вестник южно уральского-государственного_университета._серия_математика._механика._физика_№2_2013

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: . .- . ., . . ( . ), . .- . ., . . ( . ), . .- . ., . ., . . ., . ., . .- . ., . ., . .- . ., . ., . .- . ., . ., . . ., . ., . .- . ., . ., . .- . ., . ., . .- . ., . .

: . .- . ., . . ( -

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

, ), . .- . ., Ph. D., , . . (

, . , ), Ph. D., (Kim Kisik, INHA-

, . , ), Ph. D., (Kim Jaewan,

KIAS, . , ), Ph. D., - . . (

, , )

2009 .

77-26455 13 2006 .

-.

. -

«Ulrich’s Periodicals Direc-tory».

---

« -

, -

». 29211 -

« ». – 2

.

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

. . – - -.............................................................................................................................................. 6

. . 3................................................................................................................... 13

. ., . ., .- ., . ., . .

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........................................................................................................................................................ 39 . ., . ., . . -

................... 45 . ., . .

.................................................................................... 52 . . ...... 60

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. ., . . ................ 79 . .

........................................ 88

. . . 94 . ., . ., . . -

: ............................................................ 99 . ., . ., . . Ab-initio -

- .........................108 . ., . ., . .

..................................................................117

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. ., . ., . ., . . - -......................................128

. ., . ., . ., . . ......................................................133

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...................................................................................................................................................138 . ., . .

-1 ..........................................................................................................143 . ., . ., . ., . ., -

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3

. . ..............................151 . ., . .

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. . ....................166 . . ....................170 . ., . . -

...............................................................174 . . ....................................................178

© , 2013


CONTENTS Mathematics ADUKOV V.M. About Wiener–Hopf factorization of functionally commutative matrix functions .. 6KOLYASNIKOV S.A. Units of integral group rings of finite groups with a direct multiplier of order 3.................................................................................................................................................. 13LESCHEVICH V.V., PENYAZKOV O.G., ROSTAING J.-C., FEDOROV A.V., SHULGIN A.V. Experimental and mathematical simulation of auto-ignition of iron micro particles .......................... 21MARTYUSHEV E.V. Algorithmic solution of the five-point pose problem based on the cayley representation of rotation matrices....................................................................................................... 31MEDVEDEV S.V. About extension of homeomorphisms over zero-dimensional homogeneous spaces................................................................................................................................................... 39OMELCHENKO E.A., PLEKHANOVA M.V., DAVYDOV P.N. Numerical solution of delayed linearized quasistationary phase-field system of equations ................................................................. 45RUBINA L.I., UL’YANOV O.N. Towards the differences in behaviour of solutions of linear and non-linear heat-conduction equations .................................................................................................. 52SOLDATOVA N.G. About optimum on risks and regrets situations in game of two persons ........... 60TABARINTSEVA E.V. About solving of an ill-posed problem for a nonlinear differential equa-tion by means of the projection regularization method........................................................................ 65TANANA V.P., ERYGINA A.A. About the evaluation of inaccuracy of approximate solution of the inverse problem of solid state physics ........................................................................................... 72UKHOBOTOV V.I., TROITSKY A.A. One problem of pulse pursuit............................................... 79USHAKOV A.L. Updating iterative factorization for the numerical solution of two elliptic equa-tions of the second order in rectangular area ....................................................................................... 88

Physics VORONTSOV A.G. Structure amendments in heated metal clusters during its process.................... 94GUREVICH S.Yu., GOLUBEV E.V., PETROV Yu.V. Lamb waves in ferromagnetic metals: laser exitation and electromagnetic registration ........................................................................................... 99RIDNYI Ya.M., MIRZOEV A.A., MIRZAEV D.A. Ab-initio simulation of influence of short-range ordering carbon impurities on the energy of their dissolution in the FCC-iron......................... 108USHAKOV V.L., PYZIN G.P., BESKACHKO V.P. A method for monitoring small surface movements of a sessile drop during evaporation................................................................................. 117

Short communications BEREZIN I.Ya., PETRENKO Yu.O. Problems of vibration safety of operator of industrial tractor . 123BOLSHAKOV M.V., GUSEVA A.V., KUNDIKOVA N.D., POPKOV I.I. Fiber and interferential method of obtaining non-homogeneous polarized beam ..................................................................... 128BOLSHAKOV M.V., GUSEVA A.V., KUNDIKOVA N.D., POPKOV I.I. Transformation of the spin moment into orbital moment in laser beam.................................................................................. 133BOLSHAKOV M.V., KOMAROVA M.A., KUNDIKOVA N.D. Experimental determination of multimode optical fiber radiation mode composition.......................................................................... 138BRYZGALOV A.N., VOLKOV P.V. Influence of heat-treating on microhardness of silica glass KU-1 .................................................................................................................................................... 143GERASIMOV A.M., KUNDIKOVA N.D., MIKLYAEV Yu.V., PIKHULYA D.G., TERPUGOV M.V. Efficiency of second harmonic generation in one-dimentional photonic crystal from isotropic material......................................................................................................................... 147IVANOV S.A. Stability of two-layer recursive neural networks ........................................................ 151KARITSKAYA S.G., KARITSKIY Ya.D. Fluorescent impurity centers for monitoring the physical state of polymers and a computer simulation of processes of structuring of the polymer matrices .... 155


5

KATKOV M.L. Existence of the fixed point in the case of evenly contractive monotonic operator.. 160LEYVI A.Ya. Analysis of mechanisms of surface pattern formation during the high-power plasma stream processing................................................................................................................................. 162MATVEEVA L.V. Behavior of Gronwall polynomials outside the boundary of summability region 166KHAYRISLAMOV K.Z. Poiseuille flow of a fluid with variable viscosity ....................................... 170KHAYRISLAMOV M.Z., HERREINSTEIN A.W. Explicit scheme for the solution of third boundary value problem for quasi-linear heat equation....................................................................... 174EVNIN A.Yu. Polynomial as a sum of periodic functions.................................................................. 178


. « . . » 6

517.53

–- -

. . 1

- -–

. -. - -

. -.

: – , , .

1. Γ – , D+ .

D+ Γ { }= ∞ D− , , 0 D+∈ . ( )A t – -Γ - s . – ( )A t -

( ) ( ) ( ) ( ) .A t A t d t A t t− += , ∈Γ (1)

( )A t± – Γ - , D± -

, 1( ) diag [ ]sd t t tρρ= , , , 1 sρ ρ, , – , ( )A t . , 1 sρ ρ≤ ≤ . – -

- ( )A t . ( 1s = ) [1].

– ( ) – -. ,

, , , .

, . -

. --

- , . [2–5]. - . -

, . -

- - . - ( , -

- ), , - -s . ,

( )A t Γ[ ]G ( [ ])Z G

G . , -.

1 – , - , , -

. E-mail: [email protected]


. . –- -

2013, 5, 2 7

( )A t , ( )A t- . – - ,

[6]. -- - – -

.

2. 1. G – G n= , e – G , [ ]G – ,

( )g G

a g g∈

G

( )a g . G [ ]G g1 g⋅ . G – [ ]G . -

G , [ ]G. [ ]G ( )a g.

1 2{ }, , , sK e K K= – G , j jh K= – -

jK , 1, , sg g – . ( [ ])Z G G , ( )a g , -

.

j

jg K

C g∈

= , 1, ,j s= ( [ ])Z G .

1

sk

i j ij kk

C C c C=

= ,

kijc – ( [ ])Z G .

A – 1

( [ ]), ( ),s

i i i ii

a a C Z G a a g=

= ∈ =

( [ ])Z G . A 1, , sC C .

1 , 1

s sk

j i i j i ij ki i k

aC a C C a c C= =

= = ,

kjA A1

sk

kj i iji

A a c=

=

1 , 1

.ss

ki ij

i k j

A a c= =

= (2)

, G – , s n= , ( [ ]) [ ]Z G G= , A( ) [ ]

g Ga a g g G

∈= ∈ { }1 2, , , ng e g g=

1 11 1 2 1

1 12 2 2 2

1 12

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

n

n

n n n n

a g a g g a g g

a g a g g a g gA

a g a g g a g g

− −

− −

− −

= . (3)

, G .


. « . . » 8

, G n a , ,

{ }1, , , ne a a − , .

[ ]G ( ( [ ])Z G ) - (3) ( (2)). [ ]G , ( [ ])Z G -

G . – I .

[ ]G⊗ (3) ( )ka g ∈ . -, ( [ ])Z G⊗ (2) ia ∈ . sI -

s I . 1. 1, , sχ χ – -

G 1, , sn n – .

1 1 1 2 1 2 1

1 2 1 2 2 2 2

1 1 2 2

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

( ) ( ) ( )

s s

s n

s s s s s

h g I h g I h g Ih g I h g I h g I

nh g I h g I h g I

χ χ χχ χ χ

χ χ χ

= .

– ,

1 1 2 1 1

1 1 2 2 2 2

1 2

( ) ( ) ( )

( ) ( ) ( )1

( ) ( ) ( )

s

s

s s s s

g I g I g I

g I g I g In

g I g I g I

χ χ χχ χ χ

χ χ χ

− = ,

( [ ])A Z G∈ ⊗ : 1A −= Λ ,

1diag [ , ]sλ λΛ = , , 1 ( ) ( ), 1, ,j jj g G

a g g j sn

λ χ∈

= = .

. 1− :

( )1

1

,1 1( ) ( ) ( ) ( )0,| |

s

k i k j k i j ijij k g G

I i jh g g I g g I I

i jn Gχ χ χ χ δ−

= ∈

== = = =

≠. (4)

( . [7, .3, § 4, 2]) , G . -

(4) , 1sI− = . 1A − , (2)

, 1− :

( )1 1

, 1 , , 1

1 1( ) ( ) ( ).s s

kik kl lj k m ml i k j lij l k l k m

A A h a c g gn n

χ χ− −

= == =

( . [8, §109, (8)]), -:

1( ) ( ) ( )

sk m l

k ml i k i m i lik

h hh c g g g

nχ χ χ

== .

, – . ,

( )1

, 1 1 1

1 1( ) ( ) ( ) ( ) ( ) ( ).s s s

m l mm j l i m i l m i m l i l j lij i il m m l

h h hA a g g g a g h g gn n n n

χ χ χ χ χ χ−

= = == =


. . –- -

2013, 5, 2 9

, (4), :

( )1

1( ).

sij

m m i mij i mA a h g

nδ

χ−

==

– , -G . , -

( )1ij iij

A δ λ− = 1− = Λ . .

, [ ]A G∈ ⊗ , G – , 1

1 1 1s sh h n n= = = = = = | |s G= . , A , 1, , sλ λ –

. 2. (1). – -

Γ , – . -, ( )W

( )H Γ Γ ( . [1]). ( )a g ( )ga t t∈ ∈Γ . - ( )A t

, 1( ) ( ) ( ), 1, ,j g jj g G

t a t g j sn

λ χ∈

= = Γ .

ind ( )j j tρ λΓ= – Γ ( )j tλ , -2π , t Γ .

( ) ( ) ( )jj j jt t t tρλ λ λ− += – – ( )j tλ .

1 1( ) diag [ ( ) , ( )] diag [ ( ) , ( )]s st t t t tλ λ λ λ− − + +Λ = , ⋅ ,– – - ( )tΛ 1

. 2. ( ) ( [ ])A t Z G∈ ⊗ – Γ -

(2). – ( ) ( ) ( ) ( )A t A t d t A t− += :

1 1 1 2 2 1 1

1 1 2 2 2 2 2

1 1 2 2

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( ) ( ) ( )

s s

s s

s s s s s

t g t g t g

t g t g t gA t

t g t g t g

λ χ λ χ λ χ

λ χ λ χ λ χ

λ χ λ χ λ χ

− − −

− − −

−

− − −

= ,

1( ) diag [ ] ind ( ),sj jd t t t tρρ ρ λΓ= , , , =

1 1 1 1 2 1 1 2 1 1

1 2 2 1 2 2 2 2 2 2

1 1 2 2

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( ) ( ) ( )

s s

s s

s s s s s s s s

h t g h t g h t g

h t g h t g h t gA t

h t g h t g h t g

λ χ λ χ λ χ

λ χ λ χ λ χ

λ χ λ χ λ χ

+ + +

+ + +

+

+ + +

= .

3. 1. 4G V= – . -

4S : { }4 , (1 2)(3 4),(13)(2 4),(1 4)(2 3)V e= ,

2 2C C× . ( )A t 2- , 2 2× - ,

2 2×


. « . . » 10

3 41 2

4 32 1

3 4 1 2

4 3 2 1

( ) ( )( ) ( )( ) ( )( ) ( )

( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )

a t a ta t a ta t a ta t a t

A t a t a t a t a ta t a t a t a t

= .

4V ( . [7, .3, § 5])

e (1 2)(3 4) (13)(2 4) (1 4)(2 3)

1χ 1 1 1 12χ 1 1− 1 1−

3χ 1 1 1− 1−

3χ 1 1− 1− 1, ( )A t

: 1 1 2 3 4( ) ( ) ( ) ( ) ( )t a t a t a t a tλ = + + + , 2 1 2 3 4( ) ( ) ( ) ( ) ( )t a t a t a t a tλ = − + − ,

3 1 2 3 4( ) ( ) ( ) ( ) ( )t a t a t a t a tλ = + − − , 4 1 2 3 4( ) ( ) ( ) ( ) ( )t a t a t a t a tλ = − − + . ( )A t .

2. 3G S= – 3. - 3 :

{ } { } { }1 2 3, (1 2), (13), (2 3) , (1 2 3),(13 2 .K e K K= = =

jC 3( [ ])Z S :

1 2 1C 2C 3C

1C 1C 2C 3C

2C 2C 1 33 3C C+ 22C

3C 3C 22C 1 32C + (1) - ( )A t :

1 2 3

2 1 3 2

3 2 1 3

( ) 3 ( ) 2 ( )( ) ( ) ( ) 2 ( ) 2 ( ) .

( ) 3 ( ) ( ) ( )

a t a t a tA t a t a t a t a t

a t a t a t a t= +

+

3S ( . [7, .3, § 5])

e (1 2) (1 2 3)

1χ 1 1 1

2χ 1 1− 13χ 2 0 1−

, 1 1 2 3( ) ( ) 3 ( ) 2 ( )t a t a t a tλ = + + , 2 1 2 3( ) ( ) 3 ( ) 2 ( )t a t a t a tλ = − + , 3 1 3( ) ( ) ( )t a t a tλ = − ,

11 3 2 1 1 2

1 11 3 2 , 1 1 06 62 0 2 1 1 1

−= − = −− −

� � .

2


. . –- -

2013, 5, 2 11

1 2 3 1 1 1

1 2 2 2 2

1 2 3 3 3

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

( ) ( ) ( ) 0 , ( ) ( ) 3 ( ) 2 ( )

( ) ( ) ( ) 2 ( ) 0 2 ( )

t t t t t t

A t t t A t t t t

t t t t t

λ λ λ λ λ λ

λ λ λ λ λ

λ λ λ λ λ

− − − + + +

− − + + +− +

− − − + +

= − = −

− −

,

1 1 2 3ind ( ( ) 3 ( ) ( ))a t a t a tρ Γ= + + , 2 1 2 3ind ( ( ) 3 ( ) ( ))a t a t a tρ Γ= − + , 3 1 3ind ( ( ) ( ))a t a tρ Γ= − . 3. { }8 1, i, j, kG Q= = ± ± ± ± – ,

2 2 2i j k ijk 1= = = = − . 5 : { } { } { } { } { }1 2 3 4 51 , 1 , i , j , k .K K K K K= = − = ± = ± = ±

jC 8( [ ])Z Q

1 2 1C 2C 3C 4C 5C

1C 1C 2C 3C 4C 5C

2C 2C 1C 3C 4C 5C

3C 3C 3C 1 22 2C + 52C 42C

4C 4C 4C 52C 1 22 2C + 32C

5C 5C 5C 42C 32C 1 22 2C +

1 2 3 4 5

2 1 3 4 5

3 3 1 2 5 4

4 4 5 1 2 3

5 5 4 3 1 2

( ) ( ) 2 ( ) 2 ( ) 2 ( )( ) ( ) 2 ( ) 2 ( ) 2 ( )( ) ( ) ( ) ( ) 2 ( ) 2 ( )( )( ) ( ) 2 ( ) ( ) ( ) 2 ( )( ) ( ) 2 ( ) 2 ( ) ( ) ( )

a t a t a t a t a ta t a t a t a t a ta t a t a t a t a t a tA ta t a t a t a t a t a ta t a t a t a t a t a t

+=+

+

.

8Q :

1 1− i± j± k±

1χ 1 1 1 1 1

2χ 1 1 1 –1 –1

3χ 1 1 –1 1 –1

4χ 1 1 –1 –1 1

5χ 2 –2 0 0 0 ( )A t . -

. 1( ) ( ) ( ), 1, ,5j g j

j g Gt a t g j

nλ χ

∈= = , :

1 1 2 3 4 5( ) ( ) ( ) 2 ( ) 2 ( ) 2 ( )t a t a t a t a t a tλ = + + + + ,

2 1 2 3 4 5( ) ( ) ( ) 2 ( ) 2 ( ) 2 ( )t a t a t a t a t a tλ = + + − − ,

3 1 2 3 4 5( ) ( ) ( ) 2 ( ) 2 ( ) 2 ( )t a t a t a t a t a tλ = + − + − ,

4 1 2 3 4 5( ) ( ) ( ) 2 ( ) 2 ( ) ( )t a t a t a t a t a tλ = + − − + , 5 1 2( ) ( ) ( )t a t a tλ = − . ( )A t .

1. , . . / . . , . . . – M.: , 1971. – 352 .

2. Adukov, V.M. On Wiener–Hopf factorization of meromorphic matrix functions / V.M. Adukov // Integral Equations and Operator Theory. – 1991.– V. 14. – P. 767–774.


. « . . » 12

3. , . . – - / . . // . – 1992. – . 4. – . 1. – . 54–74.

4. , . . - / . . // . -. . – 1999. – . 118, 3. – . 324–336. 5. , . . – - / . .

// . – 2009. – . 200, 8. – . 3–24. 6. , . . n / . . // -

. . – 1952. – . 7. – . 4(50). – . 3–54. 7. , . . . III / . . . – M.: , 2000. –

272 . 8. , . . / . . . – .: « », 2004.– 624 .

ABOUT WIENER–HOPF FACTORIZATION OF FUNCTIONALLY COMMUTATIVE MATRIX FUNCTIONS

V.M. Adukov1

An algorithm of an explicit solution of the Wiener–Hopf factorization problem is proposed for func-tionally commutative matrix functions of a special kind. Elementary facts of the representation theory of finite groups are used. Symmetry of the matrix function that is factored out allows to diagonalize it by a constant linear transformation. Thus, the problem is reduced to the scalar case.

Keywords: Wiener–Hopf factorization, special indexes, finite groups.

References 1. Gokhberg I.Ts., Fel'dman I.A. Uravneniya v svertkakh i proektsionnye metody ikh resheniya

(Convolution equations and projection methods for their solution). Moscow: Nauka, 1971. 352 p. (in Russ.). [Gohberg I.C., Fel dman I.A. Convolution equations and projection methods for their solution.American Mathematical Society, Providence, R.I., 1974. Translations of Mathematical Monographs, Vol. 41. MR 0355675 (50 #8149) (in Eng.).]

2. Adukov V.M. On Wiener–Hopf factorization of meromorphic matrix functions. Integral Equa-tions and Operator Theory. 1991. Vol. 14. pp. 767–774. DOI: 10.1007/BF01198935.

3. Adukov V.M. Faktorizatsiya Vinera–Khopfa meromorfnykh matrits-funktsiy (Wiener-Hopf fac-torization of meromorphic matrix function). Algebra i analiz. 1992. Vol. 4. Issue 1. pp. 54–74. (in Russ.). [Adukov V.M. Wiener–Hopf factorization of meromorphic matrix functions. St. Petersburg Mathematical Journal. 1993. Vol. 4. Issue 1. pp. 51–69. (in Eng.).]

