+ +

→+∞ →+∞= . (32)

, ,

(0 ( ),0 ( ))d A Bε εε

+ +

> 0. = 1 ( + ) (32), (21).

d [0, 1] . , -

, , . , , d (A, B) (A, B).

8. A, B∈clcv(Rn) -

( , ) ( , ) ( , )D A B d A B A Bχ= + . (33) 1. (33) clcv(Rn)

( )( ) ( )( ), ( )( , ) lim 1 exp ( ), ( )

d A BD A B d A B

ε

ε εε ε

ε→∞= − + .

. (33) , [13]. 2 3 , d (A, B)

(A, B) A B (11) (21).

( )( )( ) ( )( ), ( )lim 1 exp ( ), ( ) lim

d A Bd A B

ε ε

ε εε ε

ε→∞ →∞− − + .

D(A, B) . , (A, B), – [8]. -


�� .�., �� .�. ��

2013, 5, 1 47

, , -D(A, B) d*(A, B),

, D(A, B)∈[0, 1). d*(A, B)=+ , -D(A, B)∈[1, 2], -

. . D ,

. , D -

– . , , 1{ }i iA ∞= , -

:Ai = {(x, y) : y = x tg i , i = 1/i, x∈(− , )}.

, , 1 1+sin . –

, , sin ( . [9, c. 187]). Ai i = 1/i. -

– 1{ }i iA ∞= , D .

1.

A, B ∈ clcv(R2). , 2epi 1A x= + , x ∈ (− , + ), B = epi(1/x), x ∈ (0, + ).

, ( , ) 1.d A B =

: 0+A = {(x, y) : x = r cos t, y = r cos t, t ∈ [ /4, 3 /4], r ∈ [0, + )},

0+B = {(x, y) : x = r cos t, y = r cos t, t ∈ [0, /2], r ∈ [0, + )}. /4. -

2( , ) sin( / 4)2

A Bχ π= = .

A B 2( , ) ( , ) ( , ) 1 1,707.

2D A B d A B A Bχ= + = + ≈

2.A, B ∈ clcv(R2). , A = epi(exp(x) + 1), x ∈ (− , + ), B = epi exp(x), x ∈ (− , + ).

, -

��. 3. �� A � B � �� 1 ��. 4. �� A � B � �� 2


. « . . » 48

(4), d*(A, B) = h*(B, A) = 1. -B A, 1 ,

( )( )*( , ) lim ,exp( ) , 1,t

h A B t t A→+∞

= − − = ( ),exp( ) .t t B− − ∈

A B ( , ) ( , ) ( , ) 1 exp( 1) 0 0,632.D A B d A B A Bχ= + = − − + ≈

3.A, B ∈ clcv(R2). , A = epi(exp(x) + x/4), x ∈ (− , + ), B = epi x4,x ∈ (− ,+ ).

-.

: 0+A = {(x, y) : x = r cos t, y = r cos t, t ∈ [ /4, /2 +

arctg(1/4)], r ∈ [0, + )}, 0+B = {(x, y): x = 0, y ∈ [0, + )}.

/2 + arctg(1/4) > /2.

(A, B) = 1.A B

( , ) ( , ) ( , ) 1 1 2.D A B d A B A Bχ= + = + =

1. , . . : , , / . . . – .: , 2009. – 288 .

2. , . . / . . , . . . – .: , 1974. – 456 .

3. , . . - / . . , . . , . . // . . – 1987. –

. 51. – . 2. – . 216–222. 4. , . / . . – .: , 2006. – 304 .5. , . . / . . , . . , . . . –

.; : - . ., 2004. – 511 c.6. , . / . . – .: , 1985. – 335 .7. , . / . , . . – .: , 1979. – 399 .8. , . . /

. . // . - . . – 1941. – 5. – . 1–52. 9. , . . . . . . -

/ . . , . . // . – 2009. – . 15, 3. – . 185–201.

10. , . . clcv(Rn) – - / . . , . . , . . // -

. – 2011. – . 17, 1. – . 162–177. 11. , . . / . . . – : ,

1981. – 384 .12. . . – . I /

. . // . – 1975. – . 98. – . 3. – . 450–493. 13. , . . / -

. ., . . – .: , 1976. – 544 .14. , . / . . – .: , 1973. – 472 .

��. 5. �� A � B � �� 3


�� .�., �� .�. ��

2013, 5, 1 49

A VARIANT OF A METRIC FOR UNBOUNDED CONVEX SETS

P.D. Lebedev1, V.N. Ushakov2

Convex analysis methods are used for the construction of distance function between closed (un-bounded in common case) sets of Euclidean space. It is shown that the distance satisfies all properties of metric. It is proved that this distance is invariant under motion of the sets in space. This metric space is proved to be complete.

Keywords: Hausdorff distance, metric, convex set, recessive cone.

References 1. Mesteckij L.M. Nepreryvnaya morfologiya binarnyx izobrazhenij: figury, skelety, cirkulyary

[Continuous morphology of binary images: figures, shells, circulars]. Moscow: Fizmatlit. 2009. 288 p. (in Russ.).

2. Krasovskij N.N., Subbotin A.I. Pozicionnye differencial'nye igry [Positional differential games]. Moscow: Nauka, 1974. 456 p.

3. Taras'ev A.M., Ushakov V.N., Khripunov A.P. Prikladnaya matematika i mekhanika. 1987. Vol. 51. Issue 2. pp. 216–222. (in Russ.).

4. Khausdorf F. Teoriya mnozhestv (Set Theory). Moscow: Kom kniga, 2006. 304 p. (in Russ.). [Hausdorff F. Set Theory. American Mathematical Soc., 1957. 352 p.]

5. Burago D.Yu., Burago Yu.D., Ivanov S.V. Kurs metricheskoj geometrii [The course of metric geometry]. Moscow; Izhevsk: Institut komp'yuternyx issledovanij, 2004. 511 p. (in Russ.).

6. Lejxtvejs K. Vypuklye mnozhestva [Convex sets]. Moscow: Nauka, 1985. 335 p. (in Russ.). [von K. Leichtweiss Konvexe Mengen. Berlin: VEB Doutscher Verlag der Wissenschaften, 1980. 332 p. (in Ger.).].

7. E'kland I., Temam. R. Vypuklyj analiz i variacionnye problemy [Convex analysis and variational problems]. Moscow: Mir, 1979. 399 p. [Ekeland I., Temam R. Convex analysis and variational prob-lems. SIAM, 1976. 402 p.]

8. Bebutov M.V. Bul. mex-mat. fakul'teta MGU. 1941. no. 5. pp. 1–52. (in Russ.). 9. Panasenko E.A., Tonkov E.L. Trudy Instituta Matematiki I Mekhaniki. 2009. Vol. 15, no. 3.

pp. 185–201. (in Russ.). 10. Panasenko E.A., Rodina L.I., Tonkov E.L. Prostranstvo clcv(Rn) s metrikoj Khausdorfa–

Bebutova i differencial'nye vklyucheniya. [The space clcv(Rn) with the Hausdor –Bebutov metric and di erential inclusions]. Trudy Instituta Matematiki I Mekhaniki. 2011. Vol. 17, no. 1. pp. 162–177. (in Russ.).

11. Dem'yanov V.F. Nedifferenciruemaya optimizaciya [Nondifferential optimization]. Moscow: Nauka, 1981. 384 p. (in Russ.).

12. Kružkov S.N. Generalized solutions of the Hamilton-Jacobi equations of eikonal type. I. Mathematics of the USSR – Sbornik. 1975. Vol. 27, no. 3. pp. 406–446. http://dx.doi.org/10.1070/ SM1975v027n03ABEH002522

13. Kolmogorov A.N., Fomin S.V. E'lementy teorii funkcij i funkcional'nogo analiza [Elements of theory of functions and functional analysis]. Moscow: Nauka. 1976. 544 p. (in Russ.).

14. Rokafellar R. Vypuklyj analiz [Convex analysis]. Moscow: Mir. 1973. 472 p. (in Russ.). [R. Tyrrell Rockafellar. Convex Analysis. Princeton University Press, 1997. 451 p.].

� !"#$%&' ( )*+',-%. 18 �� 2012 �.

1 Lebedev Pavel Dmitrievich is Cand. Sc. (Physics and Mathematics), Department of Dynamic Systems of the Institute of Mathematics and Mechanics of the Russian Academy of Sciences (Ural branch). E-mail: [email protected] Ushakov Vladimir Nikolaevich is Dr. Sc. (Physics and Mathematics), Department of Dynamic Systems of the Institute of Mathematics and Mechanics of the Russian Academy of Sciences (Ural branch). E-mail: [email protected]


. « . . » 50

517.948

. . 1

-. , -,

. : �� , �� , �� .

. 1963 . [1] . . . -

, . -, , . [2]

. , .

[3–11] , -.

[12–14] .

E F – , :A E F→ –

. 1. 1A− , -

:R F Eδ → , ( )00,δ δ∈ , x E∈

0lim sup 0

y AxR y xδδ δ→ − ≤

− = .

{ }Rδ1A− .

Ax y= , 1A− . -x R yδ δ δ= -

yδ δ . ( )0,1E C= , ( )2 0,1F L= . , -

A 2L - . A -M , ( ) ( )20,1 0,1C M L⊂ ⊂ .

[2] , A ( )0,1pL , 2p ≥ -

, 1A− . 1. ( ) ( )2: 0,1 0,1A C L→ -

( )0,1pL , 2p ≥ , 1A− ., . -

2p ≥ . , -

1 �� – ��!"�# $�%�!�-&�"�&�"� ��!�� '!, (#�$��#, %��')*�+ !�$��#�+ �,*�+ &�"�&�"�!�, -.��-/#�01�!�+ 2��'��#�"��3+ '��#��"�". E-mail: [email protected]


�� .�. ��

2013, 5, 1 51

, A , .

[14] . , - 1 .

[13] 1. 2. ( ) ( )2: 0,1 0,1A C L→ -

( )0,1L∞ , 1A− . , -

1 2, . 3 -

.

( ),K x t – [ ] [ ]0,1 0,1× .

( ) ( ) ( )1

0

: ,Q f x K x t f t dt→ , (1)

( )0,1C ( )2 0,1L . Q Q ( )2 0,1L . Q

, 1Q− . -Q .

3. ( )0,1C ( )2 0,1L -, ( )0,1pL , 2p ≥

, ( )0,1L∞ ., , ,

. .

11 11 ,1 , 1,2,...2 2k k kJ k+= − − = .

. ( )kh x

[ ]0,1 kJ , ( ) ( )1 0,1k kh x L +∈ , ( ) ( )2 0,1k kh x L +∉ , 1,2,...k = .

M ( )2 0,1L ,

( )kh x . ( )0 ,C a b∞ – ( ),C a b , , . . a b .

[2]. 1. ( ) ( )2 ,h t L a b∈

( ) ( ) ( ) ( )2 , : 0b

a

H f t L a b f t h t dt= ∈ = .

( )0 ,H C a b H∞ = . , N M -

( ){ }n tψ , ( ) ( )0 0,1n t Cψ ∞∈ , 1,2,...n = . ,

( )0 0,1N C N∞ = . , N

, ( )0 0,1N C∞


. « . . » 52

, ( ){ }n tψ .

f N∈ 0ε > . n ,

( )1 1

1 22

2 1 24n n

f t dt ε+ +−

≤ . (2)

( )f t ( )kh t , 1,2,...k = , 1

( ){ }, 1,2,...,kf t k n= , :

( ) ( )

( ) ( )

( )

01. , 1,2,..., ;

2. 0, 1,2,..., ;

3. , 1,2,..., ,2 1

k

k

k k

k kJ

k J

f t C J k n

f t h t dt k n

f f k nnε

∞∈ =

= =

− ≤ =+

(3)

kJf ( )f t kJ . 010,2

J = -

( )0 0 0,1 2f C∞∈ , ( )00 2 1Jf fnε− ≤+

.

( )( )

1

1

, 0,1,..., ;

2 10 ,1 .2

k k

n

n

f t t J k n

g tt

+

+

∈ =

= −∈

(2) (3) ( )0 0,1g ∞∈ , g N∈ , 1,2,...k =

( ) ( ) ( ) ( )1

0

0k

k kJ

g t h t dt g t h t dt= = ,

m M∈

( ) ( )1

0

0g t m t dt = .

,

( ) ( )1 2

2 2

00 4k

n

k J

g f g f dt g f dt ε=

− = − ≤ − + ≤

( ) ( )2

0 02 2 1 2k

n n

k kJ

g f dtn

ε ε ε ε= =

≤ − + ≤ + =+

, ( )0 0,1N C N∞ = . Q (1)

( ) ( ) ( )1

, n n nn

K x t a x tϕ ψ∞

== , (4)

( ){ }n xϕ –

[ ]0,1 , ( ){ }n tψ –


�� .�. ��

2013, 5, 1 53

( ) ( ) ( )( ) ( ) ( ) ( )( )1 1

2

0 1 0 1sup max ,..., sup max ,...,n n

n n n n nx t

a x x t t nϕ ϕ ψ ψ− −

−

≤ ≤ ≤ ≤= .

, (4) , -, , , ( ),K x t –

[ ] [ ]0,1 0,1× .

, kerQ M= . ,

( ) ( ) ( )1

10

0n n nn

a x t f t dtϕ ψ∞

== , (5)

( )1

0n n nn

a b xϕ∞

== , (6)

( ) ( )1

0n nb f t t dtψ= – n - ( )f t ( ){ }n tψ ,

(5) . (6) 0nb = , 1,2,...n = , . . ( )f t M∈ . , ( )f t M∈ ( ) kerf t Q∈ . -, kerQ M= . , Q ,

n nϕ ψ= . , Q . , -Q ( )0,1pL ,

kh , . Q ( )0,1L∞ -, M , , . .

, p - 1 2 , -

3 Q 1 1Q− , 2 1Q− .

1. , . . / . . // . . – 1963. – . 151, 3. – . 501–504.

2. , . . , / . . // . . – 1978. – . 241, 2. – . 282–285.

3. , . . / . . , . . // . . – 1976. – . 229, 6. – . 1292–1294.

4. , . . / . . , . . // . . – 1978. – . 241, 5. – . 1027–1030.

5. , . . / . . // . . . – 1979. – 11. – . 34–39.

6. , . . , - / . . // . . . – 1979. – 12. – . 35–38. 7. , . . / . . ,

. . , . . , . . // . . – 1983. – . 270, 1. – . 25–28.

8. i , . . ii i i / . . i , . . i // . . . – 1990. – . 42, 6. – . 777–781.

9. , . . / . . , . . // . . – 1998. – . 363, 5. – . 961–964.

10. , . . / . . // . – 1999. – . 40, 1. – . 130–141.


. « . . » 54

11. , . . - / . . // - .

« , , ». – 2003. – . 3. – 6(22). – . 9–16. 12. , . . , -

/ . . // . . – 1999. – . 65, 2. – . 222–229. 13. , . . -

/ . . // . . – 2007. – . 82, 2. – . 242–247. 14. , . . /

. . , . . // . – 2009. – . 1(43). – . 11–15.

ON CONNECTION BETWEEN SUFFICIENT CONDITIONS OF REGULARIZABILITY OF INTEGRAL EQUATIONS

L.D. Menikhes1

In this paper some infinite series of sufficient conditions of regularizability of integral equations is investigated. It is proved, that any two of these conditions are not equivalent, even if the integral equa-tions have the smooth kernels.

Keywords: regularizability, integral equations, smooth kernels.

References 1. Tikhonov A.N. Dokl. AN SSSR. 1963. Vol. 151, no. 3. pp. 501–504. (in Russ.). 2. Menikhes L.D. Soviet Math. Dokl. 1978. Vol. 19, no. 4. pp. 838–841. (in Russ.). 3. Vinokurov V.A., Menikhes L.D. Soviet Math. Dokl. 1976. Vol. 17, no. 4. pp. 1172–1175. (in

Russ.). 4. Menikhes L.D., Plichko A.N. Soviet Math. Dokl. 1978. Vol. 19, no. 4. pp. 963–965. (in Russ.). 5. Menikhes L.D. Izv. vuzov. Matem. 1979. no. 11. pp. 34–39. (in Russ.). 6. Menikhes L.D. Izv. vuzov. Matem. 1979. no. 12. pp. 35–38. (in Russ.). 7. Menikhes L.D., Vinokurov V.A., Domanskii E.N., Plichko A.N. Dokl. AN SSSR. 1983. Vol. 270,

no. 1. pp. 25–28. (in Russ.). 8. Menikhes L.D., Plichko A.M. Ukr. matem. zhurn. 1990. Vol. 42, no. 6. pp. 777–781. (in Ukr.). 9. Menikhes L.D., Tanana V.P. Dokl. RAN. 1998. Vol. 363, no. 5. pp. 961–964. (in Russ.). 10. Menikhes L.D. Sibirskii matematicheskii zhurnal. 1999. Vol. 40, no. 1. pp. 130–141. (in Russ.). 11. Menikhes L.D. Vestnik YuUrGU. Seriia “Matematika, fizika, khimiya”. 2003. Issue 3. no. 6(22).

pp. 9–16. (in Russ.). 12. Menikhes L.D. Matem. zametki. 1999. Vol. 65, no. 2. pp. 222–229. (in Russ.). 13. Menikhes L.D. Matem. zametki. 2007. Vol. 82, no. 2. pp. 242–247. (in Russ.). 14. Menikhes L.D., Kondrat’eva O.A. Izvestiya Chelyabinskoko nauchnogo centra. 2009. Is-

sue 1(43). pp. 11–15. (in Russ.). �� 22 �� 2012 �.

1 Menikhes Leonid Davidovich is Dr. Sc. (Physics and Mathematics), Professor, Head of General Mathematics, South Ural State University. E-mail: [email protected]


2013, 5, 1 55

517.977

-, II

. . 1

m - ( ) .

. . , . : �� , �� -�� ,

�� , �� , �� , �� .

[1–8] [9]. , , [9].

[9].

3. -- -

0

( ) ( ) ( ) ( ) ( , ) ( ) ( ) ( )t

x t f t A t x t K t s x s ds u t v t= + + + − , 0(0)x x= . (40)

[9]. [0, ]θ ,

[ ] [ ]{ }mxγ θ θ= . (41)

u P∈ -[ ]γ θ [ ]x t , 0 t θ≤ ≤ , (40), -

[ ]u t , 0 t θ≤ ≤ , u P∈ , [ ]v t , 0 t θ≤ ≤ , v Q∈ , -. (41).

0( ) ( , 0) ( ),s s x f sϕ = Φ + ( , ) ( , ) ( , )t

s

t s K t X s dτ τ τΦ = , ( , )X t s – -

( ) ( ) ( )x t A t x t= . (40)

00 0 0

( ) ( , 0) ( , ) ( ) ( , ) ( ) ( , ) ( )t t t

x t X t x x t s s ds x t s u s ds x t s v s dsϕ= + + − . (42)

( , ) ( , ) ( , ) ( , )t

s

x t s X t s X t R s dτ τ τ= + ,

( , )R t s – ( , )t sΦ . , 0t , 00 t θ≤ ≤ ,

0[ ]u t , 0[ ]v t , 00 t t≤ ≤ . [ ] 0u t ≡ , [ ] 0v t ≡t , 0t t θ≤ ≤ , (40) θ (42)

0 0

0 0

0 0 00 0

( , ) ( ) ( , ) [ ] ( , ) [ ] ( , ) [ ] ( , ) [ ] ,t tt t

t t

x t x x s u s ds x s u s ds x s v s ds x s v s dsθ θ θ θ θ θ= + + − − (43)

1 �� !"#$ %&�'!(!) *+#,!'#$!- – "�,'!'�. /!0!"#-(�.+(�.!-+ "!1 ,�2", '#3+,., "�/+')� (�.+(�.!-+ "#4# �,�&!0� ! !,/#)(�.!"!, 5) "!6 42(�,!.�),#-.+1,#&#4!-+ "!6 !, .!.2. (/!&!�& 5)+,72)4 "#4# 4# 2'�) .$+,,#4# 2,!$+) !.+.�). E-mail: [email protected]


. « . . » 56

0 00

( ) ( , 0) ( , ) ( )x X x x s s dsθ

θ θ θ ϕ= + .

3.1. { }, ( , )p t x tθ= t , 0 t θ≤ < ;

{ }0 0 0, ( , )p t x tθ= – , 0 0

0 0 0 00 0

( , ) ( ) ( , ) [ ] ( , ) [ ]t t

x t x x s u s ds x s v s dsθ θ θ θ= + − ,

(40) (43) θ

0 0

0( ) ( , ) ( , ) [ ] ( , ) [ ]t t

x x t x s u s ds x s v s dsθ θ

θ θ θ θ= + + . (44)

. 3.1. U eU ,

0p ,

00 t θ≤ < , :

0 0 0 0( [ ] , ( , ), , ) ( , ( , ))et x t U v t x tγ θ θ ε θ≤ .

3.2. V eV , 0p ,

00 t θ≤ < , :

0 0 0 0( [ ] , ( , ), , ) ( , ( , ))et x t u V t x tγ θ θ ε θ≥ . 3.3. U, V

,e eU V , 0p , 00 t θ≤ < , -

0 0 0 0( [ ] , ( , ) , ) ( , ( , ))eet x t U V t x tγ θ θ ε θ= .

0 0( , ( , ))t x tε θ 0p ,

00 t θ≤ < (43), (44)

{ } { } { }0 0

0 0 01( , ( , )) max max ( , ) [ ] max ( , ) [ ] ( , ) ,m m ml v Q u Pt t

t x t l x s v s ds l x s u s ds l x tθ θ

ε θ θ θ θ= ∈ ∈

′ ′ ′= − − (45)

, 0 0( , ( , )) 0t x tε θ = . , (45) -

0 0 0( , ( , ))l l t x tθ= , 00 t θ≤ < .

{ } { } { }0 0 0( , ) ( , ) ( , ) ( , ) [ ]em m

t

l x t l X t l X R t d x tθ

θ θ θ τ τ τ′ ′ ′= + = . (46)

3.2. m- l0 0t , 00 t θ≤ < , - (45). , 0p , 0 0 0( , ( , )) 0t x tε θ > , -

0 0( , ( , ))eU t x tθ ( 0 0( , ( , ))eV t x tθ ) eu P∈( ev Q∈ ), 0 0 0[ ] [ ] max [ ]e e e

u Px t u t x t u

∈= ( 0 0 0[ ] [ ] max [ ]e e e

v Qx t v t x t v

∈= ). -

eU ( eV ) ( ) ., 0{ ( , )}ml X tθ′ – m ( ) ( ) ( )x t A t x t′= −

l0 [1, c. 117]. l0 m- .

[1, c. 153] .


�� .. �� -�� , II

2013, 5, 1 57

3.1. 3.1 3.2 ( , ( , ))e eU U t x tθ= ( , ( , ))e eV V t x tθ= 00 t t θ≤ ≤ < . -

{ ,e eU V }, 3.3 -, 0 0( [ ] | , ( , ) , )e et x t U Vγ θ θ = 0 0( , ( , ))t x tε θ , -

0 0( [ ] | , ( , ) , )e et x t U Vγ θ θ 0 0( , ( , ))t x tθ0 0( , ( , ))t x tε θ .

. :

[ ] { }

{ } { }

{ } { }

0

0

0

0 0 0

0 0 0 0 0 0

( , ( , )) max ( , ( , ) ( , )) [ ]

max ( , ( , ) ( , )) [ ] ( , ( , ) ( , )) [ ]

( , ( , ) ( , )) [ ] ( , ( , ), ( , )) .

mv Vt

t

m mu Ut t

t

m mt

t t x t l s x s x s v s ds

l s x s x s u s ds l s x s x s v s ds

l s x s x s u s ds l t x t x t

∈

∈

′= = −

′ ′− + −

′ ′− −

θ

θ

ε ε θ θ θ

θ θ θ θ

θ θ θ θ

(47)

0[ ]u s , 0[ ]v s , 0t s t≤ ≤ – , t. - [1] , ( , ( , ))t x tε θ , 0t t θ≤ ≤ , 00 t θ≤ < , t

( , ( , )) 0t x tε θ > 0 ( , ( , ))l t x tθ t, 0t – -.

(47) [1, c. 144]

{ } { }

{ } { }0 0

0 0

[ ] max ( , ( , ) ( , ) max ( , ( , ) ( , )

( , ( , ) ( , ) ( , ( , ) ( , ) .

m mu P v Q

m m

d t l t x t x t u l t x t x t vdt

l t x t x t v l t x t x t u

ε θ θ θ θ

θ θ θ θ∈ ∈

′ ′= − +

′ ′+ − (46)

[ ] max [ ] max [ ] [ ] [ ] [ ] [ ]e e e e

v Q u P

d t x t v x t u x t v t x t u tdtε

∈ ∈= − + + − . (48)

, 0t , eU, – , (48) 3.2

[ ] max ( ) ( ) 0e e

v V

d t x t v x t vdtε

∈= − + ≤ . , [ ] ( , ( , ))t t x tε ε θ=

0[ , ]t θ . , [ ]tε 0[ , ]t θ, , 0 0( , ( , )) ( , ( , ))x t x tε θ θ θ ε θ≤ , (47) ,

( , ( , ))xε θ θ θ [ ]{ }mx θ= .

, eV .

(48) [ ] max [ ] [ ] [ ]e e

u U

d t x t u x t u tdtε

∈= − . [ ] 0d t

dtε ≥ . , -

[ ]tε , 0[ , ].t t θ∈ -, [ ]tε 0[ , ]t θ . , 0 0( , ( , )) ( , ( , )).x t x tε θ θ θ ε θ≥

, -, ,

[ ]{ } 0 0( , ( , ))m

x t x tθ ε θ= .

. . -

0

( ) ( ) ( ) ( ),t

tx t e x s ds u t v t•

= + + − (49)


. « . . » 58

( ) , ( , ) 1tf t e K t s= ≡ , (49)

0x•= . ( ) 1X t = ,

1( , ) ( ) ( ) 1X t s X t X s−= ≡ , ( ,0) 1X t = ; ( , ) ( , ) ( , )t t

s s

t s K t X s d d t sτ τ τ τΦ = = = − , -

[10, . 22] ( )

( , ) sh( )2

t s t se eR t s t s− − −−= − = ,

~( , ) 1 ch( ) 1 ch( ) 1 ch( ) 1 ch( ).

t

s

tx t s s d s t s t s

sτ

τ τ ττ=

= + − = + − = + − − = −=

- , 0 (0) 3x= = ,

0( ) ( ) ( ,0) 3 .tt f t t x e tϕ = +Φ = +t

~ ~ ~

00 0 0

0 0 0

( ) ( ,0) ( , ) ( ) ( ) ( ) ( ) ( )

3 ch( )( 3 ) ch( ) ( ) ch( ) ( ) .

t t t

t t ts

x t X t x x t s s ds x t u t dt x t v t dt

t s e s ds t s u s ds t s v s ds

ϕ= + + − =

= + − + + − − −

:

( ) ( )

0 0 0 0

1 3ch( ) 3 ch( ) ( ) ( ) .2 2

t t t ts t s t s s t s t st s e ds t s sds e e e ds e e sds− − − − − −− + − = + + +

2 ) 2

0 0

1 1 1 1 1 1( ) ( )02 2 2 2 4 4

t tt t s t t s t t tt

e e ds e s e te e e− + − + −+ = + = + − ;

0 0

3 3 3 3( ( 1 ) ( 1 )2 2 2 2

t tt s t s t te e sds e e sds t e t e− − −+ = − − + + − + .

(1)

0 0

1 7 5( ) ch( ) ( ) ch( ) ( )2 4 4

t tt t tx t te e e t s u s ds t s v s ds−= + + + − − − ,

, (0) 3.x =, 0 0t = , (3,3)

, , Oxy - = . , -

[0,1], [0,1]u v∈ ∈ . ,e eU V , [0, )t θ∈

u = 1, v = 1. 01 1,2 2

l = − − (45)

1 1 6 2(0, ( ,0)) max( ( ,0)) ( 3 3) 3 22 2 2 2

x e xε θ θ′= − = − − ⋅ − ⋅ = ⋅ = ,

, (3,3) , .