4. Adukov V.M. O faktorizatsii analiticheskikh matrits-funktsiy (About factorization of analytical matrix functions) Teoreticheskaya i matematicheskaya fizika. 1999. Vol. 118, no. 3. pp. 324–336. [Adu-kov V.M. Factorization of analytic matrix-valued functions. Theoretical and Mathematical Physics.1999. Vol. 118, no. 3. pp. 255–263. DOI: 10.1007/BF02557319].

5. Adukov V.M. Faktorizatsiya Vinera–Khopfa kusochno meromorfnykh matrits-funktsiy (Piece-wise Wiener–Hopf factorization of meromorphic matrix functions). Matematicheskiy sbornik. 2009. Vol. 200, no. 8. pp. 3–24. (in Russ.).

6. Gakhov F.D. Kraevaya zadacha Rimana dlya sistemy n par funktsiy (Riemann boundary value problem for system of n pairs of functions). Uspekhi matematicheskikh nauk. 1952. Vol.7. Issue 4(50). pp. 3–54. (in Russ.).

7. Kostrikin A.I. Vvedenie v algebru. Chast III. (Introduction into algebra. Part III). Moscow: Fiz-matlit, 2000. 272 p. (in Russ.).

8. Van der Varden B.L. Algebra. Saint Petersburg: Lan', 2004. 624 p. (in Russ.). 18 2013 .

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


2013, 5, 2 13

512.552.7

3

. . 1

A×Z3, A 3. -

(9,3), (9,3,3) (15,3). : , , -

.

1. G A b= × – , b – 3, A

a 3. ( )V G -G .

: 0 : ( ) ( ),V G V Aϕ → ( ),x x x A∀ ∈ 1b ;

1 : ( ) ( ) ( / ),V G V A V G abϕ → ≅ ( ),x x x A∀ ∈ 2b a ; 2

2 : ( ) ( ) ( / ),V G V A V G a bϕ → ≅ ( ),x x x A∀ ∈ b a ;

3 : ( ) ( ), ( ))V G V G a g g a g Gϕ → ∀ ∈ . A -

a , H . -

: , ( ).G a G h a h h H→ ∀ ∈ 1. ( )( ) ( )V G b V A= × K ,

31 2 11 12 2 2 2 23 3 31 (1 ) (1 ) (1 )K ab a b a b ab a a Gυυ υυ −− −= = + + + + + + + + + ∩ , 1 1( )ϕ υ υ= ,

2 2( ) (1 ( )) ( )I a V Aϕ υ υ= ∈ + ∩ , 3 3( ) (1 ( )) ( )I b V G aϕ υ υ= ∈ + ∩ . ( )I a – A , 1a − , 2 1a − ; ( )I b – G a , -

1b − , 2 1b − ( . [1]). . 1 [2] ( )( ) ( )V G b V A= × K ,

( )1 ( ) ( ) ( )K A A A b V G= + ∩ , ( ) ( )A A A b – , -

G ( 1)( 1)x b− − , 2( 1)( 1)x b− − \ {1}x A∈ . -, K .

Kυ ∈

2 22 21h ha h haha ha

h H h H h H h H h H h Hh ha ha h ha ha bυ α α α β β β

∈ ∈ ∈ ∈ ∈ ∈= + + ⋅ + + + ⋅ +

22 2

h ha hah H h H h H

h ha ha bγ γ γ∈ ∈ ∈

+ + + ⋅ .

1 2, ( )V Aυ υ ∈ 3 ( )V G aυ ∈

22

1 h ha hah H h H h H

h ha haυ λ λ λ∈ ∈ ∈

= + + ,

1 – , , - -



. « . . » 14

22

2 h ha hah H h H h H

h ha haυ μ μ μ∈ ∈ ∈

= + + ,

22

3 h hb hbh H h H h H

h a hb a hb aυ ν ν ν∈ ∈ ∈

= + + .

, 0 ( ) 1ϕ υ = ( . . Kυ ∈ ) 1 1( )ϕ υ υ= , 2 2( )ϕ υ υ= , 3 3( )ϕ υ υ= , .

2

2

2

1 0 0 1 0 0 1 0 00 1 0 0 1 0 0 1 00 0 1 0 0 1 0 0 11 0 0 0 1 0 0 0 10 1 0 0 0 1 1 0 00 0 1 1 0 0 0 1 01 0 0 0 0 1 0 1 00 1 0 1 0 0 0 0 10 0 1 0 1 0 1 0 01 1 1 0 0 0 0 0 00 0 0 1 1 1 0 0 00 0 0 0 0 0 1 1 1

h

ha

ha

h

ha

ha

h

ha

ha

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βββ

γγγ

2

2

2

00

h

h

ha

ha

h

ha

ha

h

hb

hb

ε

λλλ

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= ,

1, 1;0, 1.h

hh

ε=

=≠

, , 1υ ,

2υ , 3υ2h ha hha

λ λ λ ε+ + = ; 2h ha hhaλ λ λ ε+ + = ; 2h ha hha

λ λ λ ε+ + = .

, 1 2, (1 ( )) ( )I a V Aυ υ ∈ + ∩ , 3 (1 ( )) ( )I b V G aυ ∈ + ∩ . , ,

( ) 3h h h hα λ μ ν= + + ; ( ) 3ha ha ha h hα λ μ ν ε= + + − ; 2 2 2( ) 3h hha ha haα λ μ ν ε= + + − ;

2( ) 3h ha hbhaβ λ μ ν= + + ; 2( ) 3h h hb hha

β λ μ ν ε= + + − ; 2 ( ) 3ha h hb hhaβ λ μ ν ε= + + − ;

2 2( ) 3h ha ha hbγ λ μ ν= + + ; 2 2( ) 3ha h hha hb

γ λ μ ν ε= + + − ; 2 2( ) 3h ha hha hbγ λ μ ν ε= + + − .

, υ , ,

31 2 11 12 2 2 2 23 3 31 (1 ) (1 ) (1 )ab a b a b ab a aυυ υυ −− −= + + + + + + + + + . (1)

: ( ) ( ) ( )K V A V A V G aϕ → × × ,

( )1 2 3( ) ( ), ( ), ( )ϕ υ ϕ υ ϕ υ ϕ υ= . .

, ( ) ( )1 2 3 1 1 2 2 3 3( ) ( ), ( ), ( ) ( ) ( ), ( ) ( ), ( ) ( )w w w w w w wϕ υ ϕ υ ϕ υ ϕ υ ϕ υ ϕ ϕ υ ϕ ϕ υ ϕ= = .

, 2 2 2(1 ) 3(1 )a a a a+ + = + + ,

2 2 2 2 2(1 ) 3(1 )ab a b ab a b+ + = + + , 2 2 2 2 2(1 ) 3(1 )a b ab a b ab+ + = + + ,


. . 3

2013, 5, 2 15

2 2 2 2(1 )(1 ) (1 )(1 )a a b b b b a a+ + + + = + + + + =2 2 2 2 2 2 2 2(1 )(1 ) (1 )(1 )a b ab ab a b ab a b a b ab= + + + + = + + + + =

2 2 2 2 2 2(1 )(1 ) (1 )(1 )a a ab a b ab a b a a= + + + + = + + + + =2 2 2 2 2 2(1 )(1 ) (1 )(1 ).a b ab a a a a a b ab= + + + + = + + + +

, 1 21, 1 ( )w w I a− − ∈ , 2 1 ( )w I b− ∈ , 2 2 2

1 2 3( 1)(1 ) ( 1)(1 ) ( 1)(1 ) 0w a a w a a w b b− + + = − + + = − + + = .

( )( )

31 2

31 2

3 31 1 2 2

11 12 2 2 2 23 3 3

11 12 2 2 2 23 3 3

11 12 2 2 2 23 3 3

1 (1 ) (1 ) (1 )

1 (1 ) (1 ) (1 )

1 (1 ) (1 ) (1 ).

ww w

ww w

w ab a b a b ab a a

ab a b a b ab a a

ab a b a b ab a a

υυ υ

υυ υ

υ −− −

−− −

−− −

= + + + + + + + + + ×

× + + + + + + + + + =

= + + + + + + + + +

, 3 3 3 3w wυ υ≠ , , 2 23 3 3 3(1 ) (1 )w a a w a aυ υ+ + = + + .

, , ϕ -. (1), . -

.

.

2. (9,3) 9 3| 1 | 1G a a b b= = × = . ( )V a [3].

[2]. 1 2( ) ,V a a F F u u= × = × ,

8 2 71 1 ( ) ( )u a a a a= − + + + , 1 8 3 6 4 5

1 1 ( ) ( ) ( ),u a a a a a a− = − − + + + + +2 7 4 5

2 1 ( ) ( ),u a a a a= − + + + 1 2 7 3 6 82 1 ( ) ( ) ( )u a a a a a a− = − − + + + + + .

1 1 2( )V G a b u u K= × × × × ,

Kυ ∈3 6 2 6 3 21 21 11 (1 ) (1 )

3 3a b a b a b a bυ υυ − −= + + + + + + ,

31 2, (1 ( )) ( )I a V aυ υ ∈ + ∩ , 3( )V G a – ( . [3]) 3( ) 1ϕ υ = .

80

iii aλ= , 8

0i

ii aμ= ,

( 2 3 40 0 1 1 2 2 3 3 4 4

5 6 7 85 5 6 6 7 7 8 8 6 3 7 4

2 3 4 5 68 5 0 6 1 7 2 8 3 0

4 1

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

( ) ( ) ( ) ( ) ( ) ( )

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

( )

a a a a

a a a a b ab

a b a b a b a b a b

a

υ λ μ λ μ λ μ λ μ λ μ

λ μ λ μ λ μ λ μ λ μ λ μ

λ μ λ μ λ μ λ μ λ μ

λ μ

= + + ⋅ + + ⋅ + + ⋅ + + ⋅ + + ⋅ +

+ + ⋅ + + ⋅ + + ⋅ + + ⋅ + + ⋅ + + ⋅ +

+ + ⋅ + + − ⋅ + + ⋅ + + ⋅ + + − ⋅ +

+ + ⋅

)

7 8 2 2 2 25 2 3 6 4 7 5 8

3 2 4 2 5 2 6 26 0 7 1 8 2 0 3

7 2 8 21 4 2 5

( ) ( ) ( ) ( )

( 1) ( ) ( ) ( 1)

( ) ( ) .

b a b b ab a b

a b a b a b a b

a b a b

λ μ λ μ λ μ λ μ

λ μ λ μ λ μ λ μ

λ μ λ μ

+ + ⋅ + + ⋅ + + ⋅ + + ⋅ +

+ + − ⋅ + + ⋅ + + ⋅ + + − ⋅ +

+ + ⋅ + + ⋅

(2)

, 31 2, (1 ( ))u u I a∈ + ,

2 6 61 1 ( 1) ( 1)u a a a a= − − + − , 6 3 2 3

2 1 ( 1) ( 1) ( 1)u a a a a a a= − − + − + − . : K F Fϕ → × ,

( )1 2( ) ( ), ( )ϕ υ ϕ υ ϕ υ= .


. « . . » 16

1 2,u u 1

1 8a a+ 2 7a a+ 3 6a a+ 4 5a a+1u 1 –1 1 0 021u 5 –4 3 –2 131u 19 –18 15 –9 3

2u 1 0 –1 0 122u 5 1 –4 –2 332u 19 3 –18 –9 15

1 2u u –1 0 1 1 –12

1 2u u –5 –1 5 3 –421 2u u 1 1 –2 0 –12 21 2u u 7 1 –6 –3 5

. 1 (2) , 2 3 3 31 2 1 2( ,1), ( ,1), (1, ),(1, ) ( )u u u u Kϕ∈ ,

1 2 1 20 , , , 2i i j j≤ ≤ ( )1 2 1 21 2 1 2, ( )i i j ju u u u Kϕ∈ ,

1 2 1 21, 2, 2, 1i i j j= = = = 1 2 1 22, 1, 1, 2i i j j= = = = . , .

2. 2 2 3 3 31 2 1 2 2 1 2( ) ( , ) ( ,1) (1, ) (1, )K u u u u u u uϕ = × × × . 3 3 3( )F F Kϕ× ≅ × × .

, ( )V G , , ( . [1, . 45]),

. 3. ( )W G – , | ( ) : ( ) | 1V G W G = ,

1 2 3 4 5 6( )V G a b u u u u u u= × × × × × × × ,

8 2 71 1 ( ) ( )u a a a a= − + + + , 2 7 4 5

2 1 ( ) ( ),u a a a a= − + + +8 2 2 2 7

3 1 ( ) ( )u ab a b a b a b= − + + + , 2 2 7 4 5 24 1 ( ) ( )u a b a b a b a b= − + + + ,

2 8 2 7 25 1 ( ) ( )u ab a b a b a b= − + + + , 2 7 2 4 2 5

6 1 ( ) ( )u a b a b a b a b= − + + + . . ( )W G ( )V G , -

( )V C , C G . ( )V C.

3 3( )V a a= , ( )V b b= , 3 3( )V a b a b= , 6 6( )V a b a b= ,

1 2( )V a a u u= × × , 3 4( )V ab ab u u= × × , 2 25 6( )V ab ab u u= × × .

2 , 1 1 2 2

1 2 5 6 1 2 1 2( ) ( , )u u u u u u u uϕ − − = , 1 1 32 3 5 6 2( ) ( ,1)u u u u uϕ − − = ,

1 1 31 4 5 6 1( ) (1, )u u u u uϕ − − = , 1 1 3

2 3 4 5 2( ) (1, )u u u u uϕ − − = . .

3. (9,3,3) 9 3 3| 1 | 1 | 1G a a b b c c= = × = × = . A A a b= × .

( )V A 3. ( )V A a b F= × × , 1 6...F u u= × × .


. . 3

2013, 5, 2 17

1 1 6( ) ...V A a b u u K= × × × × × × , Kυ ∈

3 6 2 6 3 21 21 11 (1 ) (1 )3 3

a b a b a b a bυ υυ − −= + + + + + + ,

31 2, (1 ( )) ( )I a V aυ υ ∈ + ∩ , 3( )V G a – ( . [3]) 3( ) 1ϕ υ = .

, ( )31 ( ) , 1,...,6u I a∈ + = .

: K F Fϕ → × , ( )1 2( ) ( ), ( )ϕ υ ϕ υ ϕ υ= .

2, 3 3( )iab a= , 0,1,2i = , 3 3( ,1),(1, ) ( )u u Kϕ∈ .

( ) ( ) ( )3 3 5 6 5 61 2 1 2 4 45 51 2 1 2 3 4 3 4 6 6, , , , , ( ), , 0,1,2i j i i j ji i j j i ju u u u u u u u u u u u K i jϕ∈ =

, 1 3 5 2 4 6 1 3 5 2 4 61, 2, 2, 1i i i i i i j j j j j j= = = = = = = = = = = =

1 3 5 2 4 6 1 3 5 2 4 62, 1, 1, 2i i i i i i j j j j j j= = = = = = = = = = = = . GAP ( . [4]) ,

( )6 3 5 62 4 1 2 452 4 6 1 2 3 4 6, ( ), , 0,1,2.i j j ji i j j ju u u u u u u u u K i jϕ∉ =

, . 4. 1 12...K w w= × × ,

2 21 1 2 1 2( ) ( , )w u u u uϕ = , 2 2

2 3 4 3 4( ) ( , )w u u u uϕ = , 2 23 5 6 5 6( ) ( , )w u u u uϕ = ,

34 2( ) ( ,1)w uϕ = , 3

5 4( ) ( ,1)w uϕ = , 36 6( ) ( ,1)w uϕ = ,

37 1( ) (1, )w uϕ = , 3

8 2( ) (1, )w uϕ = , 39 3( ) (1, )w uϕ = ,

310 4( ) (1, )w uϕ = , 3

11 5( ) (1, )w uϕ = , 312 6( ) (1, )w uϕ = .

3 3 3 3 3 3 3 3 3( )F F Kϕ× ≅ × × × × × × × × . , , .

5. ( )W G – , | ( ) : ( ) | 1V G W G = .

4. (15,3) 15 3| 1 | 1G a a b b= = × = . ( )V a

[2]. . 0( )V a a u F= × × , 1 2 3F u u u= × × ,

3 120 1 ( ),u a a= − + + 1 6 9

0 1 ( )u a a− = − + + , 1 14 2 13 3 12 5 10 6 9 7 8

1 3 3( ) 2( ) ( ) 2( ) 3( ) 3( )u a a a a a a a a a a a a− = − − + − + − + + + + + + + , 2 13 3 12 4 11 5 10 6 9 7 8

2 3 2( ) ( ) 2( ) ( ) 2( ) ( )u a a a a a a a a a a a a= − + + + + + − + − + + + ,1 14 2 13 3 12 4 11 5 10 6 9

2 3 3( ) 3( ) 3( ) 2( ) 2( ) ( )u a a a a a a a a a a a a− = − + + − + + + − + + + − + , 14 3 12 4 11 5 10 6 9 7 8

3 3 ( ) 2( ) 2( ) ( ) ( ) 2( )u a a a a a a a a a a a a= + + − + − + − + + + + + ,1 2 13 3 12 4 11 5 10 6 9 7 8

3 3 3( ) ( ) 3( ) 2( ) 3( ) 2( )u a a a a a a a a a a a a− = + + − + − + + + + + − + . 1 0 1 2 3( )V G a b u u u u K= × × × × × × , Kυ ∈

5 10 2 10 5 2 5 1031 2 11 11 (1 ) (1 ) (1 )3 3 3

a b a b a b a b a aυυ υυ −− −= + + + + + + + + + ,

51 2, (1 ( )) ( )I a V aυ υ ∈ + ∩ , 5

3 (1 ( )) ( )I b V aυ ∈ + ∩ .


. « . . » 18

ψ ( )V a 5( )V G a , 5: a ab aψ → . 5 5

0( )V G a ab a u F′ ′= × × , 1 2 3F u u u′ ′ ′ ′= × × , ( ), 0,1,2,3.i iu u iψ′ = =

, 51 2 3, , (1 ( ))u u u I a∈ + , 1 2 3, , (1 ( ))u u u I b′ ′ ′ ∈ +

4 5 4 3 2 101 1 ( 1)( 1) (2 2 2 1)( 1)u a a a a a a a a= + + − − − − + − + − ,

4 3 2 5 3 2 102 1 (2 2 1)( 1) (2 2 1)( 1)u a a a a a a a a a= − − − + + − − − − + − ,

4 3 2 5 4 2 103 1 ( 2 2 1)( 1) ( 2 2 1)( 1)u a a a a a a a a a= + + + + − − − − − + − ,

5 4 5 3 5 5 51 ( 2 2 )( 1)u a a a a a a a a b′ = + + − − −−

4 5 2 5 5 5 2(2 2 )( 1)a a a a a a a b− − − + − , 5 4 5 3 5 2 5 5

2 (2 2 )( 1)u a a a a a a a a b′ = + − + − − +3 5 2 5 5 5 2( 2 2 )( 1)a a a a a a a b+ − + − − ,

5 4 5 2 5 5 53 (2 2 )( 1)u a a a a a a a a b′ = + − − + − +

4 5 3 5 5 5 2( 2 2 )( 1)a a a a a a a b+ + − − − .

( )50 1 ( )u I a∉ + , ( )0 1 ( )u I b′ ∉ + .

: K F F Fϕ ′→ × × , ( )1 2 3( ) ( ), ( ), ( )ϕ υ ϕ υ ϕ υ ϕ υ= .

, 9 3× , ,

( ) ( ) ( )3 3 3,1,1 , 1, ,1 , 1,1, ( ), 1,2,3i i iu u u K iϕ′ ∈ = .

GAP ( . [4]) ,

( )3 3 31 2 1 2 1 21 2 3 1 2 3 1 2 3, , ( ), , , 0,1,2( 1,2,3)i j ki i j j k ku u u u u u u u u K i j kϕ′ ′ ′ ∈ = = .

, . 2.