�� .. �� -�� , II

2013, 5, 1 59

4. - m , -

10

( ) ( ) ( ) ( ) ( , ) ( ) ( )t m

ii

x t f t A t x t K t s x s ds u t=

= + + + , 0(0)x x= , (50)

x – n - , ( )f t – n - [ ]0,θ -, θ > 0 – , ( ),K t s – [ ]0,θ × [ ]0,θ n n× ,

( )A t – [ ]0,θ n n× , ( ) , 1,iu t i m= – , -

, ,i i iu U U∈ – nR , -[ ] [ ], 0,iu t t θ∈ – . -

. [9], (50) [ ]x t , [ ] 00x x= .

(50) (42):

010 0

( ) ( , 0) ( , ) ( ) ( , ) [ ]t tm

ii

x t X t x x t s s ds x t s u s dsϕ=

= + + . (51)

, [8],

{ }1( , , ) ( [ ]), 1,i i m iY Y u u x i mϕ θΩ = = = . (52)

4.1. 1 , ,e emU U ,

( [ ]) ( [ ])e ii ix xϕ θ ϕ θ≤ , 1,i m= .

[ ]ex θ – [ ]x t (50), -

1 , ,e emU U ; [ ]ix θ – [ ]x t , 0 t θ≤ ≤ , (50), -

1 1 1[ ], , [ ], [ ], [ ], , [ ]e e e ei i i mu t u t u t u t u t− + , [ ]e

ju t , j i≠ , 1,j m= , ejU ; [ ]iu t – , -

i iu U∈ .

4.1 , { }1 , ,e e emU U U=

(50), (52) [8]. [8], ,

1( , , ) [ ]ii mY u u c x θ= − , (53)

ic – nR , 1,i m= . , 0t , 00 t θ≤ < ,

0[ ]iu t , 00 t t≤ ≤ ; t, 0t t θ≤ < , , t -

, [ ] 0u t ≡ . t (50)

0

01

( , ) ( , ) ( , ) [ ]tm

ii t

x t x t x s u s dsθ θ θ=

= + , 0

00 0

10 0

( , ) ( , 0) ( , ) ( ) ( , ) [ ]tm

ii

x t X x x s s ds x s u s dsθ

θ θ θ ϕ θ=

= + + .

,

0

01

( , ) ( , ) ( , ) [ ]tm

ii t

x t x t x s u s dsθ θ θ=

= + . (54)

4.1. i- , 1,i m= , { }, ( , ), ip t x t cθ= -

t , 00 t t θ≤ ≤ ≤ , { }0 0 0, ( , ), ip t x t cθ= – .


. « . . » 60

4.2. iU i- , 1,i m= , -

, { }, ( , ), ip t x t cθ= , 0t t θ≤ ≤ , 00 t θ≤ < , -

( ) ( ), ( , ), , ( , ),i ii i iU t x t c u t x t c Uθ θ÷ ⊂ [1, c. 61].

iU. -

. (50) - (54).

k- , 1,k m= .

0 0

0 0 0|| || 1 1( , ( , ), ) max[ ( ( , )) max ( { ( , ) ( )}) min( { ( , ) ( )}) ],

i ik k

mk k

k k iu Ul u U it ti k

t x t c l c x t l x s u s ds l x s u s dsθ θ

ε θ θ θ θ∈= ∈ =

≠

′ ′ ′= − − − (55)

, 0 0( , ( , ), ) 0kk t x t cε θ = .

, , -

{ }0 0, ( , ), kt x t cθ , 1 k m≤ ≤ , 00 t θ≤ < ,

(55)

0 0 0 0( , ( , ), )k k kl l t x t cθ= . . ,

0t t= 0kl , (55).

4.3. 0kl 0t , 00 t θ≤ < , -

(55). , 0p { }0 0, ( , ), kt x t cθ= , 0 0( , ( , ), ) 0kk t x t cε θ > ,

0 0( , ( , ), )e kkU t x t cθ , 1 k m≤ ≤ ,

ek ku U∈ , 0 0 0[ ] [ ] max [ ]

k k

e e ek k ku U

x t u t x t u∈

= , -

{ }'0 0 0[ ] ( , )e k

kx t l x tθ= . ekU .

. (17), (18) ( , ) ( , ) ( , ) ( , )t

s

x t s X t s X t R s dτ τ τ= + , ( , )R t s – -

( ),t sΦ , ( )ekx t -

( ){ } ( )' '0( ) , ( , )e k e

k kt

x t l X t t x dθ

θ τ τ τ= + Φ [11].

[1] , . 4.1. e

kU , 1 k m≤ ≤ , (53).

.

0

0

'

' '0 0 0

' '0 0 0

11

0 0

( , ( , ), ) ( ( , ) { ( , )} ( )

max{ ( , )} ( ) { ( , )} ( ) min{ ( , )} ( )

( ( , ) ( ) ( ) ( )

i ik k

tk k k k

k kt

t m mk k k

k k iu Pu P iit t ti k

k k e e ek k k k

t x t c l c x t l x s u s ds

l x s u s ds l x s u s ds l x s u s ds

l c x t x s u s ds x s u

θ θ

ε θ θ θ

θ θ θ

θ

∈∈ ==≠

= − − −

′− − − =

= − − −0 01,

( ) ( ) ( ) .t m

e ek i

i i kt t t

s ds x s u s dsθ θ

= ≠−

(56)


�� .. �� -�� , II

2013, 5, 1 61

4.3 , 1( ) ( ,..., )e ek k mY u uε θ = ,

( )0 1 1 1( ,..., , , ,..., )e e e ek k k k k mt Y u u u u uε − += , . . ( )kε θ – kY ,

, 0( )k tε – kY , k- -, -

.

( ) ( ) ( ) ( ) ( ) 0e e ekk k k k

d t x t u t x t u tdtε

= − + ≤ ,

, (56) kP (56) , , ,

1 1 1 1( ,..., ) ( ,..., , , ,..., )e e e e e ek m k k k k mY u u Y u u u u u− +≤ .

, .

1. , . . / . . . – .: , 1970. – 420 .

2. , . . / . . , . . . – .: , 1974. – 456 .

3. , . . / . . , . . . – .: , 1981. – 278 .

4. , . . / . . . – .: , 1985. – 520 .

5. , . . / . . // . – 1971. – . 196, 4. – . 779–782.

6. , . . / . . // . – 1971. – . 197, 5. – . 1023–1025.

7. , . . / . . // . – 1972. – . 206, 3. – . 211–213.

8. , . . / . . , . . // : . . .– , 1971. – . 10. – . 3–9.

9. , . . -, I / . . // . « .

. ». – 2012. – . 6. – 11(270). – . 33–42. 10. , . . . / . . ,

. . , . . . – .: , 1975. – 216 . 11. , . . -

, I / . . // . . – 1969. – 6 (85). – . 24–34.


. « . . » 62

EXTREME STRATEGIES IN GAME-THEORY PROBLEMS FOR LINEAR INTEGRAL DIFFERENTIAL VOLTERRA SYSTEMS, II

V.L. Pasikov1

The problems of guidance as well as game situation of m individuals in case of composed functions balance (a distance) in terms of Nash theory are studied. To solve these problems a familiar extreme construction by an academician N.N. Krasovsky, modified for the considered situations, is used.

Keywords: game-theory problem, integral differential system, control action, game positions, pro-gram maximin, balance in terms of Nash theory.

References 1. Krasovskii N.N. Igrovye zadachi o vstreche dvizhenii (Motion game problems). Moscow: Nauka,

1970. 420 p. (in Russ.). 2. Krasovskii N.N., Subbotin A.I. Pozitsionnye differentsial'nye igry (Position differential games).

Moscow: Nauka, 1974. 456 p. (in Russ.). 3. Subbotin A.I., Chentsov A.G. Optimizatsiia garantii v zadachakh upravleniia (Guarantee optimi-

zation in control problems). Moscow: Nauka, 1981. 287 p. (in Russ.). 4. Krasovskii N.N. Upravlenie dinamicheskoi sistemoi (Dynamic system control). Moscow: Nauka,

1985. 518 p. (in Russ.). 5. Osipov Yu.S. DAN SSSR. 1971. Vol. 196, no. 4. pp. 779–782. (in Russ.). 6. Osipov Yu.S. DAN SSSR. 1971. Vol. 197, no. 5. pp. 1023–1025. (in Russ.). 7. Subbotin A.I. DAN SSSR. 1972. Vol. 206, no. 3. pp. 211–213. (in Russ.). 8. Gorokhovik V.V., Kirillova F.M. O linejnykh differencialnykh igrakh neskolkikh litc [Linear dif-

ferential games of several individuals]. Upravlyaemye sistemy: sb. nauch. tr. [Controllable systems: Proceedings]. Novosibirsk, 1971. Issue 10. pp. 3–9. (in Russ.).

9. Pasikov V.L. Ekstremalnye strategii v igrovykh zadachakh dlya linejnykh integro-differencialnykh sistem Volterra, I [Extreme strategies in game-theory problems for linear integral dif-ferential Volterra systems, I]. Vestnik YuUrGU. Seriya «Matematika. Mekhanika. Fizika». 2012. Issue 6. no. 11(270). pp. 33–42. (in Russ.).

10. Krasnov M.L., Kiselev A.I., Makarenko G.I. Integral'nye uravneniia. Zadachi i uprazhneniya(Integral equations. Tasks and exercises). Moscow: Nauka, 1976. 216 p. (in Russ.).

11. Vinokurov V.R. Izvestiya vysshikh uchebnykh zavedenij. Matematika. 1969. no. 6(85). pp. 24–34. (in Russ.).

�� 15 �� 2012 �.

1 Pasikov Vladimir Leonidovich is Cand. Sc. (Physics and Mathematics), Associate Professor, Mathematical Analysis and Information Service, Orsk Humanist and Technological Institute (Branch of Orenburg State University). E-mail: [email protected]


2013, 5, 1 63

517.544.8

. . 1, . . 2

- . -.

, b(t) . : �� , �� -

��, �� , �� .

, [1]. - [2–4], .

( ) ( ) ( ) ( ) ( ) ( )t a t t b t t f tψ ψ ψ+ − += + + (1) L . ( ), ( ), ( ) ( )a t b t f t H L∈ – , ( ) 0,a t ≠

t L∈ . ( ),LInd a tκ = ( ) ( ) ( )a t a t t a tκ+ −= – ( )a t -

. (1)

1 0( ) ( ) ( ) ( ) ( ),t t t b t t f tκφ φ φ+ − += + + (2) ( )( )( )tt

a tψφ ±

±±

= , 1( )( ) ( )( )

a tb t b ta t+

+= , 0

( )( )( )

f tf ta t+

= .

1( )b t (2): ) 1( ) 1 0, ;b t t L+ ≠ ∈) 1( ) 1b t +

D− , , , ,

1κ . (2) :

11

1 1

2 ( )( ) ( ) Re ( ) ( )( ) 1 ( ) 1

b ttt t t f tb t b t

κφ φ φ+ − += + +

+ +, (3)

11

( )( )( )( ( ) 1)

f tf ta t b t+

=+

.

(3) , Re ( )tφ+ .

0 11

,( ) 1LtInd

b t

κκ κ κ= = −

+ 1 1( ( ) 1)LInd b tκ = + .

0 0κ ≥ , (3) [5]

0 1( ) ( )[ ( ) ( )]z z F z P zκφ χ −= + . (4)

11

1

2 ( )Re ( )1( ) ( )2 ( ( ) 1)L

b dF z fi b z

τ φ τ ττπ τ τ

+= ++ −

, 0 1( )P zκ − –

0 1κ − ,

1 �� !" #$!%&!' #$!%&!!"() – *+,!-�, %�.!*�� +/0!' 1��!1��(%(, 23-+-4��$5&%(' 6+&�*��&�"!--7' �-("!�&(�!�. E-mail: [email protected] 2 �� !"� 8$!-� 9�&($5!"-� – *+,!-�, %�.!*�� :�(%$�*-+' 1��!1��(%(, 23-+-4��$5&%(' 6+&�*��&�"!--7' �-("!�&(�!�.


. « . . » 64

1

1, ,( ) ( ) 1,

z Dz b z z D

zκχ

+

−

∈= + ∈

– . 0 0,κ < :

1 111 0

1

2 ( ) Re ( ) ( ) 0, 1, , .( ( ) 1)

k k

L L

b d f d kbτ φ τ τ τ τ τ τ κτ

− −+ − = = −+

(5)

L ( )zφ+ , D+ , - (3), ( )zφ+ ,

.

11

2Re ( ) ( )( ) 1

t f tb t

φ+ −+

:

01 1 11 1

2Re ( ) 2Re ( )1 1( ) ( ) ( )2 ( ( ) 1) 2 ( ( ) 1)L

t df t f d P tb t i b t κ

φ φ τ ττπ τ τ

+ +−− + − = +

+ + −. (6)

0Re ( )1

2 L

d d ci

φ τ τπ τ

+= − , 0c – .

d

1 01

2Re ( )1 ( )2 ( ) 1L

df d ai b

φ τ ττπ τ τ

+ − = ++

,

00 1(0)a Pκ −= – . 0 0,κ ≤ 0 0.a = (6) [6]

01 1 11

2Re ( ) ( ) ( ) ( )( ) 1

t f t d P t tb t κ

φ ϕ−+−− = + −

+,

1 ( )tϕ− – D− , .

01 1 11 ( )Re ( ) ( ( ) 1)[ ( ) ( )]2 2 ( )

f tt b t d P t ta tκφ ϕ−+ −+

= + + − + (7)

11

( )( )( )( ( ) 1)

f tf ta t b t+

=+

. , (7) – -

,

01 1 1 1( )Re[ ( ( ) 1) ( )] Im ( ( ) 1)[ ( )]( )

f ti b t t b t d P ta tκϕ− −+

− + = + + + , (8)

, . (8) [7]

1 1*

1 0 1 1 11

1( ) ( ) ( ) ( ) ( )( ) 1

i z F z F z Q z Q zb z κ κϕ− − −− = + + +

+. (9)

01 ( )( )( )

2 ( )L

c zF z di z

τ τ τπ τ τ

+=−

, 11 ( )( ) Im

2 ( ) ( )L

f t zF z di a t z

τ τπ τ τ+

+=−

, 1 1( )Q zκ − –

1 1κ − , { }01 1( ) Im ( ( ) 1)[ ( )]c t b t d P tκ −= + + .

1 0κ ≤ , 1 1( ) 0Q zκ − ≡ .