2

1i 2i 3i 1j 2j 3j 1k 2k 3k0 0 0 1 0 1 2 0 2 0 0 0 2 0 2 1 0 1 0 0 1 0 0 2 1 0 2 0 0 1 1 0 0 0 0 1 0 0 1 2 0 1 2 0 0 0 0 2 0 0 1 2 0 1 0 0 2 1 0 2 1 0 0 0 0 2 2 0 0 0 0 2

1i 2i 3i 1j 2j 3j 1k 2k 3k 1i 2i 3i 1j 2j 3j 1k 2k 3k0 1 0 0 2 0 2 1 2 0 2 0 0 1 0 1 2 1 0 1 0 1 2 1 1 1 1 0 2 0 1 1 1 0 2 0 0 1 0 2 2 2 0 1 0 0 2 0 2 1 2 2 2 2 0 1 1 0 2 2 0 1 1 0 2 1 0 1 2 2 2 0 0 1 1 1 2 0 2 1 0 0 2 1 1 1 0 1 2 2 0 1 1 2 2 1 1 1 2 0 2 1 2 1 1 0 2 1 0 1 2 0 2 1 1 1 0 0 2 2 0 1 1 0 2 2 0 1 2 1 2 2 0 1 2 0 2 2 1 1 2 2 2 1 0 1 2 2 2 0 2 1 1 0 2 2 2 1 0 1 2 0 1 0 0 0 0 1 1 0 0 1 1 0 0 2 1 0 1 2 1 0 0 1 0 2 0 0 2 1 1 0 1 2 2 2 1 1


. . 3

2013, 5, 2 19

1 0 0 2 0 0 2 0 1 1 1 0 2 2 0 1 1 0 1 0 1 0 0 0 2 0 2 1 1 1 0 2 0 1 1 1 1 0 1 1 0 1 1 0 1 1 1 1 1 2 1 0 1 0 1 0 1 2 0 2 0 0 0 1 1 1 2 2 2 2 1 2 1 0 2 0 0 2 0 0 1 1 1 2 0 2 2 2 1 0 1 0 2 1 0 0 2 0 0 1 1 2 1 2 0 1 1 2 1 0 2 2 0 1 1 0 2 1 1 2 2 2 1 0 1 1 1 2 0 0 1 1 2 2 1 2 0 0 0 0 2 2 0 0 1 2 0 1 1 2 1 2 0 2 0 0 1 0 0 1 0 2 1 2 0 2 1 0 0 2 2 2 0 0 2 0 1 0 0 1 1 2 1 0 1 0 0 2 0 2 0 1 0 0 1 0 0 2 1 2 1 1 1 1 2 2 2 2 0 1 1 0 2 2 0 1 1 2 1 2 1 2 1 2 1 2 0 1 2 0 0 1 0 0 1 2 2 0 1 2 1 2 2 2 0 2 0 0 0 1 0 1 1 2 2 1 1 0 0 2 1 2 0 2 1 0 1 0 0 0 1 2 2 2 1 1 2 2 0 2 0 2 2 0 2 2 0 2 2 1 0 0 2 2 1 1 2 2 2 0 0 1 2 0 2 1 2 1 0 1 2 0 0 1 1 2 2 0 1 1 0 2 2 0 2 1 0 2 2 1 2 1 0 2 2 0 2 1 1 1 2 2 2 1 1 0 2 1 2 1 1 2 2 1 0 1 1 1 2 0 2 1 1 1 2 2 1 1 0 2 2 1 1 1 2 0 2 2 2 1 1 2 2 0 0 1 2 2 2 1 2 1 0 2 2 1 2 1 2 0 2 0 0 1 0 2 2 2 0 1 0 2 2 2 2 1 2 1 2 1 2 1 2 2 2 2 1 1 1 1 2 1 2 1 2 2 2 2 1 1 1 2 2 2 2 1 2 0 2 0

, . 6. 1 9...K w w= × × ,

1 1 3 1( ) ( , , )w u u uϕ ′= , ( )2 2 22 2 2 1 2 3( ) , ,w u u u u uϕ ′ ′ ′= , ( )2 2

3 3 3 1 3( ) , ,w u u u uϕ ′ ′= ,

( )2 24 1 3 1 3( ) 1, ,w u u u uϕ ′ ′= , 3

5 2( ) (1, ,1)w uϕ = , 36 3( ) (1, ,1)w uϕ = ,

37 1( ) (1,1, )w uϕ ′= , ( )3

3 8 2( ) 1,1,w uϕ ′= , ( )39 3( ) 1,1,w uϕ ′= ,

3 3 3 3 3( )F F Kϕ× ≅ × × × × . , , -

. 7. ( )W G – , | ( ) : ( ) | 1V G W G = ,

0 1 12( ) ...V G a b u u u= × × × × × ,

3 120 1 ( )u a a= − + + , 14 2 13 3 12 4 11 5 10 6 9( ) 3 2( ) 2( ) 2( ) ( ) ( ) ( )u x x x x x x x x x x x x x= − + + + − + + + − + + + ,

1 ( )u u a= , 22 ( )u u a= , 2

3 ( )u u a= , 4 ( )u u ab= , 2 25 ( )u u a b= , 4

6 ( )u u a b= , 27 ( )u a b ,

4 28 ( ),u u a b= 8

9 ( )u u a b= , 310 ( )u u a b= , 6 2

11 ( )u u a b= , 1212 ( )u u a b= .

. ( )V C , ( )W G , C -G .

5 5( )V a a= , ( )V b b= , 5 5( )V a b a b= , 10 10( )V a b a b= , 3 30( )V a a u= × ,

0 1 2 3( )V a a u u u u= × × × × , 0 4 5 6( )V ab ab u u u u= × × × × , 2 2

0 7 8 9( )V a b a b u u u u= × × × × , 3 30 10 11 12( )V a b a b u u u u= × × × × .

6 , 1 1 1

10 11 12 1 3 1( ) ( , , )u u u u u uϕ − − − ′= , ( )1 1 1 2 2 21 2 3 4 6 7 2 2 1 2 3( ) , ,u u u u u u u u u u uϕ − − − ′ ′ ′= ,

( )1 1 1 1 1 1 2 21 7 8 9 10 12 3 3 1 3( ) , ,u u u u u u u u u uϕ − − − − − − ′ ′= , ( )1 1 1 1 1 1 2 2

1 3 7 9 10 12 1 3 1 3( ) 1, ,u u u u u u u u u uϕ − − − − − − ′ ′= ,


. « . . » 20

32 4 6 9 12 2( ) (1, ,1)u u u u u uϕ = , ( )1 1 1 1 1 1 1 3

3 4 6 7 8 9 10 12 3( ) 1, ,1u u u u u u u u uϕ − − − − − − − = ,

( )1 1 1 1 1 31 3 4 8 10 11 12 1( ) 1,1,u u u u u u u uϕ − − − − − ′= ( )3

1 3 5 9 10 2( ) 1,1,u u u u u uϕ ′= ,

( )1 1 1 1 1 31 3 6 7 8 9 11 3( ) 1,1,u u u u u u u uϕ − − − − − ′= .

.

1. , . . / . . . – : . . ., 1987. – 210 . ( . 24.09.87, 2712– 87).

2. , . . / . . . – , . . . .,

2000. – 45 . ( . 30.03.00, 860– 00). 3. , . . 7 9 / . . , . . // -

, . – 1999. – 11(450). – . 81–84. 4. Martin Schönert et al. GAP – Groups, Algorithms, and Programming. Lehrstuhl D für Mathe-

matik, Rheinisch Westfälische Technische Hochschule, Aachen, Germany, sixth edition, 1997.

UNITS OF INTEGRAL GROUP RINGS OF FINITE GROUPS WITH A DIRECT MULTIPLIER OF ORDER 3

S.A. Kolyasnikov1

The description of units of integral group rings of finite groups of type A×Z3 was obtained, where Acontains a central subgroup of order 3. For example, the unit groups of integral group rings of Abelian groups of the types (9,3), (9,3,3) and (15,3) were found.

Keywords: Abelian group, group ring, unit group of group ring.

References 1. Bovdi A.A. Mul'tiplikativnaya gruppa tselochislennogo gruppovogo kol'tsa (Multiplicative group

of integral group ring). Uzhgorod: Uzhgorodskiy gosudarstvennyy universitet, 1987. 210 p. (in Russ.). 2. Kolyasnikov S.A. O gruppe edinits tselochislennogo gruppovogo kol'tsa konechnykh grupp

razlozhimykh v pryamoe proizvedenie (About group of units of integral group ring of finite groups fac-torable into direct composition). Novosibirsk, Red. Sibirskogo Matematicheskogo Zhurnala, 2000. 45 p. (in Russ.).

3. Alev R.Zh., Panina G.A. Russian Mathematics (Izvestiya VUZ. Matematika). 1999. Vol. 43, no. 11. pp. 80–83.

4. Martin Schönert et al. GAP – Groups, Algorithms, and Programming. Lehrstuhl D für Mathe-matik. Rheinisch Westfälische Technische Hochschule, Aachen, Germany, sixth edition, 1997.

6 2013 .

1 Kolyasnikov Sergey Andreevich is Senior Lecturer, General Mathematics Department, South Ural State University. E-mail: [email protected]


2013, 5, 2 21

536.46:533.6: 662.612.32

. . 1, . . 2, .- . 3, . . 4, . . 5

-. -

. 1 125 0,5 28 500 1100 K.

--

. ,

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. . . . E-mail: [email protected]


. « . . » 22

. , .

, - ( ). [8, 9]

-, .

-, ,

. -, ,

.

500–1100 K - 0,5–28 ,

. 1. --

Kistler 6031U18 (4), - Kistler 5015A (11),

- (5) (6). -

( ), --

.

( D = 3,74).

-

. , . . -, , , -

5 . - 0,8 .

, .

1

(min–max), ( -

/ --

),

, ( -

/ ),

,

- 0,2−9,2 0,2−16,4

1−3/1−3 1−5/1−3

2−4 4−5

2,59/2,16 4,01/3,08

3,4 4,7

, -

, 45 56 63 80 125

9,0−98,1 6,7−180,3

32,8−268,1

20−40/30−40 60−90/70−90

110−140/100−110

30−60 70−90

130−160

42,92/30,56 82,76/59,01

155,48/114,8

43,8 80,6

140,2

. 1. : 1 –, 2 – , 3 – ,

4 – , 5 – , 6 – , 7 – -

, 8 – , 9 – ,

10 – ( ), 11 – , 12 –


. ., . ., .- ., . ., . .

2013, 5, 2 23

-,

1–3 1–5 . ,

, - 45, 56, 63, 80

125 . ---

«Mini-Magiscan». -. 1.

7, .

0,05 .

- ( . 2). , , , ,

. , ( ) , . ,

, , ,

, .

. , , 1–3 , 620 K 10 .

limP

limT ( ),T P .

limlim

exp .dT cP

= − (1)

c d , , -. 2, – . 2. 20–40

, : , – .

2 c d (1)

, c d 1–3 610,3383651 0,1390938854−

20–40 651,1730614,690 835, 4 30T P< ≤ < ≤

471,7423449,835 1030, 3,1 4T P< ≤ ≤ ≤

1,427623342,690 835, 4 30T P

−< ≤ < ≤

2,429618523,835 1030, 3,1 4T P

−< ≤ ≤ ≤

60–90 582,9068176 3,157251589−100–140 577,7477786 5,233353630−

, -, -

. , ,

. 2. ( ) ( -

) . : 1 – 1–3 ; 2 – 1–5; 3 – 20–40;

4 – 60–90; 5 – 100–140. — (1)


. « . . » 24

. , , , -

. -,

. , ,

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. 3 -. ,

, -.

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20–40 1–3 , -

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13,9 ( 20–40 ) 5,625 (1–3 ), -

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-, ,

« » – , -, . -

0,05 « » 31,148 10pr −= ⋅ . -, « »

. pn -. , 20–40 30 56 150pn ≈ , 1–

3 2 81,8 10pn ≈ ⋅ . - « »

[10–12]

( ) ( )31

1

3 Nu 3 exp ,2

pp b p

n p

dT q K ET T T Tdt T h T

ρλ

= − − + − − (2)

( ) ( )2 20

1exp , 0 1, 0 ,p

p

cdh K E T h hdt h T

ρλ

= − = = (3)

. 3. -

. – -: 1 – 20−−−−40 3,2–4,2 ; 2 – 20–40

19,4–28,8 ; 3 – 60–90 5,8–10,4 ; 4 – 1–3 1–1,2 ; 5 – 1–3 2–2,9 ;

6 – 1–3 3,5–4,7 ; 7 – 1–3 6,3–11,3 ; – -

20–40 ( 1 2) 1–3 ( 5 7),

–


. ., . ., .- ., . ., . .

2013, 5, 2 25

( )0/ , / , / , / ,p p n p nt t t T T T h h r E E RT= = = =

pT , 1T , bT , nT – « » , , , , Nu – , q – , -

, 2ρ , 3ρ – , h – , 1λ – -, 2c – , E – , K – -

, R – , 20 12 2 pt c rρ λ= – .

63,1 10E = ⋅ / [8, 9], -63 10q = ⋅ / [13].

(2), (3) , -, ,

[14]. , K -

1T , . -

, -ignt . -

-

( . 3).

ignt , K 1T

. 3. , ( )1K T,

. 3

ignt , 1T , K K , 2/

20–40 , 3,2–4,2 1050 990 935 900 865

390 1708 7851

26 053 99 508

391 1709 7850

26 029 99 490

113 952 33 251 9784 2425 773

20–40 , 19,4–28,8 875 840 800 770 740 710

115 372

1231 3311

10 000 30 196

116 371

1233 3311

10 001 30 187

382 432 121 743 42 541 20 294 7850 2752

1–3 , 2–2,9 720 690 670 650

1482 4576

11 212 25 707

1478 4573

11 225 25 715

37 545 15 628 7054 3213


. « . . » 26

3 1–3 , 6,3–11,3

655 635 620 615

1413 4072 9781

16 603

1417 4073 9778

16 608

38 474 17 250 8052 4875

4 Ka , Kb (4)

, ; , Ka Kb20–40; 3,2–4,2 16,1178315 240 5 10⋅ 24 201,24516

20–40; 19,4–28,8 158630520001, 100 ⋅ 18 890,19191

1–3; 2–2,9 191102057198, 100 ⋅ 20 311,21190

1–3; 6,3–11,3 151026538333, 100 ⋅ 15 690,52785

1exp K

KbK aT

= − . (4)

Ka KbK . 2 . 4.

(4) (2)

1exp K

Kp

b EaT RT

− + , (5)

. K -. 4. ,

( )1K T . ( . 3) -

(4) . , -ignt ( , -

- (4)), -

-.

( )1K T . -

5,625 13,9 . ,

( )1K T , -

( . . 3). (4) -

: 20–40 13,9 21 241,57995,Kb =

15942161303 00, 61Ka = , 1–3

. 4. -. – ( .

. 1), – (3). 1 – 20–40 3,2–4,2 , 2 – 20–40

19,4–28,8 , 3 – 1–3 2–2,9 , 4 – 1–3 6,3–11,3 , 5 – 20–40 13,9 , 6 – 1–3 5,625


. ., . ., .- ., . ., . .

2013, 5, 2 27

5,625 17 434,27789,Kb = 16,3939796530 00 1Ka ⋅= . , - ( .

. 4). . 4. , -

, . -

. (1) -

[15]

( ) ( )1 1 lim1 1

1 lim

,,

0,K T T T

K T PT T

≥=

<. (6)

( )1K T (4), -

1P . ,

1T , ( )lim 1T P , -.

. -, , -

1T . -.

( )1 1,T P . - (1). , (1) -

( )( )1 ign1000 / , logT t -

. ( )1K T . . 5 5

ignt , P, limT , K 1T , K

24 691 650 700 710 740 750 770 795 800 840 850 875

— 50 939 30 188 10 007 7708 3311 1237 1065 372 228 116

— 49 930 30 196 10 000 7632 3312 1231 1064 372 236 115

4 865 850 870 895 900 940 950 990

1000 1050

— 99 490 26 019 25 012 7850 5514 1709 1324 388

— 99 508 26 033 24 914 7851 5564 1708 1326 385


. « . . » 28

, - ( (6))

20–40 . ,

, --

. -

-.

-

(1). . 5. . 5 -

20–40 . , -, -

. -.

, -. ,

. . -

. , -, , .

, , , .

--

, 2–25 600–1100 K -

.

1. Allen, C.M. Energetic-nanoparticle enhanced combustion of liquid fuels in a rapid compression machine / C.M. Allen, T. Lee // Proc. 47th AIAA Aerospace Sciences Meeting Including The New Ho-rizons Forum and Aerospace Exposition. – 2009. – AIAA 2009-227.

2. Trunov, M.A. Ignition of aluminum powders under different experimental conditions / M.A. Trunov, M. Schoenitz, E.L. Dreizin // Propellants, Explosives, Pyrotechnics. – 2005. – Vol. 30. –P. 36–43.

3. Utilization of iron additives for advanced control of NOx emissions from stationary combustion sources / V.V. Lissianski, P.M. Maly, V.M. Zamansky, W.C. Gardiner // Ind. Eng. Chem. Res. – 2001. – Vol. 20, 15. – P. 3287–3293.

4. / . . , . . , . . . // . – 1996. – T. 32, 3. – C. 24–34.

5. Cashdollar, K.L. Explosion temperatures and pressures of metals and other elemental dust clouds / K.L. Cashdollar, I.A. Zlochower // Journal of Loss Prevention in the Process Industries. – 2007. – Vol. 20. – P. 337–348.

. 5. . (1, 3) –

, (2, 4) – - (5). 1, 2 – 30 , 3, 4 – 2 , 5 –


. ., . ., .- ., . ., . .

2013, 5, 2 29

6. Dynamics of dispersion and ignition of dust layers by a shock wave / V.M. Boiko, A.N. Papyrin, M. Wolinski , P. Wolanski // Progress in Astronautics and Aeronautics. – 1984. – Vol. 94. – P. 293–301.

7. Ignition of dust suspensions behind shock waves / A.A. Borisov, B.E. Gel’fand, E.I. Timofeev et al. // Progress in Astronautics and Aeronautics. – 1984. – Vol. 94. – P. 332–339.

8. Photoemission measurements of soot particles temperature at pyrolysis of ethylene / Y.A. Baranyshyn, L.I. Belaziorava, K.N. Kasparov, O.G. Penyazkov // Nonequilibrium Phenomena: Plasma, Combustion, Atmosphere. – Moscow: Torus Press Ltd, 2009. – P. 87–93.

9. Optical properties of gases at ultra high pressures / Yu.N. Ryabinin, N.N. Sobolev, A.M. Markevich, I.I. Tamm // Journal of Experimental and Theoretical Physics. – 1952. – Vol. 23, 5. – P. 564–575.

10. , . . / . . // . – 2004. – T. 40, 1. – . 75–77.

11. , . . / . . , . . // . – 2011. – . 47, 6. – . 98–100.

12. / . . , . . , . . . // - . – 2012. – . 85, 1. – . 139–144.

13. , . . / . . , . . // . – 2001. – . 37, 6.

– . 46–55. 14. /

. . , . . – .: , 1979. – 312 . 15. , . . / . . , . . // -

. – 2003. – . 39, 5. – . 65–68.

EXPERIMENTAL AND MATHEMATICAL SIMULATION OF AUTO-IGNITION OF IRON MICRO PARTICLES

V.V. Leschevich1, O.G. Penyazkov2, J.-C. Rostaing3, A.V. Fedorov4, A.V. Shulgin5

The experimental and numerical studies of autoignition and burning of iron microparticles in oxy-gen were introduced. Parameters of high-temperature gas environment were created by rapid compres-sion machine. Powders with a particle size 1–125 microns were studied at the oxygen pressure of 0,5–28 MPa and the temperature of 500–1100 K. Ignition delay time was measured by registration of radiation from the side wall of the combustion chamber and by measuring the pressure on the end wall. Critical auto-ignition conditions are determined according to size of the particles, oxygen temperature and pres-sure. For description of ignition mechanisms a punctual semiempirical simulator was introduced. The simulator gave the opportunity to describe the experimental data on dependence of ignition delay time of iron particles on the ambient temperature, taking into account correlations of critical ignition tempera-tures with pressure.

Keywords: iron micro particles, oxygen, rapid compression machine, ignition delay time, mathe-matical simulation.

1 Leschevich Vladimir Vladimirovich is Junior Researcher, A.V. Luikov Heat and Mass Transfer Institute of the National Academy of Sciences of Belarus. 2 Penyazkov Oleg Glebovich is Corresponding Member of the National Academy of Sciences of Belarus, director, A.V. Luikov Heat and Mass Transfer Institute of the National Academy of Sciences of Belarus. 3 Rostaing Jean-Christophe is L’Air Liquide, Centre de Recherche Claude-Delorme, France. 4 Fedorov Alexander Vladimirovich is Doctor of Physical and Mathematical Sciences, Head of Laboratory, S.A. Khristianovich Institute of Theoretical and Applied Mechanics of the Siberian Branch of the Russian Academy of Sciences. 5 Shulgin Alexey Valentinovich is Candidate of Physical and Mathematical Sciences, Senior Researcher, S.A. Khristianovich Institute of Theo-retical and Applied Mechanics of the Siberian Branch of the Russian Academy of Sciences. E-mail: [email protected]


. « . . » 30

References 1. Allen C.M., Lee T. Energetic-nanoparticle enhanced combustion of liquid fuels in a rapid com-

pression machine // Proc. 47th AIAA Aerospace Sciences Meeting Including The New Horizons Forum and Aerospace Exposition. 2009. AIAA 2009-227.

2. Trunov M.A., Schoenitz M., Dreizin E.L. Ignition of aluminum powders under different experi-mental conditions. Propellants, Explosives, Pyrotechnics. 2005. Vol. 30. pp. 36–43.

3. Lissianski V.V., Maly P.M., Zamansky V.M., Gardiner W.C. Utilization of iron additives for ad-vanced control of NOx emissions from stationary combustion sources. Ind. Eng. Chem. Res. 2001. Vol. 20, no. 15. pp. 3287–3293.

4. Zolotko A.N., Vovchuk Ya.I., Poletaev N.I., Frolko A.V., Al'tman I.S. Fizika goreniya i vzryva. 1996. Vol. 32, no. 3. pp. 24–34. (in Russ.).

5. Cashdollar K.L., Zlochower I.A. Explosion temperatures and pressures of metals and other elemen-tal dust clouds. Journal of Loss Prevention in the Process Industries. 2007. Vol. 20. pp. 337–348.

6. Boiko V.M., Papyrin A.N., Wolinski M., Wolanski P. Dynamics of dispersion and ignition of dust layers by a shock wave. Progress in Astronautics and Aeronautics. 1984. Vol. 94. pp. 293–301.

7. Borisov A.A., Gel’fand B.E., Timofeev E.I., Tsyganov S.A., Khomic S.V. Ignition of dust sus-pensions behind shock waves. Progress in Astronautics and Aeronautics. 1984. Vol. 94. pp. 332–339.

8. Baranyshyn Y.A., Belaziorava L.I., Kasparov K.N., Penyazkov O.G. Photoemission measure-ments of soot particles temperature at pyrolysis of ethylene. Nonequilibrium Phenomena: Plasma, Com-bustion, Atmosphere. Moscow: Torus Press Ltd, 2009. pp. 87–93.

9. Ryabinin Yu.N., Sobolev N.N., Markevich A.M., Tamm I.I. Optical properties of gases at ultra high pressures. Journal of Experimental and Theoretical Physics. 1952. Vol. 23, no. 5. pp. 564–575.

10. Al'tman I.S. Fizika goreniya i vzryva. 2004. Vol. 40, no. 1. pp. 75–77. 11. Fedorov A.V., Shul'gin A.V. Fizika goreniya i vzryva. 2011. Vol. 47, no. 6. pp. 98–100. 12. Leshchevich V.V., Penyaz'kov O.G., Fedorov A.V., Shul'gin A.V., Rosten Zh.-K. Inzhenerno-

fizicheskiy zhurnal. 2012. Vol. 85, no. 1. pp. 139–144. (in Russ.). 13. Bolobov V.I., Podlevskikh N.A. Fizika goreniya i vzryva. 2001. Vol. 37, no. 6. pp. 46–55. (in Russ.). 14. Kholla Dzh. (ed.), Uatta Dzh. (ed.) Sovremennye chislennye metody resheniya obyknovennykh

differentsial'nykh uravneniy (Contemporary numerical methods of solution of ordinary differential eqau-tion). Moscow: Mir, 1979. 312 p. (in Russ.). [Hall G., Watt J.M. Modern Numerical Methods for Ordi-nary Differential Equations. Oxford: Clarendon press. 312 p.]

15. Fedorov A.V., Kharlamova Yu.V. Fizika goreniya i vzryva. 2003. Vol. 39, no. 5. pp. 65–68. (in Russ.).

8 2013


2013, 5, 2 31

519.688, 519.615.5

,

. . 1

5-.

.

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. « . . » 32

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, 10- . -,

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. A ijA , TA – A , det A

– A . a b ×a b Ta b. a [ ]×a -

, [ ]× = ×a b a b b . I , 0– , ⋅ – .

2. 2.1 . -

jix , jiy , jiz iQ j - , 1, 2j = , 1, , 5i = ( . 1).

1 1 2 0j j jx y x= = = 1, 2j = . .

3 5× :

1 5

1 5

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j j

j j j

j j

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=A (1)

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1jH 2jH – , 1jx , 1jy 2jx . -:

1

1 1

2 2 21 1 1 1 1sign( )

j

j j

j j j j j

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y

z z x y z

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2

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, (2), , -. ,

20- (12) 10- (13). 2.2. . -

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π , O – ( P ). – O π , O π . -

, () .

- ( , , )j j jO π P 1, 2j = . -

[ ]1 =P I 0 , [ ]2 =P R t , SO(3)∈R –

[ ]T1 2 3t t t=t – , 1=t .


. . ,

2013, 5, 2 33

1 iO Q 2 iO Q ( . 1) 3 , -R t . [12]:

[ ]1

2 2 2 1

1

0i

i i i i

i

xx y z y

z=E , (3)

1, , 5i = , [ ]×=E t R . 2.3. .

. 1 ([10]). SO(3)∈R

2 kπ π+ , k ∈ , R :1

u uv vw w

−

× ×

= − +R I I , (4)

, ,u v w∈ . R (4) [ ]( , , , )u v w ×=E t t R – .

1.1 3 2

2 1 3

3 2 1

,

,

,

t vt wtu

t wt utv

t ut vtw

δ

δ

δ

− − +′ =

− − +′ =

− − +′ =

(5)

1 2 3ut vt wtδ = + + , ( , , , ) ( , , , )u v w u v w′ ′ ′ = −E t E t . . SO(3)′ = − ∈tR H R ,

T2= −tH I tt . [ ]×′ ′= = −E t R E . ′R (4) , ,u v w′ ′ ′ ,

, ,u v w′ ′ ′ :

' '' '' '

u u u uv v v vw w w w× × × ×

− + = − − +tI I H I I ,

, , (5). (3) t , -

: =S t 0 , (6)

i - 5 3× S [ ]2

T1 1 1 2

2

i

i i i i

i

xx y z y

z ×

R .

R (4) 3 3×S . :

4 3 2 2 3 4 3 2

2 3 2 2

[0] [0] [0] [0] [0] [1] [1]

[1] [1] [2] [2] [2] [3] [3] [4] 0,if u u v u v uv v u u v

uv v u uv v u v

= + + + + + +

+ + + + + + + + = (7)

1, ,10i = , [ ]n n w , [0] – . -, if , 4.


. « . . » 34

1. 3 3× S 3/iF Δ , 2 2 21 u v wΔ = + + + iF – 6- . , iF

i iF f= Δ , if -

iF . 2.4. . (7) :

=B m 0 , (8)

B – 10 35×T4 3 3 1u u v u w v w=m –

. (8) iuf , ivf 1, , 5i = , iwf

1, ,10i = . , : ′

′ =m

B 0m

, (9)

′B – 30 50×T4 3 3 2 2 3 4 5u w u vw u w v w vw w′ =m

– . , (9) (8). ′B –

( ). ( ):

3 2u w 3u w 3u 3 2v w 3v w 3v uv u v 11g 1 [3] [4] [4] [5]

2g 1 [3] [4] [4] [5]

3g 1 [3] [4] [4] [5]

4g 1 [3] [4] [4] [5]

5g 1 [3] [4] [4] [5]

6g 1 [3] [4] [4] [5] 28 .

1 6, ,g g :

1 21

2 32

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5 64

( )

1

g gh uvg gh u

w wg gh vg gh

≡ − = =C 0 , (10)

( )wC : [4] [5] [5] [6][4] [5] [5] [6]

( )[4] [5] [5] [6][4] [5] [5] [6]

w =C . (11)

2. ′B , « » – ′B .

24 ′B . -.

(10) , (8) , det ( ) 0w =C . det ( )W w= C . 20- -

w . 2. :


. . ,

2013, 5, 2 35

1010 10

0( )k k

kk

W p w w+ −

== + − , (12)

kp ∈ . . 1 1 0j jx y= = , 33 0E = . ,

232 1 1

13

R vw ut t tR uw v

+= =−

.

(5), 1w w−′ = − . , w1( )w−− ( ) . ,

10 101 10 10

1001

( )( ) ( )k ki i k

kiW p w w w w p w w− + −

=== − + = + −∏ .

1w w w−= − 10- : 10

0k

kW p w

== , (13)

kp 2

0( 1)

kk i i k

i

kw w

i−

== − . :

10 12

2

k ii k

i kip p

i kk'

=

+ −=

−, (14)

i k 10, mod 2 0i k− = . ,

0k = (14) 10

02 i

ip'

=.

2.5. . -, (4), – , -

. . , , [14]

[15] . -.

0w – W . 20 0 0 0/ 2 sign( ) ( / 2) 1w w w w= + +

W , 0| | 1w ≥ . , u - v --

0( )wC (11). , (4) R . R , t -

0 0 0( , , )u v wS (6). 1=t . T2= −tH I tt ′ = − tR H R . [12, 6],

: [ ]A =P R t , [ ]B = −P R t , [ ]C ′=P R t [ ]D ′= −P R t . - (3). , -

. , 2P , -

, , -, . , 1Q .


. « . . » 36

1 21 2 1 33 3

13 23,t tc c c R t

R R= − = − = + . (15)

, • 1 0c > 2 0c > , 2 A=P P ; • 1 0c < 2 0c < , 2 B=P P ; • 1 0c′ > 2 0c′ > , 2 C=P P ; • 2 D=P P .

1c′ 2c′ 1c 2c (15) R ′R . ,

12 11ini T2 22 21 2( )

1=

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.

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. . ,

2013, 5, 2 37

. 3 ( ) .

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10- W .

1. Kruppa, E. Zur Ermittlung eines Objektes aus zwei Perspektiven mit Innerer Orientierung / E. Kruppa // Sitz.-Ber. Akad. Wiss., Wien, Math. Naturw. Kl. Abt. IIa. – 1913. – Vol. 122. – P. 1939–1948.

2. Demazure, M. Sur Deux Problemes de Reconstruction / M. Demazure // INRIA. – 1988. – RR-0882. – P. 1–29.

3. Faugeras, O. Motion from Point Matches: Multiplicity of Solutions / O. Faugeras, S. Maybank // International Journal of Computer Vision. – 1990. – Vol. 4. – P. 225–246.

4. Heyden, A. Reconstruction from Calibrated Camera – A New Proof of the Kruppa-Demazure Theorem / A. Heyden, G. Sparr // Journal of Mathematical Imaging and Vision. – 1999. – Vol. 10. – P. 1–20.

5. Philip, J. A Non-Iterative Algorithm for Determining all Essential Matrices Corresponding to Five Point Pairs / J. Philip // Photogrammetric Record. – 1996. – Vol. 15. – P. 589–599.

6. Nistér, D. An Efficient Solution to the Five-Point Relative Pose Problem / D. Nistér // IEEE Transactions on Pattern Analysis and Machine Intelligence. – 2004. – Vol. 26. – P. 756–777.

7. Li, H. Five-Point Motion Estimation Made Easy / H. Li, R. Hartley // IEEE 18th Int. Conf. of Pat-tern Recognition. – 2006. – P. 630–633.

8. Kukelova, Z. Polynomial Eigenvalue Solutions to the 5-pt and 6-pt Relative Pose Problems / Z. Kukelova, M. Bujnak, T. Pajdla // Proceedings of the British Machine Conference. – 2008. – P. 56.1–56.10.

9. Stewénius, H. Recent Developments on Direct Relative Orientation / H. Stewénius, C. Engels, D. Nistér // ISPRS Journal of Photogrammetry and Remote Sensing. – 2006. – Vol. 60. – P. 284–294.

10. Cayley, A. Sur Quelques Propriétés des Déterminants Gauches / A. Cayley // J. Reine Angew. Math. – 1846. – Vol. 32. – P. 119–123.

. 3. ( ) ( ) -,


. « . . » 38

11. Faugeras, O. Three-Dimensional Computer Vision: A Geometric Viewpoint / O. Faugeras // MIT Press, 1993. – 663 p.

12. Hartley, R. Multiple View Geometry in Computer Vision. / R. Hartley, A. Zisserman. –Cambridge University Press, 2004. – 655 p.

13. Maybank, S. Theory of Reconstruction from Image Motion / S. Maybank // Springer-Verlag, 1993. – 261 p.

14. Hook, D.G. Using Sturm Sequences to Bracket Real Roots of Polynomial Equations / D.G. Hook, P.R. McAree // Graphic Gems I : . . . – Academic Press, 1990. – P. 416–422.

15. Numerical Recipes in C. Second Edition / W. Press, S. Teukolsky, W. Vetterling, B. Flannery. – Cambridge University Press, 1993. – 1020 p.

ALGORITHMIC SOLUTION OF THE FIVE-POINT POSE PROBLEMBASED ON THE CAYLEY REPRESENTATION OF ROTATION MATRICES

E.V. Martyushev1

A new algorithmic solution to the five-point relative pose problem is introduced. Our approach is not connected with or based on the famous cubic constraint on an essential matrix. Instead, we use the Cayley representation of rotation matrices in order to obtain a polynomial system of equations from epi-polar constraints. Solving that system, we directly obtain positional relationships and orientations of the cameras through the roots for a 10th degree polynomial.

Keywords: five-point pose problem, calibrated camera, epipolar constraints, Cayley representation.

References 1. Kruppa E. Zur Ermittlung eines Objektes aus zwei Perspektiven mit Innerer Orientierung. Sitz.-

Ber. Akad. Wiss., Wien, Math. Naturw. Kl. Abt. IIa. 1913. Vol. 122. pp. 1939–1948. 2. Demazure M. Sur Deux Problemes de Reconstruction. INRIA. 1988. no. RR-0882. pp. 1–29. 3. Faugeras O., Maybank S. Motion from Point Matches: Multiplicity of Solutions. International

Journal of Computer Vision. 1990. Vol. 4. pp. 225–246. 4. Heyden A., Sparr G. Reconstruction from Calibrated Camera – A New Proof of the Kruppa-

Demazure Theorem. Journal of Mathematical Imaging and Vision. 1999. Vol. 10. pp. 1–20. 5. Philip J. A Non-Iterative Algorithm for Determining all Essential Matrices Corresponding to Five

Point Pairs. Photogrammetric Record. 1996. Vol. 15. pp. 589–599. 6. Nistér D. An Efficient Solution to the Five-Point Relative Pose Problem. IEEE Transactions on

Pattern Analysis and Machine Intelligence. 2004. Vol. 26. pp. 756–777. 7. Li H., Hartley R. Five-Point Motion Estimation Made Easy. IEEE 18th Int. Conf. of Pattern Rec-

ognition. 2006. pp. 630–633. 8. Kukelova Z., Bujnak M., Pajdla T. Polynomial Eigenvalue Solutions to the 5-pt and 6-pt Relative

Pose Problems. Proceedings of the British Machine Conference. 2008. pp. 56.1–56.10. 9. Stewénius H., Engels C., Nistér D. Recent Developments on Direct Relative Orientation. ISPRS

Journal of Photogrammetry and Remote Sensing. 2006. Vol. 60. pp. 284–294. 10. Cayley A. Sur Quelques Propriétés des Déterminants Gauches. J. Reine Angew. Math. 1846.

Vol. 32. pp. 119–123. 11. Faugeras O. Three-Dimensional Computer Vision: A Geometric Viewpoint. MIT Press, 1993.

663 p. 12. Hartley R., Zisserman A. Multiple View Geometry in Computer Vision (Second Edition). Cam-

bridge University Press, 2004. 655 p. 13. Maybank S. Theory of Reconstruction from Image Motion. Springer-Verlag, 1993. 261 p. 14. Hook D.G., McAree P.R. Using Sturm Sequences To Bracket Real Roots of Polynomial Equa-

tions. Graphic Gems I (A. Glassner ed.). Academic Press, 1990. pp. 416–422. 15. Press W., Teukolsky S., Vetterling W., Flannery B. Numerical Recipes in C (Second Edition).

Cambridge University Press, 1993. 1020 p. 18 2013 .

1 Martyushev Evgeniy Vladimirovich is Cand.Sc. (Physics and Mathematics), Mathematical analysis department, South Ural State University. E-mail: [email protected]


2013, 5, 2 39

515.126

. . 1

– , . -

g:A →→→→ B -f : X →→→→ X. , -

, , gf : X →→→→ X \ .

: , , , .

. - [1]. ,

. . X ≈ Y , X Y . w(X) – -

X. ω; {0,1,2, }ω = . , -

. { : }na n ω∈a X∈ , ( ) { } { : }nS a a a n ω= ∪ ∈ .

X , a b X - :f X X→ , ( )f a b= . -

, 1- - . , . -

, -, , . , -

. [2].

1. X – , -. X

{ : }na n ω∈ { : }nb n ω∈ , limn na a→∞ = , limn nb b→∞ = ( ) ( )S a S b = ∅ . - U a -

V a :f X X→ , , : V U⊂ , ( )f a b= , ( )f V V = ∅ , Xf f id= , ( )f x x= x V∉ ( ( ))i ii a V b f V∀ ∈ ⇔ ∈ .

. X , :g X X→ , a b . - W

a , W U⊂ ( )g W W = ∅ . : { : , ( )}a i iJ i a W b g Wω= ∈ ∈ ∉ { : , ( )}b i iJ i a W b g Wω= ∈ ∉ ∈ .

aJ bJ – , .

ai J∈ - iQ ia ,

iQ W⊂ ( ) { }i iQ S a a= . , { : }i aQ i J∈, { : }na n ω∈ .

1 – - , , -

, - . E-mail: [email protected]


. « . . » 40

, ( )ig a ( )S b .

1 2a a aJ J J= ∪ , 1 { : ( ) ( )}a a iJ i J g a S b= ∈ ∉ 2 { : ( ) ( )}a a iJ i J g a S b= ∈ ∈ . : 1ai J∈ . - iD

ia , i iD Q⊂ ( ) ( )ig D S b = ∅ . ig g= . : 2ai J∈ , . . ( )i jg a b= j . –

, ( ) ( )ig Q S b – , i ic Q∈ , ( ) ( )ig c S b∉ . :i X Xψ → , ia ic . -

- i iD Q⊂ ia , ( )i i i iD D Qψ ⊂ , ( )i i iD Dψ = ∅ ( ( )) ( )i ig D S bψ = ∅ . :ig X X→ :

1

( ), ,

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

i i

i i i i

i i i

g x x D

g x g x x Dg x x X D D

ψ

ψ ψψ

−

∈

= ∈∈

, ig ( ) ( )i ig D S b = ∅ . *:g X X→ :

( ), ,* ( )

( ), \ { : }.i i a

i a

g x x D i Jg x

g x x X D i J∈ ∈

=∈ ∈

, *g – . , * \ { : }i aW W D i J= ∈ – -- a . * ( )g a b= ( ) ( ) ( ) * ( *)S b g W S b g W= .

, aJ = ∅ , *W W= *g g= . , bi J∈ - iE ib

, * ( *)iE g W⊂ , ( ) { }i iE S b b= , 1( ) ( )i iS a h E− = ∅ , { : }i bE i J∈

. :ih X X→ *g, ig g .