1

1 ( )( ) ( )Im2 ( ( ) 1) ( ) ( ) ( )L L

c z f t zd di b z z a t z

τ τ ττ τπ τ τ τ τ+

+ +++ − −

(10)


�� .., �� .�. � ��

2013, 5, 1 65

(8) , -

1( )

kkL

c dτ τ ατ + = , 1

( )Im( )k k

L

f t da t

τατ +

+= − , 10, ,k κ= − . (11)

(7), (9) Re ( )tφ+ , ( )zφ (3). -

0

0

1

11

1 ( )( ) ( ) ( ) , ,2

( )( )( ( ) 1) 1 ( ) ( ) ( ) , .

2

L

L

ga z d G z P z z Di z

za z b z g d G z P z z D

i zz

κ

κκ

τ τπ τ

ψτ τ

π τ

+ − +

−− −

+ + ∈−

=+ + + ∈

−

(12)

0 1 11

1 1 0 1 11

( )( ) ( ) ( ) ( ) ( ) ( )( ) 1

ib tg t b t d P t F t Q t Q tb tκ κ κ

−− − −= + − + +

+, 1( )1( )

2gG z d

i zτ τ

π τ=

−,

11 1

1

( ) ( )( ) ( )( ) 1 ( )b t f tg t iF t

b t a t−

+= −

+.

( ) 0f t ≡ , . ( ) 0G z ≡ . .

1. (1) ( ( ) 0f t ≡ ) ( ), ( ) ( ),a t b t H L∈ ( ) 0,a t ≠ t L∈ , ( ),LInd a tκ = 1( ) 1b t + -

L , D L− ∪ , -, , , -1κ , 0 1κ κ κ= − .

(1) ( ( ) 0f t ≡ ) - , ,

1) 1 0,κ > 0 0κ ≥ , (12) ( ( ) 0G z ≡ ),

0 12 2 2κ κ κ+ = ; 2) 1 0,κ > 0 0κ < (12) ( ( ) 0G z ≡ ), (

0 1( ) 0P zκ − ≡ ), -

1 12 rκ − , 1r – - (5) ( 1 12r κ= , );

3) 1 0,κ ≤ 0 0κ ≥ , (12) ( ( ) 0G z ≡ ,

1 1( ) 0Q zκ − ≡ ), 02 rκ − , r – (11) ( 02r κ= , ,

, ); 4) 1 0,κ ≤ 0 0,κ < 1( ) 1b t + (11) ( ( ) 0f t ≡ ) -

(5) ( ( ) 0f t ≡ ), , (12) ( ( ) 0G z ≡ ,

1 1( ) 0Q zκ − ≡ ,0 1( ) 0P zκ − ≡ ); .

2. ( ), ( ) ( )a t b t H L∈ , ( ) ( )f t H L∈ , ( ) 0a t ≠ , t L∈ , 1( ) 1b t +L , D L− ∪ , -

, . - , -

: 1) 1 00, 0κ κ> ≥ , (12),

2κ ;


. « . . » 66

2) 1 0,κ > 0 0κ < (12) 0 1( ( ) 0)P zκ − ≡ , -

0 1rκ− − , ( 1r – - (5)), 1 12 2rκ − ( 1 1r κ= , -

); 3) 1 00, 0κ κ≤ ≥ , (12) (

1 1( ) 0Q zκ − ≡ ),

1 1 rκ− + − , ( r – - (11)), 02 2rκ − -

( 0r κ= ); 4) 1 00, 0κ κ≤ < , (12)

(1 01 1( ) 0, ( ) 0Q z P zκ κ− −≡ ≡ ), , 1 1κ− + -

(11) 0κ− (5). 1. -

L : 2( ) (2 1) ( ) (3 1) ( ),t t t t t t Lψ ψ ψ+ − += + + + ∈ . (13)

- , . (13)

2(2 1) 2(3 1)( ) ( ) Re ( )3 2 3 2t tt t tt t

ψ ψ ψ+ − ++ += ++ +

. (14)

21(2 1) 2, (3 2) 1L LInd t Ind tκ κ= + = = + = ,

2

0(2 1) 1.

3 2LtIndt

κ += =+

, -

2(2 1)3 2tt++

L , D− .

, , L : (3 1)Re ( )3 1 1( ) Re ( ) ,

3 2 (3 2)( )L

dtt tt i t

τ ψ τ τψ ψ απ τ τ

++ +

++= + ++ + −

(15)

α – . ,

0Re ( ) Re ( )1 1( ) Re ( ) ,

2L L

d dt t ci t i

ψ τ τ ψ τ τψ ψπ τ π τ

+ ++ += + − +

− (16)

0c – . (15), (16) -

0Re ( ) Re ( ) Re ( )1 1 12 ,

3 2 (3 2)( ) 2 2L L

t d dd d ct i t iψ ψ τ τ ψ τ τα

π τ τ π τ+ + ++ = = + ++ + −

. (17)

(17) Re ( ) (3 2)( ( ))t t d tψ ϕ−+ = + − , (18)

( )zϕ− – D− , . (18) – ,

: Im (3 2) ( ) 0t t−+ Ψ = , (19)

( ) ( )t d tϕ− −Ψ = − – , D− , ( ) .d−Ψ ∞ = - (19) :

3(3 2) ( ) ( 2) ( ) 0t t tt

− −+ Ψ − + Ψ = ,


�� .., �� .�. � ��

2013, 5, 1 67

1 1 1( *), | | 1,(3 2)( ) ( ), ( )

3 2 ( ), | | 1.

z zt tt t zt z z

−+ −

−

Ψ <+Ψ = Ψ Ψ =+ Ψ >

, (20)

1

, | | 1,3 2( )

1 , | | 1,3 2

z zzz

zz

χ<

+=>

+, 1 1( *) ( )z zχ χ= -

(20). , : 1 1

0 1 21( ) (2 ( ) ( ))

3 2z z z i z z

zα α α− − −Ψ = + + + −

+,

0 1 2, , Rα α α ∈ . (18)

1 10 1 2Re ( ) 2 ( ) ( ).t t t i t tψ α α α− −

+ = + + + −

, , 1 2( )3iα α− +Ψ ∞ = , 1 2

0 01 ( 2 )

3 2i cα α α α+ = + + ,

1 10 1 2

0(2 ( ) ( )) Re ( )1 12

2 2L L

i d di i

α α τ τ α τ τ τ ψ τ ταπ τ π τ

− −++ + + −

= = .

(13)

0 1 2 0

0 1 22

1 2 1 20 0

2 2( ) , | | 1,4 11( )3 2( )

3 2 3 2(2 1)2( ) 2 , | | 1.

3

i z c zizz

z zzi i c z

z

α α αα α αψ

α α α α α

+ + − <++= − −

+ ++− +

− + − − >

,

0 1 2 0, , , .ic Rα α α ∈

1. , . . - / . . // . « . . -

». – 2011. – . 4. – 10(227). – . 29–37. 2. , . . -

/ . . , . . // . « ».

– 2010. – . 6. – 35(211). – . 4–12. 3. , . . /

. . // . – 2010. – 4. – . 82–97.

4. , . . / . . , . . // .

. . . . – 2011.– . 11. – . 2. – . 4–12. 5. , . . / . . . – .: , 1963. – 640 . 6. , . . /

. . // . – 1966. – . 2, 2. – . 533–544. 7. , . . / . . -. – : - - , 1977. – 301 .


. « . . » 68

A VARIANT OF THE SOLUTION OF MARKUSHEVICH BOUNDARY PROBLEM

A.A. Patrushev1, E.V. Patrusheva2

In the article an explicit method for the solution of Markushevich boundary value problem in the class of piecewise analytic functions is suggested. Boundary condition of the problem is given on the unit circle. The problem is found in a closed form under additional restriction on the coefficient b(t) of the problem.

Keywords: boundary problems for analytic functions, Riemann boundary problem, Hilbert bound-ary problem, Markushevich boundary problem.

References 1. Patrushev A.A. Zadacha Markushevicha v klasse avtomorfnykh funkciij v sluchae proizvolnoj

okruzhnosti [The Markushevich problem in the lass of automorphic functions for arbitrary circle]. Vestnik YuUrGU. Seriya “Matematika. Mexanika. Fizika”. 2011. Issue 4. no. 10(227). pp. 29–37. (in Russ.).

2. Adukov V.M., Patrushev A.A. Algoritm tochnogo resheniya chetyrekhlementnoj zadachi line-jnogo sopryazheniya s racionalnymi koefficientami i ego programmnaya realizaciya [Algorithm of exact solution of generalized four-element Riemann–Hilbert boundary problem with rational coefficients and its programm realization]. Vestnik YuUrGU. Seriya “Matematicheskoe modelirovanie i programmiro-vanie”. 2010. Issue 6. no. 35(211). pp. 4–12. (in Russ.).

3. Patrushev A.A. Izvestiya Smolenskogo gosudarstvennogo universiteta. 2010. no. 4. pp. 82–98. (in Russ.).

4. Adukov V.M., Patrushev A.A. Izvestiya Saratovskogo gosudarstvennogo universiteta. Novaya seriya. Seriya Matematika. Mekhanika. Informatika. 2011. Vol. 11. Issue 2. pp. 4–12. (in Russ.).

5. Gaxov F.D. Kraevye zadachi (Boundary problems). Moscow: Fizmatgiz, 1963. 640 p. (in Russ.). 6. Gaxov F.D. Differencialnye uravneniya. 1966. Vol. 2, no. 2. pp. 533–544. 7. Chibrikova L.I. Osnovnye granichnye zadachi dlya analiticheskikh funkcij [Basic boundary prob-

lems for analytic functions]. Kazan: Izdatelstvo Kazanskogo universiteta, 1977. 301 p. (in Russ.). �� 18 �� 2012 �.

1 Patrushev Alexey Alexeevich is Associate Professor, Department of General Mathematics, South Ural State University. E-mail: [email protected] 2 Patrusheva Elena Vasilievna is Associate Professor, Applied Mathematics Department, South Ural State University.


2013, 5, 1 69

517.95

-

. . 1

-.

. : �� , �� , � ��

�� , �� , �� -�� .

1.

. -, ,

. --

. ,

, XVIII . , -

. , , ,

. TD D R≡ ×

2 2

2 20

( , ) , , ( , ) ( , ) , ( )n T

u t x f t x K s y u s y dyds tt x

∞

−∞

∂ ∂− =∂ ∂

ϑ (1)

1 10 0( , ) ( ) , ( , ) ( ), , 1,2 1i

it tiu t x x u t x x x R i nt

ϕ ϕ += =∂= = ∈ = −∂

(2)

0( , ) ( ) , ,Tx xu t x t t Dψ= = ∈ (3)

00( ) const ,ttϑ ϕ= = = (4)

( , )u t x ( )tϑ – , ( , , , ) ( ) ,Tf t x u D R Dϑ ∈ × × ( ) ( ),i x C R∈ϕ

1,2 ,i n= ( ) ( ) ,Tt Dψ ∈0

0 ( , ) ,T

K s y dy ds∞

−∞

< [ ]0, ,TD T≡ 0 ,T< < ∞ n – -

. , -

. -

1 �� ! "#$%#& '�(��)&*!)+ – ,�&�)��- .)/),*-(�- (�-)+ %,)0 &�#,, �*1 &-, �*,-*$�&-, ,�. �$� !2%� 3 (�- (�-),), 4)5)$-%,)3 6*%#��$%-! &&23 �7$*,*%()+ %,)3 #&)! $%)- -, 6. '$�%&*8$%,. E-mail: [email protected]


. « . . » 70

. --

[1]. [2–4] -

. , -.

, ( )tϑ -. (1)–(4) -

, -

. (4) --

0t = , . . 0(0) .ϑ ϕ= [5, 6].

. (1)–(4)

{ }2 ,2( , ) ( ), ( ) ( )n nTu t x C D t C Dϑ∈ ∈ , (1) (2)–(4).

2. (1),(2) (1)

[ ]2 2

1 22 2 ,n n n

n nu u L L ut x t xt x

∂ ∂ ∂ ∂ ∂ ∂− = − + =∂ ∂ ∂ ∂∂ ∂

[ ] [ ] [ ]1 2 2 2( ) ( ) ,n n nt xL L u L u L u≡ − [ ]2L u ≡ .t xu u+

(1)

[ ]1 20

, , ( , ) ( , ) , ( ) .T

n nL L u f t x K s y u s y dyds tϑ∞

−∞

= (5)

(5) , (1) n - : 1) 1x t C− = ; 2)

2x t C+ = , 1 2,C C – . , (5) n,

[ ] ( )11 2 1

0 0

, , ( , ) ( , ) , ( ) ,t T

n nL L u x t f s x K y u y dyd s dsθ θ θ ϑ∞

−

−∞

= Φ + + (6)

[ ] ( ) ( )21 2 2 1n nL L u x t x t t− = Φ + +Φ + +

0 0

( ) , , ( , ) ( , ) , ( ) ,t T

t s f s x K y u y dyd s dsθ θ θ ϑ∞

−∞

− (7)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

[ ] ( )21 ( )!

n inn

ii

tL u x tn i

−

== Φ + +

−

1

0 0

( ) , , ( , ) ( , ) , ( ) ,( 1)!

t Tnt s f s x K y u y dyd s dsn

θ θ θ ϑ∞−

−∞

−−

(8)

( 1, )i i nΦ = – , .

(6), (2), 1 2( ) .nx ϕΦ = -

[ ] [ ] [ ] [ ]( ) [ ]( )2 2 22 2 ,

n n nn n

t x

dL u L u L u x L u L udt t x t

∂ ∂ ∂= + = −∂ ∂ ∂

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [ ] [ ]2

2 ,nn n

nn

d L uL u

t xdt∂ ∂= −∂ ∂

(9)


�� .. �� -��

2013, 5, 1 71

, (2) (7), (8) 2 2 1 1( ) ( ),..., ( ) ( ).n n nx x x xϕ ϕ− +Φ = Φ =

(8) :

[ ] ( )1

21 ( 1)!

inn

n ii

tL u x ti

ϕ−

+=

= + +−

1

0 0

( ) , , ( , ) ( , ) , ( ) ,( 1)!


θ θ θ ϑ∞−

−∞

−−

(10)

– . , - (10) n -

,

[ ] ( ) ( )1

12 1

1 0 ( 1)!

t jnn

n n jj

sL u x t x s dsj

ϕ−

−+ +

== Φ − + + +

−

0 0

( ) , , ( , ) ( , ) , ( ) ,!