1* \ { ( ) : }i i bV W h E i J−= ∈ – - a . :h X X→ , :

1

1

( ), ( ) ,( )

* ( ), \ { ( ) : }.i

i i i b

i b

h x x h E i Jh x

g x x X h E i J

−

−

∈ ∈=

∈ ∈

:f X X→ :

1

( ), ,

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

h x x V

f x h x x h Vx x X V h V

−

∈

= ∈∈

, f – . 1 . 1. X – , -

. X { : }na n ω∈ { : }nb n ω∈ , limn na a→∞ = , limn nb b→∞ = ( ) ( )S a S b = ∅ .

:f X X→ , Xf f id= , ( )f a b= ( )n nf a b=n ω∈ .

. *{ : }nU n ω∈ a , -. 1 - 0U a

0 :g X X→ , , : *0 0U U⊂ , 0 ( )g a b= , 0 0 0( )g U U = ∅ , 0 0 Xg g id= ,

0 ( )g x x= 0x U∉ 0 0 0( ( ))i ii a U b g U∀ ∈ ⇔ ∈ . -


. .

2013, 5, 2 41

, 1n ≥ - nU a -

:ng X X→ , , : *1n n nU U U −⊂ , ( )ng a b= , ( )n n ng U U = ∅ , n n Xg g id= ,

( )ng x x= nx U∉ ( ( ))i n i n ni a U b g U∀ ∈ ⇔ ∈ . , -{ : }nU n ω∈ a , { ( ) : }n ng U n ω∈

b . n ω∈ { : }n i nJ i a Uω= ∈ ∉ . j ω∈ :j X Xχ → , ( )j j ja bχ = .

- jO ja . ( )j jOχ – - -

jb . , { : }jO j ω∈

{ ( ) : }j jO jχ ω∈ , lim { }j jO a→∞ = , lim ( ) { }j j jO bχ→∞ = , 1\j k kO U U +⊂

1 1( ) ( ) \ ( )j j k k k kO g U g Uχ + +⊂ , 1\j k ka U U +⊂ k ω∈ . n ω∈ nJ , { : }n n j nV U O j J= ∪ ∈

( ) { ( ) : }n n n j j nW g U O j Jχ= ∪ ∈ - , ( ) nS a V⊂ , ( ) nS b W⊂ , 1n nV V+ ⊂ ,

1n nW W+ ⊂ n nV W = ∅ . , ( )n n nf V W= :nf X X→ , -:

1

1

( ), ,

( ), ( ),

( ) ( ), ,

( ), ( ) ,

, .

n

j

n n

n n

n j j n

j j n

g x x U

g x x g U

f x x x O j J

x x O j J

x

χ

χ χ

−

−

∈

∈

= ∈ ∈

∈ ∈

, n n Xf f id= . :f X X→ ( ) lim ( )n nf x f x→∞= . , ( )f a b=

( )f b a= . , . , { : } { }nU n aω∈ ={ ( ) : } { }n ng U n bω∈ = , \ { , }x X a b∈ n , ( )n n nx U g U∉ .

1( ) ( )n nf x f x+ = ; lim ( )n nf x→∞ . , f – . 1 .

, \ { }a A a∈ , . dA

. n ω∈ -(n) n : (0)A A= ( 1) ( )( )n n dA A+ = . ,

n, ( 1)nA − ≠ ∅ , ( )nA = ∅ . ( )nA = ∅ n ω∈ , , .

1. 2. X – .

X . :g A B→ -

:f X X→ , Xf f id= . . – , –

. f . , . –

. n . : 1n = . (1) (1)A B= = ∅ , – .

, 1{ , , }kA a a= 1{ , , }kB b b= , ( )i ib g a= 1 i k≤ ≤ . , i :if X X→ , ( )i i if a b= . -


. « . . » 42

- iU jV ia jb . -, ( )i i if U V= . :f X X→ :

1

( ), ,

( ) ( ), ,

, \ { :1 }.i

i i

i

i i

f x x U i

f x f x x V i

x x X U V i k

−

∈

= ∈

∈ ∪ ≤ ≤, , n.

1n + . (1) (1) n. , (1) (1)( )g A B= . g

:h X X→ , Xh h id= , (1) (1)( )h B A= (1) (1)( )h A B= . (1)\A A { : }ma m ω∈ . (1)\ { ( ) : }mB B g a m ω= ∈ . -

1, , ( ) ( )m kg a h a≠ m k. (1)( \ )h A A B = ∅ (1)( \ )h B B A = ∅ . m ω∈

:m X Xχ → , ma ( )mg a . ( ) ( )A h A B h B, - mU ( )m m mV Uχ= -

ma ( )mg a , mU , mV , 1( ) ( )m mh U h U−= 1( ) ( )m mh V h V−= -

, (1) (1)A B . , (1)

(1) – , , { : }jmU j ω∈

(1)*a A∈ ⇔ { : }jmV j ω∈ (1)( *) ( *)g a h a B= ∈ ⇔ -

{ ( ) : }jmh V j ω∈ (1)* ( *)a h h a A= ∈ . -

f :

1

1 1

1 1

( ), ,

( ), ,

( ) ( ), ( ) ,

( ), ( ) ,( ), .

m

m m

m

m m

m m

x x U m

x x V m

f x h h x x h V m

h h x x h U mh x

χ

χ

χ

χ

−

− −

− −

∈

∈

= ∈

∈

. 2 . :g A B→: \f X X A→ .

2. X { : }nU n ω∈ { : }nV n ω∈ - , :

1) 0 1 nU U U⊃ ⊃ ⊃ ⊃ , 2) 0U V = ∅ , { : }nV V n ω= ∈ , 3) { : }nV n ω∈ X, 4) n ω∈ :n n ng U V→ .

{ : }nA U n ω= ∈ : \f X X A→ ,

0 0( ) ( )f A g A V= ⊂ . . : \f X X A→ :


. .

2013, 5, 2 43

0 01

1 11

1

0

( ), ,

( ), ( ) ,( )

( ), \ ( ) ,, .

n n n n

n n n n

g x x U

g g x x g U nf x

g x x V g U nx x U V

ω

ω

−+ +

−+

∈

∈ ∈=

∈ ∈∉

, 0 1\ { \ : }n nU A U U n ω+= ⊕ ∈ 1 1( ) ( \ )n n n nf V U U V+ += n ω∈ , -, f X \X A . -

1( )n ng U + 1\ ( )n n nV g U + - nV ( , X),

0U V - X, ng , -f . , 0 0( ) ( )f A g A V= ⊂ . 2 .

3. X a b.

: \ { }f X X a→ , ( )f a b= . . X , X -

{ : }nW n ω∈ , - . , { : }nW W n ω= ∈ - X.

, a W∉ 0b W∈ . a *{ : }nU n ω∈ , -- *

0W U = ∅ . X , 0 :g X X→ ,

a b . - 0U a , *0 0U U⊂

0 0 0 0( )V g U W= ⊂ . , , 1n ≥ :ng X X→

- nU a : *1n n nU U U−⊂

( )n n nV g U= - nW . , { : } { }nU n aω∈ = . - 3 2.

1. X . X – .

3. - -*U { : }iW i ω∈ , -

- . -*U U⊂ { : }iV W i ω⊂ ∈ , A U⊂ .

. , { : }iA a i ω= ∈ . i

i ib W∈ :ig X X→ , ia ib . -

iU ia , : *iU U⊂ ( )i i ig U W⊂ . { : }iU i ω∈ { : }iU i k≤ .

iU \ { : }i jU U j i< , , { : }iU i k≤

- . { : }iU U i k= ≤{ ( ) : }i iV g U i k= ≤ – . 3 .

4. X – , -. X -

. : \f X X A→ , ( )f A B= .

. , , - , A B .

{ : 1, }niW n i ω≥ ∈ . { : 1, }niW W n i ω= ≥ ∈ .


. « . . » 44

2 :g X X→ , ( )g A B= . – -, - 0U ,

: 0( )B g U⊂ , 0U W = ∅ , 0( )g U W = ∅ 0 0( )U g U = ∅ . 3 1n ≥ -

nU , nV :n n ng U V→ , , nA U⊂ , 1n nU U+ ⊂ { : }n niV W i ω⊂ ∈ . , { : }nA U n ω= ∈ ,

. 4 2. . 2

, 4 . 5. X – -

, . - Z * , -

* \X X Z, * . . Z, , -

, -Z. 4 : \f X X A→ , ( )f A B= .

\X A . 1* ( \ )X f X A−= , -.

1. van Douwen, E.K. A compact space with a measure that knows which sets are homeomorphic / E.K. van Douwen // Adv. in Math. – 1984. – Vol. 52. – Issue 1. – P. 1–33.

2. Engelking, R. General topology / R. Engelking. – Berlin: Heldermann Verlag, 1989. – 540 p.

ABOUT EXTENSION OF HOMEOMORPHISMS OVER ZERO-DIMENSIONAL HOMOGENEOUS SPACES

S.V. Medvedev1

Let X be a zero-dimensional homogeneous space satisfying the first axiom of countability. We prove the theorem about an extension of a homeomorphism g: A → B to a homeomorphism f: X → X, where A and B are countable disjoint compact subsets of the space X. If, additionally, X is a non-pseudocompact space, then the homeomorphism g is extendable to a homeomorphism f: X → X \ A.

Keywords: homogeneous space, homeomorphism, first axiom of countability, pseudocompact space.

References 1. van Douwen E.K. A compact space with a measure that knows which sets are homeomorphic.

Adv. in Math. 1984. Vol. 52. Issue 1. pp. 1–33. 2. Engelking R. General topology. Berlin: Heldermann Verlag, 1989. 540 p.

30 2013 .

1 Medvedev Sergey Vasiljevich is Cand.Sc. (Physics and Mathematics), Associate Professor, Mathematical and Functional Analysis Depart-ment, South Ural State University. E-mail: [email protected]


2013, 5, 2 45

519.633

. . 1, . . 2, . . 3

, ,

. .

: , , .

1. -

[1] [ ]11 12( , ) ( , ) ( , ) ( , ) ( , ), ( , ) [0, ] 0,t t

tv x t v x t w x t v x w x x t Tπ= Δ − Δ + Φ ⋅ + Φ ⋅ ∈ × , (1)

[ ]21 220 ( , ) ( ) ( , ) ( , ) ( , ), ( , ) [0, ] 0,t tv x t w x t v x w x x t Tβ π= + + Δ + Φ ⋅ + Φ ⋅ ∈ × , (2)

[ ](0, ) ( , ) (0, ) ( , ) 0, 0,v t v t w t w t t Tπ π= = = = ∈ , (3)

[ ]( , ) ( , ), ( , ) ( , ), ( , ) [0, ] ,0v x t x t w x t x t x t rϕ ψ π= = ∈ × − , (4)

( , ) ( , ), ( , ) ( , )t tv x s v x t s w x s w x t s= + = + [ ],0 , 0.s r r∈ − >

1 1 2 2: ( , ) ( , ), : ( , ) ( , )t ti i i iv x z x t w x z x tΦ ⋅ → Φ ⋅ → [ ]0,x π∈ , [ ]0,t T∈ -

[ ]( ,0 ; )C r R− R.

[ ]( ) ( ), ,0 ,u t h t t r= ∈ − (5) -

[ ]( ) ( ) ( ), 0,tLu t Mu t u f t t T= + Φ + ∈ , (6) . . . . [2, 3]. U, F –

, ( ) ( )tu s u t s= + [ ],0s r∈ − , :L U F→ , [ ]: ( ,0 ; )C r U FΦ − →

, { }ker 0L ≠ , M: domM F→ , U,

[ ]: 0,f T F→ . (6) , ,

. - (5), (6), -

. ,

- - , (6), . . [4–6]. ,

, -

1 – , - ,

« ». E-mail: [email protected] 2 – , , - -

. E-mail: [email protected] 3 – , , . E-mail: [email protected]


. « . . » 46

, , .

[2, 3, 5, 6] (1)–(4). , -

, ( 0, , 1, 2).ij i jΦ = =

, . , -

, [5, 6] . , , -

.

2. -

[ ]( , ) ( , ) ( , ), ( , ) [0, ] 0,tv x t v x t w x t x t Tπ= Δ − Δ ∈ × , (7)

[ ]0 ( , ) ( ) ( , ), ( , ) [0, ] 0,v x t w x t x t Tβ π= + + Δ ∈ × , (8)

[ ](0, ) ( , ) (0, ) ( , ) 0, 0,v t v t w t w t t Tπ π= = = = ∈ , (9) ( ,0) ( ), [0, ],v x x xϕ π= ∈ (10)

0β < , v, w – . , ( ),0w x -w . , [7], (7)–(10) .

( ),0v x , ( ),0w x - (7)–(10) -

. [ ]0,π / ,h Nπ=

, 0, , .nx nh n N= = [ ]0,T0,τ > ,mt mτ= 0, , .m M=

v, w ( , )n mx t mnv , m

nw . -

11 1 1 1

2 22 2 , 0, , 1,

m m m m m m m mn n n n n n n nv v v v v w w w m M

h hτ

++ − + −− − + − +

= − = − (11)

1 12

20 , 0, , ,m m m

m m n n nn n

w w wv w m Mh

β + −− += + + = (12)

1, , 1n N= − , 0 ( ), 0, , ,n nv x n Nϕ= =

0 0 0, 0, , .m m m mN Nv v w w m M= = = = =

(11), (12) ( ),m m mn n nξ ηΨ = ,

1 1 12

1 12

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

( , ) 2 ( , ) ( , ) ,

m n m n m n m n m n mn

n m n m n m

v x t v x t v x t v x t v x th

w x t w x t w x th

ξτ

+ + −

+ −

− − += − +

− ++

1 12

( , ) 2 ( , ) ( , )( , ) ( , ) ,m n m n m n mn n m n m

w x t w x t w x tv x t w x th

η β + −− += + +

( , )n mv x t , ( , )n mw x t v, w (7)–(10) -. , 1 2p phτ + ,


. ., . ., . .

2013, 5, 2 47

C, τ h, 1 22 ( )p pm

n RC hτΨ ≤ + ,

1, , 1,n N= − 0, , 1m M= − . 1. v, w (7)–(10) , v -

t, v, w x. - (11), (12) 2.hτ +

. ( ),v x t , ( ),w x t

2 21 1 1( ), .2 12 12

m mn tt xxxx xxxx n xxxxv h v w h wξ τ η= − + + = −

. 1. 0β < . (11), (12) , 2.hτ ≤

. mqρ , m

qσ q- m- ,

( , ) , ( , ) , 0, 1, 2,n niqx iqxm mn m q n m qv x t e w x t e qρ σ= = = ± ±

(11), (12) 1

2 (2 (cos 1) 2 (cos 1))m m m mq q q qqh qh

hτρ ρ ρ σ+ − = − − − , (13)

2

20 (cos 1)

mqm m

q q qhh

σρ βσ= + + − .

22 (1 cos )

mqm

qqh

h

ρσ

β=

− − (14)

(13),

21

2 221 (1 cos ) 1 .

2(1 cos )m mq q

hqhh qh hτρ ρ

β+ = − − −

− − (15)

, 0β < , 2

2 221 (1 cos ) 1 1

2(1 cos )qhr qh

h qh hτ

β≡ − − − ≤

− −q N∈ . , 1qr ≥ − q N∈ . ,

2

2 2(1 cos ) 1 1.2(1 cos )

hqhh qh hτ

β− − ≤

− −2hτ ≤ .

(14) (15) 2

1 2 21

2 2

2

2

21 (1 cos ) 12(1 cos )

2 2(1 cos ) (1 cos )

2 11 (1 cos ) 1 .2 (1 cos )

mqm

qmq

mq

hqhh qh h

qh qhh h

qhh qh

h

τρρ β

σβ β

τσβ

++

− − −− −

= = =− − − −

= − − −− −


. « . . » 48

qr -2hτ ≤ .

1. 1 , 0β ≥ -

. , -

[8], 1 1. 2. 0β < (11), (12) ,τ h , 2hτ ≤ . -

(7) – (10) 2hτ + .

3. - (1)–(4).

[ ]0,π /h Nπ= , 0, ,nx nh n N= = .

[ ],r T− . , /T r – -, / /T M r Lτ = = , ,L M N∈ ,

, , ,0, , .mt m m L Mτ= = − ,m mn nv w v, w

nx kt :

( ){ } ( ){ }, , : , 0, , , 0, , .m m m mn n n n

kv w v w k L m k n N k M= − ≤ ≤ = =

,

( ){ } ( ) [ ] [ ]: , , ,0 ,0m m k kn n n n

kI v w g h Q r Q r→ ∈ − × − . 0, , ,n N= [ ],0Q r− – -

[ ],0r− -

. : [ ] [ ],0,0

sup ( )Q rs r

g g s−∈ −

= [ ],0 .g Q r∈ − ,

ijΦ [ ],0Q r− . [4, 5], , -

pτ v, w, C1, C2,0, , ,n N= 0, ,k M= [ ],k kt t r t∈ −

1 2( ) ( , ) max ( , ) ,k m pn n n n m

k L m kg t v x t C v v x t C τ

− ≤ ≤− ≤ − +

1 2( ) ( , ) max ( , ) .k m pn n n n m

k L m kh t w x t C w w x t C τ

− ≤ ≤− ≤ − +

11 1 1 1

11 122 22 2 ,

m m m m m m m mm mn n n n n n n nn n

v v v v v w w w g hh hτ

++ − + −− − + − +

= − + Φ + Φ (16)

1 121 222

20 ,m m m

m m m mn n nn n n n

w w wv w g hh

β + −− += + + + Φ + Φ (17)

1, , 1,n N= − 0, , 1m M= − (16) 0, ,m M= (17), 0 0( ,0), ( ,0), 0, , ,n n n nv x w x n Nϕ ψ= = = (18)

[ ]0 0( ) ( , ), ( ) ( , ), 0, , , ,0n n n ng t x t h t x t n N t rϕ ψ= = = ∈ − , (19)

0 0 0, 0, , .m m m mN Nv v w w m M= = = = = (20)

[2, 3] (1) – (4) ϕ ψ

21 220 ( ,0) ( ) ( ,0) ( , ) ( , ), (0, ).x x x x xϕ β ψ ϕ ψ π= + + Δ + Φ ⋅ + Φ ⋅ ∈


. ., . ., . .

2013, 5, 2 49

, . (16), (17) ( , )m m m

n n nψ ξ η= ,

1 1 12

1 111 122

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

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

m n m n m n m n m n mn

t tn m n m n mn n

v x t v x t v x t v x t v x th

w x t w x t w x t v x w xh

ξτ

+ + −

+ −

− − += − +

− ++ − Φ ⋅ − Φ ⋅

1 121 222

( , ) 2 ( , ) ( , )( , ) ( , ) ( , ) ( , ).m mt tm n m n m n mn n m n m n n

w x t w x t w x tv x t w x t v x w xh

η β + −− += + + + Φ ⋅ + Φ ⋅

, 4 [5], 1 (11) , (12), 1 3 [5] .

3. 0β < , 2hτ ≤ , [ ]2 2: ,0LI R Q r+ → −0pτ , m

nψ 1 2p phτ + .

(1)–(4) { }0 1 2min ,p p phτ + . , ijΦ

, .

4. ( 1)ij ijg a gΦ = − [ ]1,0 ,g Q∈ − i, j = 1, 2. , (1)–(4)

[ ]11 12( , ) ( , ) ( , ) ( , 1) ( , 1), ( , ) [0, ] 0,tv x t v x t w x t a v x t a w x t x t Tπ= Δ − Δ + − + − ∈ × , (21)

[ ]21 220 ( , ) ( ) ( , ) ( , 1) ( , 1), ( , ) [0, ] 0,v x t w x t a v x t a w x t x t Tβ π= + + Δ + − + − ∈ × , (22) (3), (4). (16), (17)

, , - -1

1 1 1 111 122 2

2 2 ,m m m m m m m m

m L m Ln n n n n n n nn n

v v v v v w w w a v a wh hτ

+− −+ − + −− − + − +

= − + + (23)

1 121 222

20 ,m m m

m m m L m Ln n nn n n n

w w wv w a v a wh

β − −+ −− += + + + + (24)

1, , 1, 0, , 1n N m M= − = − (23) 0, ,m M= (24), (18), (19) (20). -

τ , - – 2τ [9]. 1, .