θ θ θ ϑ∞

−∞

−+ (11)

[ ] ( ) ( ) ( )1

22 2 1

1 0

( )( 1)!

t jnn

n n n jj

sL u x t x t t t s x s dsj

ϕ−

−+ + +

== Φ − +Φ − + − + +

−1

0 0

( ) , , ( , ) ( , ) , ( ) ,( 1)!


θ θ θ ϑ∞+

−∞

−++

(12)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1 12

1 1 0

( )( , ) ( ) ( )(2 )! ( 1)! ( 1)!

tn i n jn n

i n ji n j

t t s su t x x t x t dsn i n j

ϕ− − −

+= + =

−= Φ − + + +− − −

2 1

0 0

( ) , , ( , ) ( , ) , ( ) ,(2 1)!


θ θ θ ϑ∞−

−∞

−+−

(13)

( 1,2 )i i n nΦ = + – , -.

(11) (2) 1( ) .n nx ϕ+Φ = - (9)

,t xdu u u x u udt t x t

∂ ∂ ∂= + = +∂ ∂ ∂

... , ,nn

nd u u

t xdt∂ ∂= +∂ ∂

(14)

(2) (12) (13) , 2 1 2 1( ) ( ),..., ( ) ( ).n n nx x x xϕ ϕ+ −Φ = Φ =

, (1), (2) -:

1 1 1

11 1 0

( )( , ) ( , ; , ) ( ) ( )( 1)! ( 1)! ( 1)!

ti n jn n

i n ji j

t t s su t x t x u x t x t dsi n j

ϑ ϕ ϕ− − −

+= =

−≡ Θ = − + + +− − −

2 1

0 0

( ) , , ( , ) ( , ) , ( ) ,(2 1)!


θ θ θ ϑ∞−

−∞

−+−

(15)

– . (15) , 1 2( ), ( ),..., ( )nx t x t x tϕ ϕ ϕ− − −

0n

ut x∂ ∂+ =∂ ∂

.

.


. « . . » 72

1 2 2( ), ( ),..., ( )n n nx t x t x tϕ ϕ ϕ+ ++ + +

0n

ut x∂ ∂− =∂ ∂

.

. , , - (15) -

(1). 2n - (15)

2

20

, , ( , ) ( , ) , ( ) .Tn

nd u f t x K s y u s y dyds tdt

ϑ∞

−∞

= (16)

(9) - – (14), (16)

[ ]2

1 22 .n nn

n nn

d u u L L ut x t xdt∂ ∂ ∂ ∂= − + =∂ ∂ ∂ ∂

, (16) - (1).

3. (3), (15)

1 1 1

0 01 1 0

( )( ) ( ) ( )( 1)! ( 1)! ( 1)!

ti n jn n

i n ji j

t t s st x t x t dsi n j

ψ ϕ ϕ− − −

+= =

−= − + + +− − −

2 1

00 0

( ) , , ( , ) ( , ) , ( )(2 1)!


θ θ θ ϑ∞−

−∞

−+−

0 0

, , ( , ) ( , ) , ( ) ( ),t T

h t s K y u y dyd s ds g tθ θ θ ϑ∞

−∞

= (17)

1 1 1

0 01 1 0

( )( ) ( ) ( ) ( ) ,( 1)! ( 1)! ( 1)!

ti n jn n

i n ji j

t t s sg t t x t x t dsi n j

ψ ϕ ϕ− − −

+= =

−= − − − +− − −

0

, , ( , ) ( , ) , ( )T

h t s K y u y dyd sθ θ θ ϑ∞

−∞

=

2 1

00

( ) , , ( , ) ( , ) , ( ) .(2 1)!

Tnt s f s x K y u y dyd sn

θ θ θ ϑ∞−

−∞

−=−

( )tϑ (17) -. -

, -. (17)

( )0

( ) ( ) ( ) ( )t

t G s s ds t g tϑ ϑ ϑ+ = + + ( )0 0

( ) , , ( , ) ( , ) , ( ) ,t T

G s s h t s K y u y dyd s dsϑ θ θ θ ϑ∞

−∞

− (18)

0 ( )G t< – , 0

exp ( ) 1.t

G s ds−

(18) [ ]( )G s− ,

( )0

( ) ( ) ( )exp ( , ) ( ) ,t

t h t G s t s h s dsϑ μ= − − ⋅ (19)


�� .. �� -��

2013, 5, 1 73

( )0 0

( ) ( ) ( ) ( ) , , ( , ) ( , ) , ( ) ,t T

h t t g t G s s h t s K y u y dyd s dsϑ ϑ θ θ θ ϑ∞

−∞

= + + −

( , ) ( ) , ( ,0) ( ).t

s

t s G d t tμ θ θ μ μ= =

(19) , {2( ) ( ; , ) ( ) ( )t t u t g tϑ ϑ ϑ≡ Θ = + +

( ) ( )0 0

( ) , , ( , ) ( , ) , ( ) exp ( )t T

G s s h t s K y u y dyd s ds tϑ θ θ θ ϑ μ∞

−∞

+ − − +

( ){0

( )exp ( , ) ( ) ( ) ( ) ( )t

G s t s g t g s t sμ ϑ ϑ+ − − + − +

( )0 0

( ) , , ( , ) ( , ) , ( )t T

G s s h t s K y u y dyd s dsϑ θ θ θ ϑ∞

−∞

+ − −

( )0 0

( ) , , ( , ) ( , ) , ( ) .s T

G h s K y u y dyd d dsθ ϑ θ θ ζ ζ ζ ϑ θ θ∞

−∞

− − (20)

(17) (4) (20).

4. (1)–(4) , , (1)–(4)

: 1

2

( , ) ( , ; , ),( ) ( ; , ).

u t x t x ut t u

ϑϑ ϑ

≡ Θ≡ Θ

(21)

D ( , )h t x -:

( , )( , ) max ( , ) .

t x Dh t x h t x

∈= -

TD . . :

1. 12

01

( ) ,0 ;( 1)!

in

i i ii

Tx M Mi

ϕ−

=≤ < ≤ Δ < ∞

−

2. { }0 0 1( , , , ) ( ( )) ( ) , ( ) ;uf t x u Bnd M t Lip L t L t ϑϑ ∈ ∩

3. ( )2 1

0 10

( )0 ;(2 1)!

T nT s M s dsn

−−< ≤ Δ < ∞−

4. ( )2 1

0 20

( )0 ;(2 1)!

T nT s L s dsn

−−< ≤ Δ < ∞−

5. ( )2 1

1 30

( )0 ;(2 1)!

T nT s L s dsn

−−< ≤ Δ < ∞−

6. 1,ρ < 1 2max{ ; },ρ β β= 2 3 3 0max{ ; },Mβ = Δ Δ 1 2 2 00

max ; 1 ( ) ,T

G t dt Mβ α= Δ + + Δ

0

( , ) ,T

K t x dxdtα∞

−∞

= ( ) ( )00

max exp ( ) 2 ( )exp ( , ) 1.t

tM t G s t s dsμ μ= − + −


. « . . » 74

(1)–(4)

{ }2 ,2( , ) ( ), ( ) ( ) .n nTu t x C D t C Dϑ∈ ∈

. . -: 0 0 1 1

1 2

( , ) 0, ( , ) 0, ( , ) ( , ; , )( , ) ( ; , ), 0,1,2,...

k k k

k k k

u t x t x u t x t x ut x t u k

ϑ ϑϑ ϑ

+

+

= = ≡ Θ≡ Θ =

, (22)

(22) , :

( )1 2 12

1 0 0 0 11 0

( )( , ) ( , ) ;( 1)! (2 1)!

ti nn

ii

T t su t x u t x M M s dsi n

− −

=

−− ≤ + ≤ Δ + Δ− −

(23)

( )(2 1

1 0 10

( )( , ) ( , ) ( , ) ( , )(2 1)!

t n

k k k kt su t x u t x L s u s x u s x

n

−

+ −−− ≤ − +−

( ) )1 1 2 1 3 1( ) ( ) ( , ) ( , ) ( ) ( ) ;k k k k k kL s s s ds u t x u t x t tϑ ϑ ϑ ϑ− − −+ − ≤ Δ − + Δ − (24)

( )1 00

( ) ( ) ( ) ( ,0,0) exp ( )t

t t g t h t,s ds tϑ ϑ μ− ≤ + − +

( )0 0

2 ( )exp ( , ) ( ) ( ,0,0)t t

G s t s g t h t,s ds ds+ − +μ 00

( ) ( ,0,0) ,t

g t h t,s ds M≤ + (25)

( ) ( )00

max exp ( ) 2 ( )exp ( , ) 1,t

tM t G s t s dsμ μ= − + −

10

( ) ( ) 1 ( )t

k kt t G s dsϑ ϑ+ − ≤ + +

( )2 1

0 0 10 0

( ) ( , ) ( , ) ( , )(2 1)!

t Tn

k kt s L s K y dyd ds M u t x u t x

nθ θ

∞−

−−∞

−+ − +−

( )2 1

1 0 10

( ) ( ) ( )(2 1)!

t n

k kt s L s ds M t t

nϑ ϑ

−

−−+ − ≤−

2 0 1 3 0 10

1 ( ) ( , ) ( , ) ( ) ( ) ,T

k k k kG t dt M u t x u t x M t tα ϑ ϑ− −≤ + + Δ − + Δ − (26)

0

( , ) .T

K s y dydsα∞

−∞

=

1 2 2 00

max ; 1 ( ) ,T

G t dt Mβ α= Δ + + Δ 2 3 3 0max{ ; },Mβ = Δ Δ 1 2max{ ; }.ρ β β=

(24) (26) 1 1( , ) ( , ) ( , ) ( , ) ,k k k kU t x U t x U t x U t xρ+ −− ≤ − (27)

1 1 1( , ) ( , ) ( , ) ( , ) ( ) ( ) .k k k k k kU t x U t x u t x u t x t tϑ ϑ− − −− ≡ − + − (23), (25) (27) , (21)

, , (1)–(4)

{ }2 ,2( , ) ( ), ( ) ( ) .n nTu t x C D t C Dϑ∈ ∈


�� .. �� -��

2013, 5, 1 75

1. . ., . ., . . . – .: , 1999. – 95 .

2. . ., . . - // . – 1989. –

. 23, 3. – . 465–477. 3. . ., . . -

// . . – 1992. – . 323, 3. – . 410–411. 4. . ., . . -

// . . – 1992. – . 325, 6. – . 111–115.

5. . . // . . – 2012. – 2. – . 56–62.

6. . . - // . « . . ». – 2012.

– . 6. – 11(270). – . 35–41.

INVERSE PROBLEM FOR NONLINEAR INTEGRAL DIFFERENTIAL EQUATION WITH HYPERBOLIC OPERATOR OF A HIGH DEGREE

�.�. Yuldashev1

In this paper a method of studying an inverse problem for nonlinear integral differential equations with hyperbolic operator of arbitrary natural degree is given. The theorem on the existence and unique-ness of the solution of this problem is proved.

Keywords: inverse problem, nonlinear equation, hyperbolic operator of a high degree, method of characteristics, the existence and uniqueness of the solution.

References 1. Gorickij A.Yu., Kruzhkov S.N., Chechkin G.A. Uravneniya s chastnymi proizvodnymi pervogo

poryadka [Partial differential equations of the first order]. Moscow: Mexmat MGU, 1999. 95 p. (in Russ.).

2. Imanaliev M.I., Ved' Yu.A. Differencial'nye uravneniya. 1989. Vol. 23, no. 3. pp. 465–477. (in Russ.).

3. Imanaliev M.I., Alekseenko S.N. Doklady RAN. 1992. Vol. 323, no. 3. pp. 410–411. (in Russ.). 4. Imanaliev M.I., Alekseenko S.N. Doklady RAN. 1992. Vol. 325, no. 6. pp. 111–115. (in Russ.). 5. Yuldashev T.K. Ob obratnoj zadache dlya kvazilinejnogo uravneniya v chastnyx proizvodnyx

pervogo poryadka [On the inverse problem for the quasilinear partial differential equation of the first order]. Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika. 2012. no. 2. pp. 56–62. (in Russ.).

6. Yuldashev T.K. Ob obratnoj zadache dlya sistemy kvazilinejnyx uravnenij v chastnyx proizvod-nyx pervogo poryadka [On an inverse problem for a system of quazilinear equations in partial deriva-tives of the first order]. Vestnik YuUrGU. Seriya “Matematika. Mekhanika. Fizika”. 2012. Issue 6. no. 11(270). pp. 35–41. (in Russ.).

�� 22 �� 2012 �.

1 Yuldashev Tursun Kamaldinovich is Cand. Sc. (Physics and Mathematics), Associate Professor, Doctoral Candidate, Department of Higher Mathematics, Siberian state aerospace university, Krasnoyarsk. E-mail: [email protected]


. « . . » 76

538.915

AB INITIO -

. . 1, . . 2

WIEN2k 5 - . -

-.

: �� , -��, �� .

, -, [1, 2]. -

, , , , ,

, , . [3, 4]. - ( ) .

, - [5]. - -

, , . , . , ,

. --

, . -, , [6]. -

, -:

, . , -

, . , - - ,

WIEN2k.

(DFT) - (LAPW) -

(GGA’96) WIEN2k 8×4×1 k-, – [7]. -Fe =

2,847 Å, ( = 2,867 Å), Kmax = 5,0 a.e.–1, Ecut = –7,0 [8]. 0,01 .

- = 5 -: 36,9° (310) 53,1° (210).

1 �� !�� – ��"��, #$��-%��&�'�( )��* ��( *��. E-mail: [email protected] 2 +��,�� '�� !��*��- – "��.��, �'�� .�,�'�-!��!��-��'�� *', '�.� �� /0�( � ��-��'�( .�,�'�, #$-��-%��&�'�( )��* ��( *��. E-mail: [email protected]


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2013, 5, 1 77

[9, 10]. 40 , ,

OY OZ. , python ( . 1).

, -. -

: 1) - [11]:

2f gb fsE Eγ = −

gbE – ; fsE – . -. 2) ab initio:

2gb bulk

gbE E

Sγ

−= ,

gbE – ; bulkE – , ; S – -

. / 2. , fγ ,

, , , . . .