2. v, w (21), (22), (3), (4) , vt, v, w

x. (23), (24) 2hτ + . 3

1. 0β < , 2hτ ≤ , - - .

(1)–(4) 2hτ + . . 1 (v,w) β = –0,75, T = 10, M = 1000, N = 16

c ( ,0) sin , [0, ]v x x x π= ∈ (7)–(10).


. « . . » 50

(1)–(4) 11 1a = , 12 21 22 0a a a= = = , β = –0,75, T= 10, M = 1000, N = 16, [ ]( , ) ( 1)sin , ( , ) [0, ] 1,0v x t t x x t π= + ∈ × − ( , )wν

. 2.

1. , . . / . . , . . // . . . – 2001. – . 42, 3. – . 651–669.

2. Fedorov, V.E. On solvability of some classes of Sobolev type equations with delay / V.E. Fedorov, E.A. Omelchenko // Functional Differential Equations. – 2011. – Vol. 18, 3–4. – P. 187–199.

3. , . . / . . , . . // . . . – 2012. – . 53, 2. – . 418–429.

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

. – 2009. – 5. – . 62–67. 5. , . .

/ . . // . - . – 2010. – . 16, 5. – C. 151–158.

6. , . . - / . . , . . // . - . –

2011. – . 17, 1. – C. 178–189. 7. , . . -

/ . . , . . // . . . . . : . – 2004. – . 161–172.

8. , . / . , . . . – M.: , 1972. – 420 .

9. , . . i- - / . . , . . . – : , 2004.– 256 .

. 1

. 2


. ., . ., . .

2013, 5, 2 51

NUMERICAL SOLUTION OF DELAYED LINEARIZED QUASISTATIONARY PHASE-FIELD SYSTEM OF EQUATIONS

E.A. Omelchenko1, M.V. Plekhanova2, P.N. Davydov3

For delayed linearized quasistationary phase-field system of equations the numerical method of so-lution was proposed. The convergence of explicit difference scheme that takes account of delay in the system under investigation was thoroughly studied. On the basis of the results obtained the implementa-tion of the method was realized.

Keywords: Sobolev type equation, quasistationary phase-field system of equations, difference scheme.

References 1. Plotnikov P.I., Klepacheva A.V. Siberian Mathematical Journal. 2001. Vol. 42, no. 3. pp. 551–

567. 2. Fedorov V.E., Omelchenko E.A. On solvability of some classes of Sobolev type equations with

delay. Functional Differential Equations. 2011. Vol. 18, no. 3–4. pp. 187–199. 3. Fedorov V.E., Omel’chenko E.A. Siberian Mathematical Journal. 2012. Vol. 53, no. 2. pp. 335–

344. 4. Lekomtsev A.V., Pimenov V.G. Russian Mathematics (Izvestiya VUZ. Matematika). 2009.

Vol. 53, no. 5. pp. 54–58. 5. Pimenov V.G. Trudy Inst. Mat. i Mekh. UrO RAN. 2010. Vol. 16, no. 5. pp. 151–158. (in Russ.). 6. Pimenov V.G., Lozhnikov A.B. Proceedings of the Steklov Institute of Mathematics (Supplemen-

tary issues). 2011. Vol. 275. Suppl. 1. pp. 137–148. 7. Fedorov V.E., Urazaeva A.V. Trudy Voronezhskoy zimney matematicheskoy shkoly (Proc. of the

Voronezh Winter Mathematical School). Voronezh: VGU. 2004. pp. 161–172. 8. Richtmyer R.D., Morton K.W. Difference Methods for Initial-Value Problems. Interscience, New

York, 1967. 405 p. 9. Kim A.V., Pimenov V.G. i-Gladkiy analiz i chislennye metody resheniya funktsional'no-

differentsial'nykh uravneniy (i-differential analysis and numerical methods of solutions of functional and differential equations). Izhevsk: RKhD, 2004. 256 p. (in Russ.).

15 2013 .

1 Omelchenko Ekaterina Aleksandrovna is Senior Lecturer, Department of Humanities and socio-economic disciplines, Ural Branch of Russian Academy of Justice. E-mail: [email protected] 2 Plekhanova Marina Vasilyevna is Associate Professor, Differential and Stochastic Equations Department, South Ural State University. E-mail: [email protected] 3 Davydov Pavel Nikolaevich is Post-graduate student, Mathematical Analysis Department, Chelyabinsk State University. E-mail: [email protected]


. « . . » 52

517.977

1

. . 2, . . 3

--

. , -

. -, -

. : ,

, , .

( ., , [1–5]), , -

, , -. ,

[1]. -

[1], -. [6, 7]

, , - ( ), -

. -, -

. , -

, . -, -

, , , .

( ) [1] ,t xxu ku qu= + const,k = const .q = (1)

3,t xxu ku qu uα= + − const .α = (2) ,

. (1) (2) [6] xu Q= ,

xt xxx xQ kQ qQ= + , (3) 3

xt xxx x xQ kQ qQ Qα= + − . (4)

1 ( 12- -1-1001). 2 – - , , . E-mail: [email protected] 3 – - , , ,



. ., . .

2013, 5, 2 53

, [6, 7], Q: ( ( , ))Q Q x tψ= , ( , ) constx tψ = – -

( , )Q x t . (3) (4) ''' 3 '' '( 3 ) ( ) 0x x t x xx xt xxx xkQ Q k Q k qψ ψ ψ ψ ψ ψ ψ ψ+ − + + − + + = , (5)

''' '3 3 '' '( ) ( 3 ) ( ) 0x x t x xx xt xxx xkQ Q Q k Q k qα ψ ψ ψ ψ ψ ψ ψ ψ− + − + + − + + = . (6) ( )' ψ . (5) (6), -

, ( , )x tψ , , 0xψ ≠ [6, 7]

123 ( ),t xx

x

k fψ ψ ψψ

− += 23 ( )xt xxx x

x

k q fψ ψ ψ ψψ

− + += . (7)

1( ),f ψ 2 ( )f ψ – . , (7) . 2

1( ) /(3 ).xx t xf kψ ψ ψ= + -x xxxψ -

. xtψ , 2

1 ,3

t xxx

fk

ψ ψψ += 3

3 13 12 3xt x x x tq f f

kψ ψ ψ ψ ψ= + + , ' 2

3 1 1 21 1 32 3 2

f f f fk

= + − . (8)

(8) , ( xxt xtxψ ψ= ). ,

2 2 41 3 4

3 1 3 ,2 3tt t t t x xq f f f

kψ ψ ψ ψ ψ ψ= + + + '

4 1 3 32 3f f f kf= + . (9)

xtψ , . , xtt ttxψ ψ= . -

, 2 2 ' 4

4 3 4 1 46 6 3 ( / ) 0t x t xf f k f f f kψ ψ ψ ψ+ + + = . (10) , 3 0f = , (10) . -

: 1. (7) , ( , )x tψ -

(8), (9) ' 22 1 1/ 3 2 /(9 )f f f k= + , 1( )f ψ – .

(5), (6) ''' '' ' ' 2

1 1 1[ / 3 2 /(9 )] 0kQ Q f Q f f k+ + + = ; ''' '3 '' ' ' 21 1 1( ) [ / 3 2 /(9 )] 0kQ Q Q f Q f f kα− + + + = .

3 0f ≠ , , (10) - (8), (9), , 1( )x gψ ψ= , 2 ( ).t gψ ψ= -

, 2 1( ) ( ), const .g Cg Cψ ψ= = -

2. 3 0f ≠ , ( ),ax btψ ψ= + consta = , constb = . , ' ( ) ( )x xu Q Q ψ ψ ψ= = , ( )u u ax bt= + (1) (2)

2 0,z zzbu ka u qu− + + = 2 3 0,z zzbu ka u qu uα− + + − = .z ax bt= +

, 1 constf = 22 12 /(9 ) constf f k= = . (10) -

. (9) tψ , , x – . , 0tψ ≠ , (9) 1(1/ ) 1,5 (1/ ) /(3 )t t tq f kψ ψ+ = . -, 11,5 ( )exp(3 / 2) /[1 ( )exp(3 ) /(3 )]t qC x qt f C x qt kψ = − . (8)

xψ . ,


. « . . » 54

1exp(3 / 2) /[1 ( )exp(3 / 2) /(3 )].x xC qt f C x qt kψ = − (8), -

( )C x : /(2 )xxC qC k= . 1 2( ) exp[ /(2 )] exp[ /(2 )].C x A x q k A x q k= ± +

1 const,A = 2 constA = . , 1 1A = , 2 0A = - tψ xψ

1

3 exp[(3 / 2) /(2 )]/ 2,

1 exp[(3 / 2) /(2 )]/(3 )tq qt x q kf qt x q k k

ψ ±=

− ±

1

/(2 ) exp[(3 / 2) /(2 )]1 exp[(3 / 2) /(2 )]/(3 )x

q k qt x q kf qt x q k k

ψ ± ±− ±

.

xψ. , ,

1 13 ln{1 exp[(3 / 2) /(2 )]/(3 )}/k f qt x q k k fψ = − − ± . (11)

2 0f = . 1( ) 3 /(2 )of kψ ψ ψ= + , constoψ = ( . 1). , ( , )x tψ (8), (9).

(9) (8) , x – , , 2 [ ( ) 3 2 / 2],t o oM x qψ ψ ψ ψ ψ= + + + ( ) 2 exp(3 / 2)x oN x qtψ ψ ψ= + .

(8), 22

22 3exp ,

2 2 2 3 2ok q q Mqt x

k k kqψψ = ± ± − − const,M = constoψ = . (12)

, . 2 0f = , constt xxk q Aψ ψ ψ− − = = . ,

2 24 /(9 )o M qψ = , ( , )x tψ (2) (1) 0A = . , 1. 2 2 2( , ) (2 / ){ /(2 ) exp[3 / 2 /(2 )] /(3 )} / 2ou x t k q q k qt x q k M k ψ= ± ± − −

t xxu ku qu C= + + , constM = , constoψ = , ( , , )oC C q M ψ= , (2).

2. 2 2 2( , ) (2 / ){ /(2 ) exp[3 / 2 /(2 )] /(3 )} / 2ou x t k q q k qt x q k M k ψ= ± ± − − (1) (2),

2 24 /(9 )o M qψ = . , '2

2 ( )f Qα ψ= , 1 (7) -,

' 2 '21 1 2/ 3 2 /(9 )f f k f Qα+ = = . (13)

, (6) ''' ''1 0kQ Q f+ = . -

1f (13), '''' '' '''2 '' 2 '25 / 3 3 / 0Q Q Q Q Q kα− + = .

, '02 / (3 )Q k α ψ ψ= + , 0 constψ = . -

(13) , 1 06 /(3 )f k ψ ψ= − . , (8), (9). ,

301 2 3exp ,

3 3 2 2 3qqt x c

q kψψ = ± − − const .c = (14)

( )Q ψ ( 1 constf = ) ''' '' ' ' 2

1 1 1[ / 3 2 /(9 )] 0kQ Q f Q f f k+ + + = ; ''' '3 '' ' ' 21 1 1( ) [ / 3 2 /(9 )] 0kQ Q Q f Q f f kα− + + + = .

( )Q ψ , ,


. ., . .

2013, 5, 2 55

'1 1 2 1exp[ /(3 )] exp[ 2 /(3 )]Q C f k C f kψ ψ= − + − , 1 const,C = 2 constC = .

, (1) 'x xu Q Qψ= = ,

1

/(2 ) exp[3 / 2 /(2 )]1 exp[3 / 2 /(2 )]/(3 )

q k qt x q ku

f qt x q k k± ±

=− ± 1 1 2 1{ exp[ /(3 )] exp[ 2 /(3 )]}.C f k C f kψ ψ− + −

ψ (11),

11 2

3 3exp 1 exp .2 2 2 3 2 2

fq q q q qu t x C C t xk k k k

= ± ± + − ±

( )Q ψ . , ' ( )Q p ψ= , , ' ( )p r p= . 3 2

1 12 /(9 )pkrr f r p f p kα+ = − . 2r ap bp= + , const,a = constb = . ,

/(2 )a kα= ± , 1 /(3 )b f k= . ,

' 1

1

/(3 ) ,/(2 ) exp[ /(3 )]

f kQk C f kα ψ

= −± −

const .C =

, (2) 'x xu Q Qψ= = , -

(2)

1

0 1

[ /(3 )] / exp[3 / 2 /(2 )],

1 [ /(3 )]exp[3 / 2 /(2 )]f k q qt x q k

uC f k qt x q k

α± ±=

− − ± 0 .2

C Ck

α= ± (15)

, -

1

0 1

[ /(3 )] / exp[3 / 2 /(2 )],

1 [ /(3 )]exp[3 / 2 /(2 )]f k q qt x q k

uC f k qt x q k

α +=

− − ±1

0 1

[ /(3 )] / exp[3 / 2 /(2 )],

1 [ /(3 )]exp[3 / 2 /(2 )]f k q qt x q k

uC f k qt x q k

α− +=

− − ±

1

0 1

[ /(3 )] / exp[3 / 2 /(2 )],

1 [ /(3 )]exp[3 / 2 /(2 )]f k q qt x q k

uC f k qt x q k

α −=

− − ±1

0 1

[ /(3 )] / exp[3 / 2 /(2 )].

1 [ /(3 )]exp[3 / 2 /(2 )]f k q qt x q k

uC f k qt x q k

α− −=

− − ± (16)

, 2 0f = (5) ''' ''3 /(2 ) 0.oQ Q ψ ψ+ + = -, , '

1 2 / 2 oQ C C ψ ψ= − + , 1 const,C = 2 constC = . -, (12),

1 22 3 3 3exp exp exp .

2 2 2 2 2 3 2 2k q q q M qu C qt x qt x C qt x

q k k k k k= ± ± ± − − ±

2 0f = , (6) ''' '3 ''( / ) 3 /(2 ) 0.oQ Q k Qα ψ ψ− + + = -: ' 2 / [1/(2 )]oQ k α ψ ψ= ± + .

(6) 1

2 3 3exp exp .2 2 2 2 2 2

q k q q q Mu qt x qt xk k k k kα

−

= ± ± ± − (17)

(14), (2) 1/ exp[3 / 2 /(2 )]{exp[3 / 2 /(2 )] } ,u q qt x q k qt x q k cα −= ± ± ± − const .c = (18)

u( , , ).

, (1) (2) constt = . , (2) ( , )u x t

( , )t u x [8].


. « . . » 56

2 2 2 3 3( 2 ) ( ) 0u u xx x u xu x uu ut k t t t t t t t qu u tα+ − + − − = . (19) (19) ( ) ,u xϕ ξ− = x η= :

2 2 2 2[ ( 2 ) 2 ( )( ) ( ) ]x x xx x x xt k t t t t t t t t t t t t tξ ξ ηη ξη ξξ ξ ξ η ξ ξη ξξ ξξ η ξϕ ϕ ϕ ϕ ϕ ϕ+ − + − − − − + −3 3[ ( ) ( ) ] 0.q tξξ ϕ α ξ ϕ− + − + =

, constξ = – (19), -tξξ . , 0tη = . ,

constξ = constt = . , (19) , 2 3 3 3[ ( ) ( ) ] 0.xxt kt q tξ ξ ξϕ ξ ϕ α ξ ϕ− − + − + =

, , 0α = ( (1)), uξ ϕ+ = , , -constt =

0 0 1( , ) { ( ) ( )sin[ ( ( )) / ]}/ ,u t x w t c t x c t q k q= + ± +

0 ( ) const,c t = 0 ( ) const,w t = 1( ) const .c t = (20) (2). , ,

( , )x u t – 2 3 3( ) 0t u uu ux x kx qu u xα− + − = . (21)

0α ≠ , constt = , -

1 4 20 0/(4 ) /(4 )

dux cu k qu k w u cα

= ±− + +

. (22)

, , 1 1 2 2 0 0( ), ( ), ( )c c t c c t w w t= = = , (21),

1 1 2 2 0 0( ), ( ), ( )c c t c c t w w t= = = . -( , )u x t , (22):

4 2 2 2 20 0/(4 ) /(4 ) [ /(4 )][ 2 ( ) ( / 2 ( ) ( ))][ 2 ( )u k qu k w u c k u a t u q a t b t u a t uα α α− + + = + − − − −

2( / 2 ( ) ( ))]q a t b tα− − + (22) -

, [9]. -a constt = ( , ) ( )u x t x=℘ ( ( )x℘ – -

). [10], , -, , , (20)

(1).

(2) , , -, , (

(15)–(18) ) . . 1 (1:

t = 0; 2: t = 0,5; 3: t = 1; 4: t = 1,5; 5: t = 2; 6: t = 2,5). . 2 (1: t = 0; 2: t = 0,1; 3: t = 0,5; 4: t = 1; 5: t = 1,5).

(15), (17), (18) , , , –

( . . 3, ( , )u x tc 1t = (1: c = 100; 2: c = 1000; 3: c = 2500;

4: c = 4000; 5: c = 7000).

{ 0, 0}x t= = . , (18) [ / (0,0)]/ (0,0).c q u uα= ± − (0,0) / ,u q α= 0c =


. ., . .

2013, 5, 2 57

( , ) const /u x t q α= = . , ( . 4, ( , )u x t

c , 1: c = 1600; 2: c = 2800; 3: c = 4900). , (15)

, , (15) ( . (16)). (0,0)u

( . 1, . 2).

0,00 2,00 4,00 6,00 8,00 x

0,00

0,04

0,08

0,12

u 6

5

4

3 2 1

. 1.

1 2 3 4 5

30,0

u

20,0

10,0

0,0

–4,00 –2,00 0,00 2,00 x

. 2.

0,00 4,00 8,00 x

0,00

4,00

8,00

u

12,00

1

2 3

4

5

. 3.

0,0

100,0

200,0

u

300,0

1

2

3

7,0 8,0 9,0 x 10,0

. 4.


. « . . » 58

. 5 q . - (1: q = 1; 2: q = 2; 3: q = 3; 4: q = 4; 5: q = 5). α , ( . . 6; 1:

α = 1; 2:α = 40; 3: α = 200; 4: α = 500; 5: α = 1500).

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

. . . – .: , 1996. – 111 . 2. / -

. ., . . . – .: , 1987. – 477 . 3. Vazquez, J.L. A Stability Technique for Evolution Partial Differential Equations. A Dynamical

System Approach / J.L. Vazquez, V. – Birkhauser Verlag, 2004. – 377 p. 4. , . . / . .

// . . – 2005. – 2(36). – . 32–64. 5. , . . div( grad )tu u u uσ β= + /

. . , . . // . -. : , 100-

. – , 2008. – . 512. 6. , . . -

/ . . , . . // . : , 2010. – . 16, 2. – C. 209–225.

7. , . . / . . , . . // -. : , 2008. – .14, 1. –

C. 130–145. 8. , . .

/ . . // . – 2005. – . 69. – . 5. – . 829–836. 9. , . . . /

. . ; . . . , . . . – .: , 1967. – 88 . 10. , . . , , / . . ,

. . . – .: , 1962. – 1100 .

0,00 2,00 4,00 6,00 x 8,00

0,12

u

0,08

0,04

0,00 1

2

4

5

3

. 5. q10,00 12,00 14,00 16,00 x

0,3

u

0,2

0,1

0,0

1 2

4

5

3

. 6. α


. ., . .

2013, 5, 2 59

TOWARDS THE DIFFERENCES IN BEHAVIOUR OF SOLUTIONS OF LINEAR AND NON-LINEAR HEAT-CONDUCTION EQUATIONS

L.I. Rubina1, O.N. Ul’yanov2

The linear and non-linear heat-conduction equations are analyzed by the previously initiated geo-metrical method of analyzing linear and non-linear equations in partial derivatives. The reason of the difference in behavior of solutions of equations under consideration was stated, as well as the reason of aggravation of the non-linear equation. A class of solutions of linear equations that represents the sur-faces of the levels of non-linear heat-conduction equations was excluded.

Keywords: non-linear equations in partial derivatives, heat-conduction equations, exact solutions, surfaces of the level.