�) �)

�) �)

�� 1. !"#$%&�'# ()# �*)+,%-#.�# $)/#0�,)1%.�2 3,%.�45 *#,.%: �) 6)1),)& /17" 60) () &#8 )&.)- �&#09.) /,73 /,73% .% 73)0 ,%*),�#.&�,)1(� (: – 1 60) () &� z = 0,5; ; – 1 60) () &� z = 0,0);

�) ,#<#&(% )16%/%=>�" 7*0)1; �) 3,%.�4% .%(0).% ?5(210); �) 3,%.�4% .%(0).% ?5(310)


. « . . » 78

XY, . 2, .

. , 60–80

, 0,04 . ( )

12 Å. , 20 - 12 Å .

- - 1 2 ( . 2, ) 53,1°. , - LAPW,

. -.

Rmt, -.

, Rmt . ( -

Rmt) OY ( , -). 0,2 Å OY,

OZ 0,5 , . . .

( . 3). 0,4 Å OX, 0,6 Å OY 0,5 OZ ( . 4, ).

5(310) - ( . 4, ). :

Rmt = 1,68 a.e., 2,00 . ., . , -

.

��. 2. �� -�� :�) �5(210); �) �5(310) (� – � �� z = 0,5; – � �� z = 0,0)

�) �)

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�� .�., Ab initio �� .�. �� -��

2013, 5, 1 79

5(210) ( . 4, ) 5(310) ( . 4, ) -.

.

�� !"#$%��&' ��($#)*�*"�

5(310) 5(210) 5(310) 5(210)

fγ , –17,6 –9,6 − –5,41 [12], –3,43 [4]

gbγ , / 2 0,61 1,14 1,1 [13], 1,63 [14] 6 [9] 0,77 [15],

0,985 [16]

, 53.1° . Rmt

fγ = –4,86 , , - [12,4]. , fγ

- , , , k- muffin-tin .

, -, -

.

5(210) 5(310), -, . -

53,1° : 0,4 Å OX, 0,6 Å OY 1,4235 Å OZ. 5(210) 5(310).

, 36,9°. .

+ ,. 4. �'�-�* %�,."� ("/��0�� 1�� 2& (�� 35.53,1°: 4) ,*�$.*$�� /�( 05,*. ' ,6� 1"�; 7) 05,*. 8 ,6� 1 0,4 Å !" OX 0,6 Å !" OY;

9) "*��#�., �"��: ,*�$.*$��, "*��%�;<�: - � -$-$ =��1 ; 36,9°: >) (��%�#)��: ,*�$.*$��; ?) "*��#�., �"��: ,*�$.*$��, "*��%�;<�: - � -$-$ =��1

4)

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. « . . » 80

1. , . . Ab initio - / . . , . . , . . , . . // . -

« . . ». – 2011. – . 4. – 10(227). – . 114–119. 2. , . . Ab initio- - (VHn) - /

. . , . . , . . // . « » – 2011. – . 17. – 36(253). – . 51–53. 3. Grain boundary diffusion of Fe in ultrafine-grained nanocluster-strengthened ferritic steel /

R. Singh, J.H. Schneibel, S. Divinski, G. Wilde // Acta Materialia. – 2011. – V. 59. – P. 1346–1353. 4. Wachowicz, E. Effect of impurities on grain boundary cohesion in bcc iron / E. Wachowicz,

A. Kiejna // Computational Materials Science. – 2008. – V. 43. – P. 736–743. 5. Sutton, A.P. Interfaces in crystalline solids / A.P. Sutton, R.W. Balluffi. – Oxford: Clarendon,

1995. – 856 c. 6. , . . -

/ . . , . . , . . . – .: , 1991. – 231 . 7. Monkhorst, H.J. Special points for brillouin-zone integrations / H.J. Monkhorst, J.D. Pack //

Phys. Rev. B. – 1976. – V.13. – P. 5188–5192. 8. , . . -

- / . . , . . , . . // . « -. . ». – 2010. – . 2. – 9(185). – . 97–101.

9. Zhang, J.M. Energy calculation for symmetrical tilt grain boundaries in iron / J.-M. Zhang, Y.-H. Huang, X.-J. Wu, K.-W. Xu // Applied Surface Science. – 2006. – V. 252. – P. 4936–4942.

10. Lejcek, P. Solute segregation and classification of [100] tilt grain boundaries in -iron: conse-quences for grain boundary engineering / P. Lejcek, S. Hofmann, V. Paidar // Acta Materialia. – 2003. – V. 51. – P. 3951–3963.

11. Rice, J.R. Embrittlement of interfaces by solute segregation / J.R. Rice, J.-S. Wang // Materials Science and Engineering: A. – 1989. – V. 107. – P. 23–40.

12. Braithwaite, J.S. Grain boundary impurities in iron / J.S. Braithwaite, P. Rez //Ada Materialia. –2005. – V. 53. – P. 2715–2726.

13. Hyde, B. Atomistic sliding mechanisms of the =5 symmetric tilt grain boundary in bcc iron / B. Hyde, D. Farkas, M.J. Caturla // Philosophical Magazine. – 2005. – V. 85. – P. 3795–3807.

14. Cak, M. First-principles study of magnetism at grain boundaries in iron and nickel / M. Cak, M. Sob, J. Hafner // Phys. Rev. B. – 2008. – V. 78. – P. 054418.

15. Vlack, L.H.V. Intergranular energy of iron and some iron alloys / L.H. Van Vlack // Transac-tions. American Institute of Mining, Metallurgical and Petroleum Engineers. – 1951. – V. 191. – P. 251.

16. Roth, T.A. The surface and grain boundary energies of iron, cobalt and nickel / T.A. Roth // Ma-terials Science and Engineering. – 1975. – V. 18 – P. 183–192.


�� .�., Ab initio �� .�. �� -��

2013, 5, 1 81

AB INITIO MODELING OF THE GRAIN BOUNDARY FORMATION ENERGY IN BCC IRON

A.V. Verkhovykh1, A.A. Mirzoev2

First-principles modeling of the grain boundary 5 bcc iron are carried out by WIEN2k code. Opti-mal parameters and the grain boundary formation energy for the two tilt angles are calculated.

Keywords: first-principles modeling, bcc iron, grain boundary.

References 1. Ursaeva A.V., Rakitin M.S., Ruzanova G.E., Mirzoev A.A. Ab initio modelirovanie vzaimodej-

stviya vodoroda s tochechnymi defektami v OCK-zheleze [Ab initio modeling of vacancy-point defects interaction in BCC iron]. Vestnik YuUrGU. Seriya “Matematika. Mekhanika. Fizika”. 2011. Issue 4. no. 10(227). pp. 114–119. (in Russ.).

2. Ursaeva A.V., Mirzoev A.A., Ruzanova G.E. Ab initio-modelirovanie kompleksov vodorod-vakansiya (VHn) v OCK-zheleze [Ab initio modelling of hydrogen-vacancy (VHn) complexes in BCC iron]. Vestnik YuUrGU. Seriya «Metallurgiya». 2011. Issue 17. 36(253). pp. 51–55. (in Russ.).

3. Singh R., Schneibel J.H., Divinski S., Wilde G. Grain boundary diffusion of Fe in ultrafine-grained nanocluster-strengthened ferritic steel. Acta Materialia. 2011. Vol. 59. pp. 1346–1353.

4. Wachowicz E., Kiejna A. Effect of impurities on grain boundary cohesion in bcc iron. Computa-tional Materials Science. 2008. Vol. 43. pp. 736–743.

5. Sutton A.P., Balluffi R.W. Interfaces in crystalline solids. Oxford: Clarendon, 1995. 856 p. 6. Valiev R.Z., Vergazov A.N., Gercman V.Yu. Kristallogeometricheskij analiz mezhkristallit-nyx

granic v praktike e'lektronnoj mikroskopii [Crystal geometric analysis of intercrystalline boundaries in electron microscopy]. Moscow: Nauka, 1991. 231 p. (in Russ.).

7. Monkhorst H.J., Pack J.D. Special points for brillouin-zone integrations. Phys. Rev. B. 1976. Vol. 13. pp. 5188–5192.

8. Ursaeva A.V., Ruzanova G.E., Mirzoev A.A. Vybor optimalnykh parametrov dlya postroeniya maksimal'no tochnoj modeli OCK-zheleza [Selection of optimal parameters for formation the most ac-curate model of BCC iron]. Vestnik YuUrGU. Seriya “Matematika. Mexanika. Fizika”. 2010. Issue 2. no. 9(185). pp. 97–101.

9. Zhang J.-M., Huang Y.-H., Wu X.-J., Xu K.-W. Energy calculation for symmetrical tilt grain boundaries in iron. Applied Surface Science. 2006. Vol. 252. pp. 4936–4942.

10. Lejcek P., Hofmann S., Paidar V. Solute segregation and classification of [100] tilt grain boundaries in -iron: consequences for grain boundary engineering. Acta Materialia. 2003. Vol. 51. pp. 3951–3963.

11. Rice J.R., Wang J.-S. Embrittlement of interfaces by solute segregation. Materials Science and Engineering: A. 1989. Vol. 107. pp. 23–40.

12. Braithwaite J.S., Rez P. Grain boundary impurities in iron. Ada Materialia. 2005. Vol. 53. pp. 2715–2726.

13. Hyde B., Farkas D., Caturla M.J. Atomistic sliding mechanisms of the =5 symmetric tilt grain boundary in bcc iron. Philosophical Magazine. 2005. Vol. 85. pp. 3795–3807.

14. Cak M., Sob M., Hafner J. First-principles study of magnetism at grain boundaries in iron and nickel. Phys. Rev. B. 2008. Vol. 78. pp. 054418.

15. Vlack L.H.V. Intergranular energy of iron and some iron alloys. Transactions. American Insti-tute of Mining, Metallurgical and Petroleum Engineers. 1951. Vol. 191. p. 251.

16. Roth T.A. The surface and grain boundary energies of iron, cobalt and nickel. Materials Science and Engineering. 1975. Vol. 18. pp. 183–192.

�� !" # $%&"'( ) 11 �� 2013 �.

1 Verkhovykh Anastasia Vladimirovna is Post-graduate student of South Ural State University. E-mail: [email protected] 2 Mirzoev Aleksandr Aminulaevich is Dr.Sc. (Physics and Mathematics), Professor, General and Theoretical Physics Department, South Ural State University. E-mail: [email protected]


. « . . » 82

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ANALYSIS OF CARBON DISSOLUTION IN TITANIUM UNDER ELECTRON BEAM TREATMENT

V.E. Gromov1, S.V. Raykov2, D.A. Sherstobitov3, Yu.F. Ivanov4, B.B. Khaimzon5, S.V. Konovalov6

Analysis of structure phase state of titanium alloy BT6 subjected to electroexplosive alloying by carbon graphite fibers and following electron beam treatment is carried out by methods of scanning elec-tron microscopy. The carbon graphite fibres dissolution process on each stage of treatment is considered. The comparison with the results of theoretical model representations is realized.

Keywords: carbon dissolution, titanium, fibres, scanning electron microscopy.

1 Gromov Viktor Evgenievich is Dr. Sc. (Physics and Mathematics), Professor, Head of Physics Department, Siberian State Industrial Univer-sity, Novokuznetsk. E-mail: [email protected] 2 Raykov Sergey Valentinovich is Cand. Sc. (Engineering), Senior Researcher, Siberian State Industrial University, Novokuznetsk. 3 Sherstobitov Sergey Aleksandrovich is Junior Research Fellow, Siberian State Industrial University, Novokuznetsk. 4 Ivanov Yury Fedorovich is Dr.Sc. (Physics and Mathematics), Professor, Chief Scientist Officer, Institute of High Current Electronics SB RAS. E-mail: [email protected] 5 Khaimzon Boris Bernardovich is Cand. Sc. (Physics and Mathematics), Docent, Department of Physics and Methodic of Teaching Physics, Kuzbass State Pedagogical Academy. E-mail: [email protected] 6 Konovalov Sergey Valerievich is Cand. Sc. (Engineering), docent, Physics Department, Siberian State Industrial University. E-mail: [email protected]


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References 1. Budovskikh E.A., Vashhuk E.S., Gromov V.E., Ivanov Yu.F., Koval' N.N. Formirovanie struk-

turno-fazovykh sostoyanij metallov i splavov pri elektrovzryvnom legirovanii i elektronno-puchkovoj obrabotke [Formation of structural-phase states of metals and alloys at electroexplosive doping and elec-tron-beam processing]. Novokuznetsk: Inter-Kuzbass, 2011. 207 p. (in Russ.).

2. Ivanov Yu.F., Gromov V.E., Budovskikh E.A. et al. Struktura, fazovyj sostav i svojstva poverkhnostnykh sloev titanovykh splavov posle elektrovzryvnogo legirovaniya i elektronno-puchkovoj obrabotki [The structure, phase composition and properties of the surface layers of titanium alloys after electroexplosive doping and electron-beam processing]. Novokuznetsk: Inter-Kuzbass, 2012. 434 p. (in Russ.).

3. Ivanov Yu.F., Karpij S.V., Morozov M.M., Koval' N.N., Budovskix E.A., Gromov V.E. Struk-tura, fazovyj sostav i svojstva titana posle e'lektrovzryvnogo legirovaniya i e'lektronno-puchkovoj obrabotki: monografiya [The structure, phase composition and properties of titanium after electroexplo-sive doping and electron-beam processing: Monograph]. Novokuzneck: NPK, 2010. 173 p. (in Russ.).

4. Karpij S.V., Morozov M.M., Budovskikh E.A. et al. Strukturno-fazovye sostoyaniya titana posle e'lektrovzryvnogo legirovaniya i posleduyushhej e'lektronno-puchkovoj obrabotki [Structural and phase states of titanium after electroexplosive doping and subsequent electron-beam processing]. Uspekhi fiziki metallov. 2010. Vol. 11, no. 3. pp. 273–293. (in Russ.).

5. Sarychev V.D., Granovskij A.Yu., Starovackaya S.N., Gromov V.E. Gidrodinamicheskaya model obrazovaniya nanostrukturnykh sloev [Hydrodynamic model of nanostructured layers forming] / Izves-tiya vuzov. Chernaya metallurgiya. 2012. no. 6. pp. 87–89. (in Russ.).