References 1. Kurdyumov S.P., Malinetskiy G.G., Potapov A.B. Nestatsionarnye struktury, dinamicheskiy

khaos, kletochnye avtomaty. Novoe v sinergetike. Zagadki mira neravnovesnykh struktur (Unsteady structures, dynamic chaos, cellular automata. New in synergetics. Mysteries of the world of nonequilib-rium structure). Moscow: Nauka, 1996. 111 p. (in Russ.).

2. Samarskiy A.A., Galaktionov V.A., Kurdyumov S.P., Mikhaylov A.P. Rezhimy s obostreniem v zadachakh dlya kvazilineynykh parabolicheskikh uravneniy (Blow-up regimes in problems for quasilin-ear parabolic equations). Moscow: Nauka, 1987. 477 p. (in Russ.).

3. Vazquez J.L., Galaktionov V. A Stability Technique for Evolution Partial Differential Equations. A Dynamical System Approach. Birkhauser Verlag, 2004. 377 p. (ISBN: 0-8176-4146-7)

4. Berkovich L.M. Vestnik SamGU – Estestvennonauchnaya seriya. 2005. no. 2(36). pp. 32–64. (in Russ.).

5. Kurkina E.S., Nikol'skiy I.M. O rezhimakh s obostreniem v uravneniyakh div( grad )tu u u uσ β= + (About blow-up regimes in equations div( grad )tu u u uσ β= + ). Different-

sial'nye uravneniya. Funktsional'nye prostranstva. Teoriya priblizheniy: Tezisy dokladov mezhdunarod-noy konferentsii, posvyashchennoy 100-letiyu so dnya rozhdeniya Sergeya L'vovicha Soboleva. (Ab-stracts of the International Conference dedicated to the 100th anniversary of the birth of Sobolev “Dif-ferential Equations. Functional Space. Theory of Approximation”) Novosibirsk, 2008. p. 512.

6. Rubina L.I., Ul’ianov O.N. Trudy Inst. Mat. i Mekh. UrO RAN. 2010. Vol. 16, no. 2. pp. 209–225. (in Russ.).

7. Rubina L.I., Ul’yanov O.N. Proceedings of the Steklov Institute of Mathematics (Supplementary issues). 2008, Vol. 261. Suppl. 1. pp. 183–200.

8. Rubina L.I. Journal of Applied Mathematics and Mechanics. 2005. Vol. 69. Issue 5. pp. 829–836. (in Russ.).

9. Lomkatsi Ts.D. Tablitsy ellipticheskoy funktsii Veyershtrassa. Teoreticheskaya chast' (Weier-strass elliptic function charts. Theoretical part). Moscow: VTs AN SSSR, 1967. – 88 p. (in Russ.).

10. Gradshteyn I.S., Ryzhik I.M. Tablitsy integralov, summ, ryadov i proizvedeniy (Table of inte-grals, sums, series and compositions). Moscow : Fizmatlit, 1962. 1100 p.

17 2012 .

1 Rubina Liudmila Ilinichna is Cand. Sc. (Physics and Mathematics), Senior Staff Scientist, Institute of Mathematics and Mechanics of the Russian Academy of Sciences (Ural branch). E-mail: [email protected] 2 Ul’yanov Oleg Nikolaevich is Cand. Sc. (Physics and Mathematics), Senior Staff Scientist, University’s academic secretary, Institute of Mathematics and Mechanics of the Russian Academy of Sciences (Ural branch). E-mail: [email protected]


. « . . » 60

519.8

. . 1

. --

, ( ), . - «

» . . , ,

. -.

: , , , , .

. , - [1, 2]:

{1,2} 1 2 1 2 {1,2}{ } ,{ ( , ), ( , ), ( ), ( )}i ii i i i V i S i iX f x x x x R x R xϕ∈ ∈ . (1)

(1) , i- « » ini ix X comp∈ ⊆ R ; -

1 2 1 2( , )x x x X X= ∈ × ; 1 2( , )if x x ( {1,2}i ∈ ) -

1 2X X× . i-

1 2 1 2( ) max ( , ) ( , )i i

i i ix Xx f x x f x xϕ

∈= − ( {1,2}i ∈ ).

i-

1 2 1 2( ) max min ( , ) min ( , )k k k ki i

iV i i ix X x Xx X

R x f x x f x x∈ ∈∈

= − ( , {1,2}i k ∈ , i k≠ ).

1 2 1 2( ) max ( , ) min max ( , )i ik k k k

iS i i ix Xx X x X

R x x x x xϕ ϕ∈∈ ∈

= − ( , {1,2}i k ∈ , i k≠ ).

, ( )iV iR x ( )i

S iR x( {1,2}i ∈ ) ,

i ix X∈ . 1 2( , )if x xXi , i(x1, x2),

( )iV iR x , ( )i

S iR x ( {1,2}i ∈ ) . (1) i (x1, x2)

( )1 2 1 2( , ), ( , ), ( ), ( )i ii i V i S if x x x x R x R xϕ , (2)

fi(x1, x2) ( xi ∈ Xi) -, . 1. . :

(2) ? -

( ). , -:

{1,2} 1 2 {1,2}{ } ,{ ( , )}i i i iX f x x∈ ∈ , (3)

i- 1 2( , )x x « »

1 2( , )if x x .

1 – , ,



. .

2013, 5, 2 61

1. (3) 1 2( , )e ex x - ( ),

1 1 2 1 1 2 1 1

2 1 2 2 1 2 2 2

( , ) ( , ) ,

( , ) ( , ) ,

e e e

e e e

f x x f x x x X

f x x f x x x X

≥ ∀ ∈

≥ ∀ ∈

1 2( , )e ex x . , * *

1 2( , ) 0i x xϕ = ( {1,2}i ∈ ), * *1 2( , )x x .

-.

2. (3) *1x

, 2 2 2 21 1

1 1 2 1 1 2max min ( , ) min ( , )x X x Xx X

f x x f x x∗

∈ ∈∈= .

11( ) 0VR x∗ = .

, *2x ,

1 1 1 12 2

*2 1 2 2 1 2max min ( , ) min ( , )

x X x Xx Xf x x f x x

∈ ∈∈= ,

2 *2( ) 0VR x = .

, .

3. , {1,2} 1 2{ } , ( , )i iX f x x∈

1 1 2 1 2( , ) ( , )f x x f x x= ,

2 1 2 1 2( , ) ( , )f x x f x x= − 1 2( , )e ex x ( ). 1) ,

1 2( , ) 0e ei x xϕ = ( {1,2}i ∈ );

2) , ( ) 0i e

V iR x = ( {1,2}i ∈ ); 3) 11 21( , )e ex x 12 22( , )e ex x – ( ), -

« » ,

11 21 12 22( , ) ( , )e e e ef x x f x x= . , ( ) 1 2( , )e ex x -

( )1 2 1 2( , ), ( , ), ( ), ( )e e e e i e i ei i V i S if x x x x R x R xϕ

( {1,2}i ∈ ) 1 2( , )e ex x (

2 1 2 1 1 2( , ) ( , )e e e ef x x f x x= − ). « » -

. ( )

{ } { } ( )1 11 1 1 2 2 2,..., , ,..., ,m n

ijX x x X x x A a= = = (4)

m n× -

11 12 1

21 22 2

1 2

n

n

m m mn

a a aa a a

A

a a a

= ,

1 1ix X∈ i- A, 22

jx X∈j- ,


. « . . » 62

1 2( , )jix x ija ( )ijA a= , ,

1 1 1 2 1 12 2 2 2( , ) ( , ) , ( , ) ( , )j j j ji i i iij ijf x x f x x a f x x f x x a= = = − = − . (5)

{ } { } ( ) ( ){ } { } { }1 2 1 21 12 21,2 1,2

, , , ( ) ( ), ( ), ( )j jk i iA k ij k ij V V S Sk k

X A a R x R x R x R xϕ∈ ∈Γ = = Φ = ,

{ }11 1 1,..., mX x x= – , { }1

2 2 2,..., nX x x= – -

, 1 2( , )jix x -.

-. ( )1

1 ijϕΦ = -

:

{ }1 1

11 1 1 1 1 12 2 2 1,...,( , ) max ( , ) ( , ) max

i

j j ji i iij ij ij

i mx Xx x f x x f x x a aϕ ϕ

∈∈= = − = − ,

( )22 ijϕΦ =

{ }22

22 1 2 1 2 12 2 2 1,...,( , ) max ( , ) ( , ) min

j

j j ji i iij ij ijj nx X

x x f x x f x x a aϕ ϕ∈∈

= = − = − .

1ix

{ } { } { }1 1 2 22 2

11 1 1 1 12 2 1,..., 1,...,1,...,

( ) max min ( , ) min ( , ) max min minj ji

j ji i iV ij ijj n j ni mx X x X x X

R x f x x f x x a a∈ ∈∈∈ ∈ ∈

= − = − ,

2jx

{ } { } { }1 1 1 122

22 1 2 12 2 2 1,...,1,..., 1,...,

( ) max min ( , ) min ( , ) max min maxi ij

j j ji iV ij ijj ni m i mx X x Xx X

R x f x x f x x a a∈∈ ∈∈ ∈∈

= − = − .

1ix

{ } { } { }1 12 22 2

1 1 11 1 1 1 12 2 1,...,1,..., 1,...,

( ) max ( , ) min max ( , ) max min maxij j

j ji i iS ij iji mj n j nx Xx X x X

R x x x x xϕ ϕ ϕ ϕ∈∈ ∈∈∈ ∈

= − = − ,

2jx

{ } { } { }1 1 1 122

2 2 22 1 2 12 2 2 1,...,1,..., 1,...,

( ) max ( , ) min max ( , ) max min maxji i

j j ji iS ij ijj ni m i mx X x Xx X

R x x x x xϕ ϕ ϕ ϕ∈∈ ∈∈ ∈∈

= − = − .

(4). (4),

1 2( , )jix x (5).

( )1 1 22, jix x X X∗∗

∈ × ( ),

1 1 1 22 2( , ) ( , ) ( , )j jk i i tf x x f x x f x x∗ ∗∗ ∗

≤ ≤

1 1kx X∈ 2 2

tx X∈ .

kj i j i ta a a∗ ∗ ∗ ∗≤ ≤

{ }1,..., ,k m∈ { }1,...,t n∈ . . ( ) ,

. : 7 1 4 14 2 3 22 2 5 24 3 7 2

A

− −

=

− −

.


. .

2013, 5, 2 63

2 2 2 4 3 2 3 41 2 1 2 1 2 1 2( , ),( , ),( , ),( , )x x x x x x x x , -

( ) . :

1 21( ),SR x 1 3

1( ),SR x 2 22( ),SR x 2 4

2( )SR x .

( )1 1S ijϕΦ = ( )

: 1 maxij ij ij

ia aϕ = − .

1

0 3 11 13 0 4 05 0 2 03 5 0 4

SΦ = .

( )2 2S ijϕΦ = :

2 max( ) ( ) minij ij ij ij ijjja a a aϕ = − − − = − ,

, 2 1 2( , )jiijf x x a= − .

2

11 3 0 52 0 1 00 0 3 07 0 10 1

SΦ = .

: 1min max 4,iji j

ϕ = 2min max 3ijj iϕ = .

1 1 1

1( ) max min maxiS ij ijij j

R x ϕ ϕ= − 2 2 22( ) max min maxj

S ij ijji iR x ϕ ϕ= − ,

: 1 2

1( ) 0,SR x = 1 31( ) 1,SR x = 2 2

2( ) 0,SR x = 2 42( ) 2SR x = .

, 2 21 2( , )x x ,

.

2. S- (1) :

1 11 1 1 1,{ ( ), ( )}e

V SX R x R xΓ = − − ,

1eX – ,

2 22 2 2 2,{ ( ), ( )}e

V SX R x R xΓ = − − ,

2eX – .

1. 1 1e ex X∈ 2 2

e ex X∈ (1) s- , -

1Γ 2Γ . 1. (1) :

1) 1X 2X ; 2) 1 1 2( , )f x x 1 2X X× 1x

2 2x X∈ ;


. « . . » 64

3) 2 1 2( , )f x x 1 2X X× 2x 1 1x X∈ . (1) s- .

, -.

2. 1 1e ex X∈ , 2 2

e ex X∈ s- -.

1ex [3]

* 1 11 1 2 1 1 2 1 1 2 1 1, ,{ ( , ), ( , ), ( ), ( )e

V SX X f x x x x R x R xϕΓ = − − − .

2ex

* 2 22 1 2 2 1 2 2 1 2 2 2, ,{ ( , ), ( , ), ( ), ( )e

V SX X f x x x x R x R xϕΓ = − − − .

. : - -

.

1. , . . / . . // 3-- . . - . – 2010. – . 86–

89. 2. , . . / . . , . . //

« ». . – 2012. – . 22. – . 2–5.

3. Zhukovskiy, V.I. The Vector-Valued Maximin / V.I. Zhukovskiy, M.E. Salukvadze. – N.Y. etc.: Academic Press, 1994. – 282 p.

ABOUT OPTIMUM ON RISKS AND REGRETS SITUATIONS IN GAME OF TWO PERSONS

N.G. Soldatova1

Non-cooperative game of two persons is considered. Quality of functioning of players is estimated by four-component vector criterion. Here an outcome (payoff), risk and regrets of players are consid-ered. Concepts of optimum situations of non-cooperative game from the point of view of the stated vec-tor estimates are considered. The example is given. Solution of game is based on the principle of guar-anteed result and function of regret used by Savage. Existence of this solution is established.

Keywords: strategy, Nash equilibrium, maximin, Savage regret, Wald risk.

References 1. Bardin A.E. Riski i sozhaleniya igrokov v igrovykh modelyakh. Materialy 3-ey mezhdunarodnoy

nauchno-prakticheskoy konferentsii (Risks and regret players in the game models. Proceedings of the 3rd International Scientific Conference). MGOGI. Orekhovo-Zuevo, 2010. pp. 86–89. (in Russ.).

2. . ., . . Mezhdunarodnyy nauchnyy zhurnal «Spektral'nye i evolyutsionnye zadachi». Simferopol'. 2012. Vol. 22. pp. 2–5. (in Russ.).

3. Zhukovskiy V.I., Salukvadze M.E. The Vector-Valued Maximin. N.Y. etc.: Academic Press, 1994. 282 p.

1 2013 .

1 Soldatova Natalya Gennadevna is Senior Lecturer, Department of Mathematics and Physics, Moscow State Regional Institute of Humanities. E-mail: [email protected]


2013, 5, 2 65

517.948

. . 1

-. -

. . -.

. : , -

, , .

-- .

, -

. -

-, , . . , . . ,

( ., ., [1–3]) . - [1].

[4, 5]. H – , M U⊂ , C[H] – -

, H .

0A u f u H f H= ; ∈ ; ∈ , (1)

0 [ ]A C H∈ – - . , 0f f= 0u (1), -

M , , f Hδ ∈ , 0f fδ δ− ≤ .

M fδ (1) .

1. 0{ : 0 }Tδ δ δ< ≤ - (1) M , 0(0 ]δ δ∈ ; Tδ -

H H 0T f uδ δ → 0δ → M

0 0A u fδ δ− ≤ . , 00{T }δ δ δ: < ≤

M : 0( ) sup{ : , }.T u T f u M A u fδ δ δ δ δΔ = − ∈ − ≤

1 1 2 1 2 0 1 0 2( ; ) sup{ , : , , }M u u u u M A u A uω τ τ= ∈ − ≤

1 – - , , , -



. « . . » 66

, 0A , 0A M . 2. 0{ : 0 }Tδ δ δ< ≤ -

M , k , 0(0 ]δ δ∈ ;

1( ) ( )T k Mδ ω δΔ ≤ ; .-

, , [6–9] [10, 11]. -

( ) -.

1. - . H – , A – -

( )D A , H . Hϕ ∈ , ,

0( ( )) ( )du Au f t u t t t Tdt

= − + , ; ∈ ; , (2)

0 0( ) 0u t H t Tϕ ϕ= , ∈ , < < , ( )u T χ= . 0[ ]f t T H H: ; × → – , -

u t , . . - 0L > , 0K > 0 1,α< < ,

1 2 1 2( ) ( ) H Hf u t f u t L u u, − , ≤ −

1 2 0, [ ]t t t T∈ ; , 1 2,u u H∈ . 0r > .

0 0{ ( ) : }.At AtM D e e rϕ ϕ= ∈ ≤, Hχ ∈ Hϕ ∈

, M . Hχ ∈ , Hδχ ∈ , ,

δχ χ δ− < . δϕ.

(2)

0

0

( ) ( )( ) ( ( ))t

A t t A t

t

u t e e f u dτϕ τ τ τ− − − −= + , (3)

( .. ., [12]). :

1 2 1 2 1 2( , ) sup{ : , , }M Mω δ ϕ ϕ ϕ ϕ χ χ δ= − ∈ − ≤ −,

1 2 1 2 1 2ˆ ˆ ˆ( , ) sup{ : , , }M Mω δ ϕ ϕ ϕ ϕ δχ χ= − ∈ − ≤ − ,

, . . Hϕ ∈ ,

0( )dv Av t t Tdt

= − ; ∈ ; , (2)

0 0( ) 0v t H t Tϕ ϕ= , ∈ , < < , ( )v T χ= , Hχ ∈ – .

: 1 2 1 2 1 2( , ) sup{ : , , }M Mω δ ϕ ϕ ϕ ϕ χ χ δ= − ∈ − ≤ −

, 1 2 1 2 1 2ˆ ˆ ˆ( , ) sup{ : , , }M Mω δ ϕ ϕ ϕ ϕ δχ χ= − ∈ − ≤ −


. . …

2013, 5, 2 67

. ( . [13 ).

. 0 0δ > , , 0δ δ<

ˆ ˆ( ) ( ) ( )LTLT LTeM e M M eω δ ω δ ω δ−, ≤ , ≤ , .

3. . { : 0}Eλ λ ≥ , A . Aα –

H ,

0

A u dE uα

α λλ= .

(2)

0( )u tα αϕ = , ( )u tα

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

( ) .

du t A u t E f t u t t t Tdt

u T E

αα α

α α

αα χ

= − + , ∈ , ;

= (4)

0

0

( ) ( ) ( , ( )) ( )

( ) ,

du t A u t E f t u t t t Tdt

u t

αα α

α α

ααϕ

= − + , ∈ , ;

= (5)

Eα α αϕ ϕ= . Aα – H , (5) -

0

0

( ) ( )( ) ( ( ))t

A t t A t

t

u t e E e E f u dα α τα α αα αϕ τ τ τ− − − −= + , . (6)

. Hχ ∈ E Hα αϕ ϕ= ∈ , , -

( )u t (5) ( )u T Eαα χ= .

. (5), - (6) (6) , ( )u tα

( ) ( )( ) ( ( ))T

A T t A t

t

u t e E e E f u dα α τα αα αχ τ τ τ− −= − , (7)

0([ ] )C t T H; → [0 ]T;H

0max ( )kt

k Ht t Tu e u t−

≤ ≤= ,

0max ( )C Ht t T

u u t≤ ≤

=

0([ ] )C t T H; → . Pα , 0([ ] )C t T H; → -

( ) ( ) ( ( ))T

A T t A t

t

P u e E e E f u dα α τα α αχ τ τ τ− −= − , .

( 1 2 ([0 ] )u u C T H, ∈ ; → )


. « . . » 68

0

( )1 2 1 2max ( ) ( )

TT k t k

k Ht t Tt

P u P u Le e e u u dα τ τα α τ τ τ− − −

≤ ≤− ≤ − ≤

0

( )1 2 1 2

1maxT kT

T k t Tk kt t T

t

eLe u u e d Le u uk

α τ ατ−

− −

≤ ≤

−≤ − ≤ − .

Tk Leα> , , Pα

0([ ] )C t T H; → k⋅ . , (7)

0([ ] )C t T H; → . :B H E Hα α→ ,

0( )B u tαα χ = ,

( )u tα – (7).

0( )u t Bα ααϕ χ= = − (8)

, χ . .