6. Sarychev V.D., Voloshina M.S., Gromov V.E. Matematicheskaya model generacii termou-prugikh voln pri vozdejstvii koncentrirovannyx potokov energii na materialy [A mathematical model for the generation of thermoelastic waves under the influence of concentrated energy flows of materials]. Fundamentalnye problemy sovremennogo materialovedeniya. 2011. Vol. 8, no. 4. pp. 71–76. (in Russ.).

7. Sarychev V.D., Khaimzon B.B., Soskova N.A., Gromov V.E. Matematicheskaya model rast-voreniya chastic ugleroda v titane pri vozdejstvii koncentrirovannyx potokov e'nergii [A mathematical model of dissolution of carbon particles in titanium when exposed to concentrated energy flows]. Titan. 2012. no. 1. pp. 4–8. (in Russ.).

8. Khaimzon B.B., Sarychev V.D., Soskova N.A., Gromov V.E. Diffuzionnaya model rastvoreniya chastic ugleroda v titane pri vozdejstvii koncentrirovannykh potokov energii [Diffusion model of disso-lution of carbon particles in titanium when exposed to concentrated energy flows]. Fundamentalnye problemy sovremennogo materialovedeniya. 2012. no. 2. pp. 19–21. (in Russ.).

9. Engel L., Klingele G. Rastrovaya elektronnaya mikroskopiya. Razrushenie: spravochnik [Raster electron microscopy. Destruction: A Handbook]. Moscow: Metallurgiya, 1986. 230 p. (in Russ.). [Engel L., Klingele H. Rasterelektronenmikroskopische Untersuchungen von Metallschaden. Munchen, Wien, 1982].

10. Krishtall M.M., Yasnikov I.S., Polunin V.I., Filatov A.M., Ulyanenko A.G. Skaniruyushhaya elektronnaya mikroskopiya i rentgenospektral'nyj mikroanaliz v primerax prakticheskogo primeneniya[Scanning electron microscopy and X-ray microanalysis in the practical application of]. Moscow: Tex-nosfera, 2009. 208 p.

11. Brandon D., Kaplan U. Mikrostruktura materialov. Metody issledovaniya i kontrolya [The mi-crostructure of materials. Methods of research and monitoring]. Moscow: Tekhnosfera, 2006. 384 p. (in Russ.).

12. Arzamasov B.N., Makarova V.I., Mukhin G.G. et al. Materialovedenie [Materials Science]. Moscow: izdatelstvo MGTU im. N.E. Baumana, 2003. 648 p. (in Russ.).

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SIMULATION OF EXPERIMENTS TO MEASURE SURFACE TENSION BY THE SHAPE OF A DROP SURFACE AT THE EXISTANCE OF IRREGULARITY IN ITS HANGER OR BEARING

V.G. Rechkalov1, V.P. Beskachko2

We suggest a technique to determine surface shape for a drop suspended to a hanger of an arbitrary form and orientation in space. This technique allows researchers to obtain geometrical characteristics (such as volume, surface area, cross-sectional areas) of the drop suspended to a flawed hanger, and to find experimental errors in determining surface tension that are associated with using such hanger. Our numerical experiments show that the main source of error in determining surface tension is braking of circular symmetry of the hanger.

Keywords: surface tension measurements, the shape of a drop, numerical methods.

References 1. Babichev A.P., Babushkina N.A., Bratkovskij A.M. et al. Fizicheskie velichiny: spravochnik

[Physical values: a handbook]. Moscow: Energoatomizdat, 1991. 1232 p. (in Russ.). 2. Adams J.C. An attempt to test the theories of capillary action. Cambridge: Deighton, Bell and

CO, 1883. 60 p. 3. Rechkalov V.G., Pyzin G.P., Ushacov V.L., Beskachko V.P. Kompyuternaya obrabotka izobraz-

heniya v metode opredeleniya koefficienta poverkhnostnogo natyazheniya zhidkosti po forme poverkhnosti kapli [Computer-driven processing of the liquid drop shape image for determination of surface tension coefficient]. Vestnik YuUrGU. Seriya «Matematika. Mexanika. Fizika». 2010. Issue 3. no. 30(206). pp. 83–88. (in Russ.).

�� 24 �� 2012 �.

1 Rechkalov Viktor Grigorevich is Cand. Sc. (Pedagogical), Associate Professor, General and Teoretical Physics Department, South Ural State University. E-mail: [email protected] 2 Beskachko Valeriy Petrovich is Dr. Sc. (Physics and Mathematics), Professor, General and Theoretical Physics Department, South Ural State University.


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1. Krishnan, A. Young’s modulus of single-walled nanotubes / A. Krishnan, E. Dujardin, T.W. Ebb-sen, P.N. Yianilos, M.J. Treacy // Physical Review B. – 1998. – V. 58. – P. 14013–14019.

2. Wong, E.W. Nanobeam mechanics: elasticity, strength, and toughness of nanorods and nanotubes / E.W. Wong, P.E. Sheehan, C.M. Lieber // Science. – 1997. – V. 277. – P. 1971–1975.

3. Giannopoulos, G.I. Evaluation of the effective mechanical properties of single walled carbon nanotubes using a spring based finite element approach / G.I. Giannopoulos, P.A. Kakavas, N.K. Ani-fantis //Computational Materials Science. – 2008. – V. 41. – P. 561–569.

4. Kalamkarov, A.L. Analytical and numerical techniques to predict carbon nanotubes properties / A.L. Kalamkarov, A.V. Georgiades, S.K. Rokkam, V.P. Veedu, M.N. Ghasemi-Nejhad // International Journal of Solids and Structures. – 2006. – V. 43. – P. 6832–6854.

5. Gupta, S.S. Continuum structures equivalent in normal mode vibrations to single-walled carbon nanotubes / S.S. Gupta, R.C. Batra // Computational Materials Science. – 2008. – V. 43. – P. 715–723.

6. Avila, A.F. Molecular Mechanics Applied to Single-Walled Carbon Nanotubes / A.F. Avila, G.S.R. Lacerda // Materials Research. – 2008. – V. 11. – . 325–333.

7. Alex, A. Granovsky, Firefly version 7.1.G, www http://classic.chem.msu.su/gran/ firefly/index.html


�� .�., �� . ., ��-�� .�., �� .�., �� .�. ��!�� "#�� "$��

2013, 5, 1 99

8. , . . (7,7) - Li, Na, S Se / . . , . . // . « -

. . ». – 2012. – . 7. – 34(293). – . 182–185. 9. , . . -

/ . . , . . , . . // . « . -. ». – 2012. – . 7. – 34(293). – . 191–194. 10. , . . . /

. . , . . , . . // . . – 2010. – 7. – C. 19–23.

QUANTUM-CHEMICAL MODELING OF DEFORMATION PROCESSES OF CHIRAL CARBON NANOTUBES

S.A. Sozykin1, E.R. Sokolova2, K.A. Telnoy3, V.P. Beskachko4, G.P. Vyatkin5

The paper presents the results of quantum-mechanical calculations of the mechanical properties of chiral carbon nanotubes. We assess the conditions under which the boundary effects can be neglected in the simulation. Tensile strength, Young's modulus and shear energy for optimized nanotubes geometry were identified. The ranges of strains of nanotubes in which they have not experienced fracture. were defined

Keywords: carbon nanotubes, Young modulus, shear modulus, ultimate stress.

References 1. Krishnan A., Dujardin E., Ebbsen T.W., Yianilos P.N., Treacy M.J. Young’s modulus of single-

walled nanotubes. Physical Review B. 1998. Vol. 58. pp. 14013–14019. 2. Wong E.W., Sheehan P.E., Lieber C.M. Nanobeam mechanics: elasticity, strength, and toughness

of nanorods and nanotubes. Science. 1997. Vol. 277. pp. 1971–1975. 3. Giannopoulos G.I., Kakavas P.A., Anifantis N.K. Evaluation of the effective mechanical proper-

ties of single walled carbon nanotubes using a spring based finite element approach. Computational Ma-terials Science. 2008. Vol. 41. pp. 561–569.

4. Kalamkarov A.L., Georgiades A.V., Rokkam S.K., Veedu V.P., Ghasemi-Nejhad M.N. Analyti-cal and numerical techniques to predict carbon nanotubes properties. International Journal of Solids and Structures. 2006. Vol. 43. pp. 6832–6854.

5. Gupta S.S., Batra R.C. Continuum structures equivalent in normal mode vibrations to single-walled carbon nanotubes. Computational Materials Science. 2008. Vol. 43. pp. 715–723.

6. Avila A.F., Lacerda G.S.R. Molecular Mechanics Applied to Single-Walled Carbon Nanotubes. Materials Research. 2008. Vol. 11. pp. 325–333.

7. Granovsky A.A. Firefly version 7.1.G, www http://classic.chem.msu.su/gran/firefly/index.html 8. Sozykin S.A., Beskachko V.P. Mekhanicheskie svojstva kompleksov uglerodnoj nanotrubki (7,7)

s odinochnymi atomami Li, Na, S i Se [Mechanical properties of the complexes of carbon nanotube (7,7) with single Li, Na, S and Se atoms] Vestnik YuUrGU. Seriya “Matematika. Mekhanika. Fizika”. 2012. Issue 7. no. 34(293). pp. 182–185. (in Russ.).

9. Telnoj K.A., Sozykin S.A., Beskachko V.P. Struktura i mekhanicheskie svojstva ftorirovannykh uglerodnykh nanotrubok [Structure and mechanical properties of fluorinated carbon nanotubes] Vestnik YuUrGU. Seriya “Matematika. Mekhanika. Fizika”. 2012. Issue 7. no. 34(293). pp. 191–194. (in Russ.).

10. Beskachko V.P., Sozykin S.A., Sokolova E.R. Mekhanicheskie svoistva odnosloinykh uglerod-nykh nanotrubok. Vse materialy. Entsiklopedicheskii spravochnik (The mechanical properties of single-walled carbon nanotubes. All materials. Encyclopedic Handbook.). 2010. no. 7. pp. 19–23. (in Russ.).

�� 11 �� 2013 �. 1 Sozykin Sergey Anatolevich is Post-graduate student, General and Theoretical Physics Department, South Ural State University. E-mail: [email protected] 2 Sokolova Ekaterina Romanovna is student, General and Theoretical Physics Department, South Ural State University. 3 Telnoy Konstantin Aleksandrovich is student, Department of Optics and Spectroscopy, South Ural State University. 4 Beskachko Valeriy Petrovich is Dr.Sc. (Physics and Mathematics), Professor, General and Theoretical Physics Department, South Ural State University. 5 Vyatkin German Platonovich is Dr.Sc. (Chemistry), Professor, Corresponding Member of the Russian Academy of Sciences, General and Theoretical Physics Department, South Ural State University.


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1. , . . / . . , . . , . . // . – 2010. – . 2, 37. – . 122–125.

2. Bradt, H.V. Astrophysics Processes / H.V. Bradt. – Cambridge University Press, 2008. – 536 . 3. , . . / . . ,

. . , . . // . . – 2008. – . 3. – 25. – . 74–77.

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5. , . . D / . . , . . , . . // I -

« ». – : . – 2012. – . 297–299.

6. , . . / . . . – .: , 1991. – 720 .

RADIO PHYSICAL PROPERTIES OF COLLISIONAL PLASMA IN GAS DISCHARGE

I.A. Shevyakov1, V.I. Tambovcev2, A.A. Kuchurkin3

The paper presents the research of radio physical properties of low-temperature plasma in the transi-tion region using the computer: from the state of the ionized gas to the state gas plasma. The analysis of simulative dependence proves the significant effect of electron collision frequency on radio wave propa-gation through the medium. The collision ionized low-pressure gas of discharge lamp is considered as physical model. The critical frequency for radio transparency is determined by microwave diagnostics.

Keywords: ionized gas, collisional plasma, radio transparency, microwave diagnostics, wave vec-tor, the complex conductivity, low pressure discharge.

References 1. Kuchurkin A.A., Tambovcev V.I., Teplyakov A.V. SVCh diagnostika gazorazryadnoj plazmy

[Microwave diagnostics of gas-discharge plasma]. Trudy MFTI. 2010. Vol. 2, no. 37. pp. 122–125. (in Russ.).

2. Bradt H.V. Astrophysics Processes. Cambridge University Press, 2008. 536 . 3. Tambovcev V.I., Teplyakov A.V., Usachyov V.K. Ionnyj zvuk v chastichno ionizovannom gaze

[Ion sound in a partially ionized gas]. Vestnik Chelyabinskogo gosudarstvennogo universiteta. Fizika.2008. Issue 3. no. 25. pp. 74–77. (in Russ.).

4. Golant V.E. SVCh metody issledovaniya plazmy [Ultra-high frequency methods of plasma analysis]. Moscow: Nauka, 1968. 328 p. (in Russ.).

5. Shevyakov I.A., Tambovcev V.I., Kuchurkin A.A. O radioprozrachnosti estestvennogo i voz-mushhyonnogo gaza sloya D osnovaniya ionosfery [Dynamoelectric processes in electric power systems and in lower ionosphere]. Trudy XI mezhdunarodnoj konferencii «Fizika i texnicheskie prilozheniya vol-novyx processov» [Physics and engineering applications of wave processes. Proceedings of XI Interna-tional Conference.]. Ekaterinburg: UrFU. 2012. pp. 297–299. (in Russ.).

6. Rokhlin G.N. Razryadnye istochniki sveta [Discharge sources of light]. Moscow: Energoatomiz-dat, 1991. 720 p. (in Russ.).

�� 18 �� 2012 �.

1 Shevyakov Igor Andreevich is design engineer, Experimental Design Office, Open Joint-Stock Company Chelyabinsk Radio Manufacturing Plant “Polet” (“Flight”). E-mail: [email protected] 2 Tambovcev Vladymir Ivanovich is Dr.Sc. (Physics and Mathematics), Professor of Instrument Engineering Department, South Ural State University. E-mail: [email protected] 3 Kuchurkin Artyom Alexandrovich is Engineer, Mechel Service Global. E-mail: [email protected]


2013, 5, 1 107

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1 �� . ��-�� – � �� , ��-�� , ��-�� ; Al-Mustansiriyah University, ��, ��. E-mail: [email protected]


. « . . » 108

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2013, 5, 1 109

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. -mp .

. . .

1. , . , , / . . – M: , 1980. – 664 c.

2. Al-Delfi, J.K. Quasi-Banach space for the sequence space p , where 0 < p < 1 / J.K. Al-Delfi // Journal of college of Education (Iraq – Baghdad). Mathematics. – 2007. – 3. – P. 285–295.

�� ! 28 "��# 2013 $.

QUASI-SOBOLEV SPACES mp

Jawad K. Al-Delfi1

Firstly, the notion of quasi-Banach spaces for the sequence spaces mp , m R∈ , (0, )p∈ +∞ has

been considered and we have been proved analogs of the Sobolev embedding theorem. Also, the notion quasi-operator Laplace has been considered.

Keywords: quasi-norm, Quasi-Banach space, Quasi-Sobolev spaces, Laplace’ Quasi-operator, Green’ Quasi-operator.

Reference 1. Triebel H. Interpolation theory, function spaces, differential operators. Moscow: Mir, 1980. 664

p. (in Russ). 2. Al-Delfi J.K. Quasi-Banach space for the sequence space p , where 0 < p < 1. Journal of college

of Education (Iraq – Baghdad). Mathematics. 2007. no. 3. pp. 285–295.

1 Jawad K. Al-Delfi is Post-graduate student, Equations of Mathematical Physics Department, South Ural State University; Al-Mustansiriyah University, Baghdad, Iraq. E-mail: [email protected]


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. 19. – . 41–46. 4. // . – 1990. – . 26. – . 22–29. 5. // . –

1991. – . 29. – . 56. 6. Judd–Ofelt analysis of the Er3+ (4f11) absorption intensities in Er3+−doped garnets / Sardar D.K.,

Bradley W.M., Perez J.J. et al. // J. Appl. Phys. – 2003. – V. 93, 5. – . 2602–2607. 7. Bertrand, J.A. Polynuclear complexes with hydrogen-bonded bridges. 4. Structure and magnetic

properties of dinuclear copper(II) complexes of amino alcohols / J.A. Bertrand, E. Fujita, D.G. VanDerveer // Inorg. Chem. – 1980. – V. 19, 7. – P. 2022–2028.

8. , . / . ; . . – .: , 1989. – 400 . 9. Crystal Structure and Magnetic Properties of catena- -Sulfato-[N,N'-bis(2-

hydroxyethyl)dithiooxamido(2-)-N,O,S:N',O',S']bis[aquacopper(II)]: A Chain of Copper(II) Dinuclear Units with a 594-cm–1 Singlet-Triplet Separation and a 5.61-Å Copper-Copper Distance / J.J. Girerd, S. Jeannin, Y. Jeannin, O. Kahn // Inorg. Chem. – 1978. – V. 17, 11. – P. 3034–3040.

10. , . . . - / . . , . . . – .: , 1980. – 302 .

11. , . . / . . . – .: , 1971. – 1032 . 12. , . . - : / . . . – .:

, 1978. – 472 . 13. , . / . . – .: , 1978. – 791 .


�� .�., �� .�., �� .�., �� .�., �� .�., �� !.". �� (Y1–xNdx)3Al5O12

2013, 5, 1 113

THE MAGNETIC PROPERTIES OF SOLID SOLUTIONS (Y1–xNdX)3Al5O12

E.N. Rybina1, A.N. Bryzgalov2, L.M. Svirskaya3, V.V. Viktorov4, P.V. Volkov5, D.E. Zhivulin6

Samples of solid solutions (Y1–xNdx)3Al5O12 system obtained by directional crystallization are ana-lyzed. The parameters of the lattice and the magnetic susceptibility of the crystals as function of the Nd3+

concentration are determined. Neodymium replaces yttrium and due to the proximity of the ionic radii of these elements there is no change in the lattice parameters of solid solutions. The magnetic susceptibility obeys the Curie’s law. It is suggested that there is an antiferromagnetic ordering of the cations closest f-felectrons within Nd3+ pairs of the nearest neighbors.

Keywords: solid solution, Yttrium Aluminum Garnet doped with neodymium, magnetic susceptibility of solid solutions.

References 1. Kaminskij A.A. Lazernye kristally [Laser crystals]. Moscow: Nauka, 1975. 256 p. (in Russ.). 2. Kaminskij A.A. Fizika i spektroskopiya lazernykh kristallov [Physics and Spectroscopy of Laser

Crystals]. Moscow: Nauka, 1986. 272 p. (in Russ.). 3. Lazer na kristallakh ittrij-e'rbij-alyuminievogo granata [Yttrium-erbium-aluminum garnet laser].

Trudy IOFAN. 1989. Vol. 19. pp. 41–46. (in Russ.). 4. Opticheski plotnye aktivnye sredy [Optically dense active media]. Trudy IOFAN. 1990. Vol. 26.

pp. 22–29. (in Russ.). 5. Spektroskopiya oksidnykh kristallov dlya kvantovoj elektroniki [Spectroscopy of oxide crystals

for quantum electronics]. Trudy IOFAN. 1991. Vol. 29. p. 56. (in Russ.). 6. Sardar D.K., Bradley W.M., Perez J.J., Gruber J.B., Zandi B., Hutchinson J.A., Trussell C.W.,

Kokta M.R. Judd–Ofelt analysis of the Er3+ (4f11) absorption intensities in Er3+−doped garnets. J. Appl. Phys. 2003. Vol. 93, no 5. pp. 2602–2607. http://dx.doi.org/10.1063/1.1543242

7. Bertrand J.A., Fujita E., VanDerveer D.G. Polynuclear complexes with hydrogen-bonded bridges. 4. Structure and magnetic properties of dinuclear copper(II) complexes of amino alcohols. Inorg. Chem. 1980. Vol. 19, no. 7. pp. 2022–2028.

8. Karlin R. Magnetokhimiya [Magnetochemistry]. Moscow: Mir, 1989. 400 p. (in Russ.). [Carlin R.L. Magnetochemistry. Springer-Verlag, 1986. 328 p.].

9. Girerd J.J., Jeannin S., Jeannin Y., Kahn O. Crystal Structure and Magnetic Properties of catena--Sulfato-[N,N'-bis(2-hydroxyethyl)dithiooxamido(2-)-N,O,S:N',O',S']bis[aquacopper(II)]: A Chain of

Copper(II) Dinuclear Units with a 594-cm–1 Singlet-Triplet Separation and a 5.61-Å Copper-Copper Distance. Inorg. Chem. 1978. Vol. 17, no. 11. pp. 3034–3040.

10. Kalinnikov V.T., Rakitin Yu.V. Vvedenie v magnetokhimiyu. Metod statisticheskoj magnitnoj vospriimchivosti v khimii [Introduction to Magnetochemistry. Statistical method of magnetic susceptibil-ity in Chemistry]. Moscow: Nauka, 1980. 302 p. (in Russ.).

11. Vonsovskij S.V. Magnetizm [Magnetism]. Moscow: Nauka, 1971. 1032 p. (in Russ.). 12. Samsonov G.V. Fiziko-ximicheskie svojstva okislov: spravochnik [Physico-chemical properties

of oxides: A Handbook]. M.: Metallurgiya, 1978. 472 p. (in Russ.). 13. Kittel Ch. Vvedenie v fiziku tvyordogo tela [Introduction to Solid State Physics]. Moscow:

Nauka, 1978. 791 p. (in Russ.). #$%&'()*+ , -./+01)2 29 3$45-4 2012 6.

1 Rybina Elvira Nafizovna is Post-Graduate Student, General and Theoretical Physics Department, Chelyabinsk State Pedagogical University. E-mail: [email protected] 2 Bryzgalov Aleksandr Nikolaevich is Dr. Sc. (Physics and Mathematics), Professor, General and Theoretical Physics Department, Chelyabinsk State Pedagogical University. E-mail: [email protected] 3 Svirskaya Lyudmila Moiseevna is Cand. Sc (Physics and Mathematics), Associate Professor, General and Theoretical Physics Department, Chelyabinsk State Pedagogical University. E-mail: [email protected] 4 Viktorov Valeriy Viktorovich is Dr. Sc (Chemistry), Professor, General and Theoretical Physics Department, Chelyabinsk State Pedagogical University. E-mail: [email protected] 5 Volkov Petr Vyacheslavovich is Post-Graduate Student, General and Theoretical Physics Department, Chelyabinsk State Pedagogical Univer-sity. E-mail: [email protected] 6 Zhivulin Dmitry Evgenievich is Post-Graduate Student, General and Theoretical Physics Department, Chelyabinsk State Pedagogical Univer-sity. E-mail: [email protected]


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1. , . . / . . , . . , . . . – .: , 1977. – 236 .

2. Bain, E.C. The Nature of Martensite / E.C. Bain // Trans. AIME, Steel Div. – 1924. – V. 70. – P. 25–46.

3. Zener, C. Theory of Strain Interaction of Solute Atoms / C. Zener // Phys. Rev. – 1948. – V. 74. – Issue 6. – P. 639–647.

4. , . . / . . , . . // – 1971. – . 32, 1 – C. 5–

13. 5. , . . Fe-

Ni-C Fe-Mn-C / . . , . . , . . // . – 1977. – . 22. – 10 – C. 1728–1730.

6. / . . , . . , . . . // – 1977. – . 43, 3. – C. 582–590.

7. Interplay between long-range elastic and short-range chemical interactions in Fe-C martensite formation / A. Udyansky, J. von Pezold, V.N. Bugaev et al. // Phys. Rev. – 2009. – V. 79. – Issue 12. – P. 224112.

8. Lau, T. Many-Body Potential for Point Defect Clusters in Fe-C Alloys / T. Lau, C.J.F. Forst // Phys. Rev. Lett. – 2007. – V. 98. – Issue 21. – P. 215501.

9. Atomistic modeling of an Fe system with a small concentration of C / C.S. Becqaurt, J.M. Raulot, G. Bencteux et al. // Comp. Mat. Sc. – 2007. – V. 40. – Issue 1. – P. 119–129.

10. Hepburn, D.J. Metallic-covalent potential for carbon in iron / D.J. Hepburn, G.J. Ackland // Phys. Rev. B – 2008. – V. 78. – Issue 16. – P. 165115

11. Ackland, G.J. Development of an interatomic potential for phosphorus impurities in -iron // J. Phys: Condens. Matter – 2004. – V. 16. – P. 2629–2642.

12. Hoover W.G. Canonical dynamics: Equilibrium phase-space distributions / W.G Hoover // Phys. Rev. A – 1985. – V. 31. – Issue 3. – P. 1695–1697.

13. Liu, X. Lattice-parameter variation with carbon content of martensite. I. X-ray-diffraction ex-perimental study / X. Liu, F. Zhong // Phys. Rev. B – 1995. – V. 52. – Issue 14. – P. 9970.


. « . . » 118

INERATOMIC POTENTIAL FOR IRON-CARBON SYSTEM AND MARTENCITIC PHASE TRANSITION PROBLEM

P.V. Chirkov1, A.A. Mirzoev2

Computer simulation of tetrahedral lattice distortion of iron-carbon system was performed. Modern embedded atom potentials for Fe-C system were used. It was shown that existing potentials can repro-duce tetrahedral lattice distortion and don’t allow to describe concentration dependence of martensite thermal stability.

Keywords: martensitic phase transition, molecular dynamic, embedded atom potential

References 1. Kurdyumov G.V., Utevskij L.M., Entin R.I. Prevrashheniya v zheleze i stali [Transformation in

iron and steel]. Moscow: Nauka, 1977. 236 p. (in Russ.). 2. Bain E.C. The Nature of Martensite. Trans. AIME, Steel Div. 1924. Vol. 70. pp. 25–46. 3. Zener C. Theory of Strain Interaction of Solute Atoms. Phys. Rev. 1948. Vol. 74. Issue 6.

pp. 639–647. 4. Khachaturyan A.G., Shatalov G.A. K teorii uporyadocheniya atomov ugleroda v kristalle

martensita. Fizika metallov i metallovedenie. 1971. Vol. 32, no. 1. pp. 5–13. (in Russ.). 5. Gavrilyuk V.G., Nadutov V.M., Razumov O.N. Messbauerovskoe issledovanie martensitnogo

prevrashheniya v splavakh Fe-Ni-C i Fe-Mn-C. Ukrainian Journal of Physics. 1977. Vol. 22, no. 10. pp. 1728–1730. (in Russ.).

6. Gridnev V.N., Gavrilyuk V.G., Nemoshkalenko V.V., Polushkin Yu.A., Razumov O.N. Fizika metallov i metallovedenie. 1977. Vol. 43, no. 3. pp. 582–590. (in Russ.).

7. Udyansky A., von Pezold J., Bugaev V.N., Friak M., Neugebauer J. Interplay between long-range elastic and short-range chemical interactions in Fe-C martensite formation. Phys. Rev. 2009. Vol. 79. Issue 12. p. 224112.

8. Lau T., Forst C.J.F. Many-Body Potential for Point Defect Clusters in Fe-C Alloys. Phys. Rev. Lett. 2007. Vol. 98. Issue 21. p. 215501.

9. Becqaurt C.S., Raulot J.M., Bencteux G., Domain C., Perez M., Garruchet S., Nguyen H. Atom-istic modeling of an Fe system with a small concentration of C. Comp. Mat. Sc. 2007. Vol. 40. Issue 1. pp. 119–129. http://dx.doi.org/10.1016/j.commatsci.2006.11.005

10. Hepburn D.J., Ackland G.J. Metallic-covalent potential for carbon in iron. Phys. Rev. B. 2008. Vol. 78. Issue 16. p. 165115.

11. Ackland G.J. Development of an interatomic potential for phosphorus impurities in -iron. J. Phys: Condens. Matter. 2004. Vol. 16. p. 2629–2642.

12. Hoover W.G. Canonical dynamics: Equilibrium phase-space distributions. Phys. Rev. A. 1985. Vol. 31. Issue 3. p. 1695–1697.

13. Liu X., Zhong F. Lattice-parameter variation with carbon content of martensite. I. X-ray-diffraction experimental study. Phys. Rev. B. 1995. Vol. 52. Issue 14. p. 9970.

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1 Chirkov Pavel Vladimirovich is Master Student, General and Theoretical Physics Department, South Ural State University. E-mail: [email protected] 2 Mirzoev Aleksandr Aminulaevich is Dr.Sc. (Physics and Mathematics), Professor, General and Theoretical Physics Department, South Ural State University. E-mail: [email protected]


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