4. . 0( )u tα αϕ = , ( )u tα – (4); 0( )tα α

δ δϕ ϕ= , ( )u tαδ –

(4) . ( )

( )Pα δα δ δδϕ χ=

( )α α δ= .

( , ) sup{ : , },M Mαδ δα δ ϕ ϕ ϕ χ χ δΔ = − ∈ − ≤

(2) - M .

1 2( ) ( ) ( )M α δ α α δΔ , ≤ Δ + Δ , ,

2 ( ) supδ

α αδ

χ χ δα δ ϕ ϕ

− ≤Δ , = − ;

1 ( ) supM

α

ϕα ϕ ϕ

∈Δ = − .

2 ( )α δΔ , . 0( )( ) ( )t tv t e u tαα α

−= . ( )v tα , -,

0( ) ( ) ( ) ( ( )) ( )dv t A E v t E g t v t t t T

dt

αα α

α α αα= − − + , , ∈ , , (9)

0 0( ) ( )0( ) ( ) [ ]t t t tg t v E e f t e v t T H Hα α

α α− − −, = , : ; × → – ,

v t , 0[ ]t t T∈ ; . ( )v tα

0( )v tααϕ= ; (10) 0( )( ) T tv T e αα χ−= . (11)

(9), (10)

0

0

( )( ) ( )( )( ) ( ( ))t

A E t t A E t

t

v t e E e E g v dα αα α ταα α α α αϕ τ τ τ− − − − − −= + , . (12)

(12) t T= (11)

0 0

0

( ) ( )( ) ( )( ) ( ( ))T

T t A E t t A E T

t

e E e E e E g v dα αα α α τα α α α α αχ ϕ τ τ τ− − − − − − −= + , . (13)


. . …

2013, 5, 2 69

(13)

0 0

0

( )( ) ( ) ( )( ) ( ( ))T

A E t t T t A E t

t

e E e E e E g v dα αα α α τα α α α α αϕ χ τ τ τ− − − − − − −= − , . (14)

(14) (12) ,

0( )( ) ( ) ( )( )( ) ( ( ))T

A E T t T t A E t

t

v t e e E e E g v dα αα α α ταα α α αχ τ τ τ− − − − −= − , . (15)

, ( )v tαδ

0( )( ) ( ) ( )( )( ) ( ( ))T

A E T t T t A E t

t

v t e e E e E g v dα αα α α τα δδ α δ α α αχ τ τ τ− − − − −= − , . (16)

(15) (16)

0( )( ) ( ) ( )( )( ) ( )T

A E T t T t A E tlta

t

v v t e e L e v v dα αα α α τα α α αδ δχ χ τ τ− − − − −− ≤ − + − . (17)

(17) ( )( ) ( )( )

0max 1A E T t T te eα α λ α

λ α− − − −

≤ ≤≤ ≤

0( ) ( )( ) ( ) T t L T tv t v t e eαα αδ δ− −− ≤ ,

0t t= 0 0( ) ( )T t L T te eαα α

δϕ ϕ δ− −− ≤ ., 2 ( )α δΔ ,

0( )2 ( )

LTT t LTee eαα δ δ−Δ , ≤ (18)

1( )αΔ . ( )u t , (3) ( )u tα , (6) ( ) ( )u t E u tα= .

( ) ( )( ) ( ( ))T

A T t A t

t

u t e E e E f u dα α τα αχ τ τ τ− −= − , . (19)

(3)

0

0

( ) ( ) ( ( ))T

A T t A T

t

e e f u dτχ ϕ τ τ τ− − − −= + , . (20)

(20) (6) , ( ) ( )A T t A T te E e Eαα α

− −= ,

0

0

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

A t t A t A t

t t

u t e E e E f u d e E f u f u dα τ τ αα α αϕ τ τ τ τ τ τ τ τ− − − − − −= + , + , − , . (21)

(19) (21)

( )( ) ( ) ( ) ( ) ( ( ( )) ( ( ))T

A t

t

u t u t E u t u t e E f u f u dα τ αα α τ τ τ τ τ− −− = − + , − , (22)

(22) ( ) ( )A t te E eτ α τα

− −≤

( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )T

T t T t T

t

e u t u t e E u t u t L e u u dα α α α τ αα τ τ τ− − − − − −− ≤ − + − . (23)

(23) 0( )( ) ( ) ( ) ( )L T tu t u t e E u t u tα

α−− ≤ − .


. « . . » 70

, 0 0 0 0( ) ( ) ( )

1 1( ) sup ( )L T t L T t L T t te { E M} e e re ααα ϕ ϕ ϕ α− − − −Δ ≤ − : ∈ = ≤ .Δ

( )α α δ∗= . 0 0( )T t tLTe e reα αδ− −= (24)

( . [14]). (24) , 0( )1( ) ln( )L T tre

Tα δ δ−∗ = / .

, M (24)

0 00( )( ( ) )

T t tL T t T T

M e rα δ δ δ−

−∗Δ , ≤ . (25) M -

0 00( )( )

T t tL T t T TM e rω δ δ

−−, ≤

( . [13]), (25) . .

, (8), M .

1. , . . / . . , . . , . . . – .: , 1978. – 208 c.

2. , . . / . . // . – 1970. – . 6, 8. – . 1490–1495.

3. , . . - / . . -, . . , . . . – .: , 1995. – 176 c. 4. , . . -

/ . . // . – 1976. – . 16, 2. – . 503–507. 5. , . . - / . . // . . . – 2003. – . 6, 3. – . 119–133. 6. , . . / . . , . . , . . .

– .: , 1995. – 312 c. 7. , . . / . . , . . . –

: , 1993. – 262 c. 8. , . . -

// . . . – - , . . - , 1998. – 292 . 9. Tanana, V.P. Methods for solving of nonlinear operator equations / V.P. Tanana. – Utrecht, VSP,

1997. – 241 p. 10. , . .

/ . . // -. . – 2005. – . 8, 3. – . 259–271.

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

. – 2005. – . 8, 1(21). – . 129–142. 12. , . / . . –

.: , 1985. – 376 . 13. , . .

/ . . // . – 2013. – . 19, 1. –C. 253–257

14. , . . // . . . – : , 1962. – 92 .


. . …

2013, 5, 2 71

ABOUT SOLVING OF AN ILL-POSED PROBLEM FOR A NONLINEAR DIFFERENTIAL EQUATION BY MEANS OF THE PROJECTION REGULARIZATION METHOD

E.V. Tabarintseva1

A retrospective inverse problem for a semi-linear differential equation is studied. The projection regularization method with the choice of the regularization parameter by means of M.M. Lavrentiev scheme is used to find a stable approximate solution to the ill-posed problem under consideration. An explicit evaluation of inaccuracy of this method was measured on one of the cases of robustness.

Keywords: inverse problem, nonlinear differential equation, approximate method, evaluation of in-accuracy.

References 1. Ivanov V.K., Vasin V.V., Tanana V.P. Teoriya lineynykh nekorrektno postavlennykh zadach i ee

prilozheniya (Theory of linear ill-posed problem and its applications). Moscow: Nauka, 1978. 208 p. (in Russ.).

2. Strakhov V.N. Differentsial'nye uravneniya. 1970. Vol. 6, no. 8. pp. 1490–1495. 3. Ivanov V.K., Mel'nikova I.V., Filinkov A.I. Differentsial'no-operatornye uravneniya i nekor-

rektnye zadachi (Differential and operator equations and ill-posed problems). Moscow: Nauka, 1995. 176 p. (in Russ.).

4. Tanana V.P. Optimal order methods of solving non-linear ill-posed problems. USSR Computa-tional Mathematics and Mathematical Physics. 1976. Vol. 16, no. 2. pp. 219–225.

5. Tanana V.P. Sibirskiy zhurnal industrial'noy matematiki. 2003. Vol. 6, 3. pp. 119–133. (in Russ.).

6. Tikhonov A.N., Leonov A.S., Yagola A.G. Nelineynye nekorrektnye zadachi (Non-linear ill-posed problems). Moscow: Nauka, 1995. 312 p. (in Russ.).

7. Vasin V.V., Aneev A.L. Nekorrektnye zadachi s apriornoy informatsiey (Ill-posed problems with prior information). Ekaterinburg: Nauka, 1993. 262 p. (in Russ.).

8. Kokurin M.Yu. Operatornaya regulyarizatsiya i issledovanie nelineynykh monotonnykh zadach(Operator regularization and the study of non-linear monotonic problem). Yoshkar-Ola, Izd. Mariyskogo gosudarstvennogo universiteta, 1998. 292 p.

9. Tanana V.P. Methods for solving of nonlinear operator equations. Utrecht, VSP, 1997. 241 p. 10. Tabarintseva E.V. Sib. Zh. Vychisl. Mat. 2005. Vol. 8, no. 3. pp. 259–271. (in Russ.). 11. Tanana V.P., Tabarintseva I.V. Sib. Zh. Ind. Mat. 2005. Vol. 8, no. 1. pp. 129–142. (in Russ.). 12. Khenri D. Geometricheskaya teoriya polulineynykh parabolicheskikh uravneniy (Geometrical

theory of semilinear parabolic equations). Moscow: Mir, 1985. 376 p. (in Russ.). [Henry D. Geometric Theory of Semilinear Parabolic Equations. Springer-Verlag, 1981. 348 p.]

13. Tabarintseva E.V. Trudy Inst. Mat. i Mekh. UrO RAN. 2013. Vol. 19, no. 1. pp. 253–257. (in Russ.).

14. Lavrent'ev M.M. O nekotorykh nekorrektnykh zadachakh matematicheskoy fiziki (Some improperly-posed problems in mathematical physics). Novosibirsk: Sibirskoe otdelenie AN SSSR, 1962. 92 p. (in Russ.). [Lavrentiev M.M. Some Improperly Posed Problems in Mathematical Physics. Springer-Verlag Berlin and Heidelberg GmbH & Co. K, Berlin, 2012. 88 p.]

15 2013 .

1 Tabarintseva Elena Vladimirovna is Cand. Sc. (Physics and Mathematics), Associate Professor, Computational Mathematics Department, South Ural State University. E-mail: [email protected]


. « . . » 72

517.948

. . 1, . . 2

-. . .

, . : , , -

, .

. . . - [1] α , [2, 3] -

, . [4] -

, .

. 1. . - , -

,

0

( )( ) ( ) ; 0 ,d CSn S nε ε ε θε ε θθ θ ε θ

∞

= = ≤ < ∞ (1)

( )2

2( ) , ( )2sh 2

xS x Cx

θ= – , kTθ = , T – , k –

, , ( )n ε – [4]. H [0,∞) f(x) ,

2 2

0

( ) ( ) .Hdxf x f xx

∞

= (2)

, (2) .

, 0 ( )( ) Hθθθ θ

= ∈ n0( )∈H (1),

0 ( ) ,rn Gε ∈2

2 2

0 0

( )( ) : ( ) , [ ( )] ,rnG n n H d n d rεε ε ε ε ε ε

ε

∞ ∞′= ∈ + ≤ (3)

( )n ε′ – ( )n ε , 0 ( )θθ

-

(1) ( ) Hδ θθ

∈ 0δ > ,

0( ) ( ) .H

C Cδ θ θ δθ θ

− ≤

( )n Hδ ε ∈ (1) -

0( ) ( ) Hn nδ ε ε− 0 ( )n ε H.

1 – - , , , –

. E-mail: [email protected] 2 – , , -



. ., . .

2013, 5, 2 73

, ( ) , ( )n Hθ εθ

∈ , (1) .

2. . . . . . [1] - (1)

22 2 1

0 0 0 0

( )inf ( ) [ ( )] ( ) : ( ) [0, ) ,Cd d dS n n d n n Hδ θε ε ε θ εε α ε ε ε α ε εθ θ ε θ θ ε

∞ ∞ ∞ ∞′− + + ∈ ∞ (4)

H1[0, ) – ,

1

22 2

[0, )0 0

( )( ) [ ( )] ,Hnn d n dεε ε ε ε ε

ε

∞ ∞

∞′= + > 0.

[5], ( ) Hδ θθ

∈ -

(4). α (4),

[2, 3], 2

2

0 0

( )( ) Cd dS nα δδ

θε ε ε θε δθ θ ε θ θ

∞ ∞

− = (5)

α .

[3], 2

0

( )C dδ θ θ δθ θ

∞

> , (5) -

( , )Cδα δ . ( )nδ ε (1)

( , )( ) ( )Cn n δα δδ δε ε= ,

0{ : 0 }Rδ δ δ< ≤ -H H

( )( ), ,( )

( )0, .

H

H

CnCR

C

δδ

δδ

δ

θε δθθ

θ θ δθ

>=

≤

3. }0:{ 0δδδδδδδδδδδδ ≤≤≤≤<<<<R Gr. -

0{ : 0 }Rδ δ δ< ≤

0{ ( ) : 0 }Rδ δ δ δΔ < ≤ [6]

0 0 0( ) ( ) ( )

( ) sup ( ) : ( ) , , ( )rHH

C C CR R n n G H Snδ δ δ

δ δ δθ θ θε ε ε δ

θ θ θΔ = − ∈ ∈ − ≤ . (6)

w( ,r) S-1 S[Gr] ( , ) sup{|| ( ) || : ( ) ,|| ( ) || }H rw r n n G Snδ ε ε ε δ= ∈ ≤ . (7)

( )Rδ δΔ [7]

0( ) 2 ( , );0R w rδ δ δ δ δΔ ≤ < ≤ , (8) w( ,r) (7), ( )Rδ δΔ (6). 4. w( ,r), (7).

=et e= ; t ,−∞ < < ∞ −∞ < < ∞ (9) S A


. « . . » 74

( ) ( ) ( ) ; ,Au t K t u t dt t ,τ∞

−∞

= − ∞ < < ∞ −∞ < < ∞ (10)

( ) ( ),tu t n e=3

2( ) ,

2sh2

x

xeK x

e

−

−=

u(t), Au(t)∈L2(– , ). , (9) Gr, (3) Mr

21 2 22{ ( ) : ( ) ( , ), ( ) ( ) }.rM u t u t W u t dt u t dt r

∞ ∞

−∞ −∞

′= ∈ −∞ ∞ + ≤ (11)

A-1 Nr=AMr -

2 2( , ) sup{|| ( ) || : ( ) ,|| ( ) || }.L r Lw r u t u t M Au tδ δ= ∈ ≤ (12)

1. ( , )w rδ (7), ( , )w rδ (12). ( , ) ( , )w r w rδ δ= .

5. ( , )w rδ , (12) , 1 2( ) ( , ) ( , )u t L L∈ −∞ ∞ ∩ −∞ ∞ , F

1[ ( )] ( )2

iptF u t u t e dtπ

∞

−∞

= . (13)

F L2(– , )., 2 ( , )L −∞ ∞ .

, F, (13) 2 ( , )L −∞ ∞ 1 2( , ) ( , )L L−∞ ∞ ∩ −∞ ∞ 2 ( , )L −∞ ∞ .

, 1( , )L −∞ ∞ 2 ( , )L −∞ ∞ , F

2 ( , )L −∞ ∞ . F . F 2 ( , )L −∞ ∞ 2 ( , )L −∞ ∞ .

F Y , Y - 2 ( , )L −∞ ∞ .

F A ˆ ˆˆ ˆ( ) ( ) ( );Au p K p u p= ˆ( ) ,u p Y∈ 2

ˆ ˆ( ) ( , ),Au p L∈ −∞ ∞ (14) ˆ( ) [ ( )],u p F u t= , 1( ) ( , )K x L∈ −∞ ∞

1ˆ ( ) ( ) .2

ixpK p K x e dxπ

∞

−∞

=

K(x) , (2 ) (2 )

2

1 2ˆ ( ) . ( ).2 ch( ) 1 ( 1)

x

x

ip x x ip x ex

x e

e e e eK p dx d ee eππ

−

−

∞ ∞− − − − − −−

− −−∞ −∞

= = −− −

z = e-x, (2 ) (2 )

2 20

2 2ˆ ( ) .( 1) ( 1)

ip z ip z

z zz e z eK p dz dz

e eπ π

∞ ∞− − − −

−∞

= − =− −

- [8]


. ., . .

2013, 5, 2 75

1

0

( ) ( )1

s

zzs s dz

eζ

∞ −Γ =

−,

, 2 2ˆ ( ) (2 ) (2 ) (2 ) (3 ) (2 )K p ip ip ip ip ipζ ζπ π

= − Γ − − = Γ − − ,

( )zΓ – , ( )zζ – - . ˆ ( )K p p -

- , [8, . 16 19]: (z+1) = z (z), (15)

( ) ( )z zΓ = Γ , (16) z z, ( )zΓ (z)

( ) (1 )sin

z zz

ππ

Γ Γ − = . (17)

, (15) , 2 2(3 ) 1 4 (1 )ip p p ipΓ − = + + Γ − , (18)

(16) (17),

(1 )sh

pipp

ππ

Γ − = . (19)

(18) (19) p 2

2(3 ) 2p

ip eπ

π−

Γ − ≥ . (20) - )2( ip−ζ .

1

1( ) sn

sn

ζ∞

== , (21)

(21) , ln

21

(2 )ip n

n

eipn

ζ∞

=− = . (22)

, ln 1ip ke = , (22)

22

1 1(2 ) 13n

ipn

ζ∞

=− ≥ − ≥ . (23)

, (20) (23) , 2p ≥

22ˆ ( )3

PK p e

π−≥ . (24)

1A A , (14) -

2 ( , )L −∞ ∞

1ˆ ˆˆ ˆ( ) ( ) ( );A u p K p u p= 1 2

ˆˆ ˆ( ), ( ) ( , )u p A u p L∈ −∞ ∞ . (25)

2ˆ ( , )rM L⊂ −∞ ∞

22 22

ˆ ˆ ˆ ˆ ˆ{ ( ) : ( ), ( ) ( , ), (1 ) ( ) }.rM u p u p pu p L p u p dp r∞

−∞

= ∈ −∞ ∞ + ≤ (26)

(11) (26) , ˆ[ ] .r rF M M⊂ (27)

,


. « . . » 76

2 2

ˆˆ ˆ ˆ ˆ( , ) sup{ ( ) : ( ) [ ], ( ) }rL Lw r u p u p F M Au pδ δ= ∈ ≤ , (28)

{ }2 21 1

ˆˆ ˆ ˆ ˆ( , ) sup ( ) : ( ) , ( )rL Lw r u p u p M A u pδ δ= ∈ ≤ . (29)

F (10), (12), (14) (28) , ˆ ( , ) ( , ),w r w rδ δ= (30)

(14), (25), (27)–(29), 1ˆ ˆ( , ) ( , ).w r w rδ δ≥ (31)

, (30) (31) , 1ˆ( , ) ( , ).w r w rδ δ≤

1A , (25) 1

1A− , 1T1

1 1ˆ ˆˆˆ ( ) ( );T f p A f p−= 1

ˆ ˆ( ) ( ),f p R A∈ 1 2ˆˆ ( ) ( , ),T f p L∈ −∞ ∞ (32)

1ˆ( )R A – 1A .

ˆrM , (27) B

2ˆ ˆ( ) 1 ( );Bu p p u p= + 2ˆ ˆ( ), ( ) ( , ),u p Bu p L∈ −∞ ∞ (33) 1ˆ

r rM B S−= , (34)

22ˆ ˆ ˆ{ ( ) : ( ) ( , ), ( ) }r LS u p u p L u p r= ∈ −∞ ∞ ≤ .

2 ( , )L −∞ ∞ 1ˆrN

1 11

ˆ ˆ ˆ( ).r rN T M−= (35) (26), (29), (32)–(34), (35) ,

{ }2

11 1

ˆ ˆ ˆˆ ˆˆ ( , ) sup ( ) : ( ) , ( )r Lw r T f p f p N f pδ δ= ∈ ≤ .

1ˆ ( , )w rδ .

T , 2 ( , )L −∞ ∞ 2 ( , )L −∞ ∞

ˆ ˆˆ ( ) ( ) ( ),Tf p g p f p= (36)

g(p) ∈C(– , ), g(–p) = g(p), g(0) > 0, (37) lim ( )p

g p→∞

= ∞ g(p) [0, ).

2ˆ ( , )w rδ T1ˆ ˆ ˆ( ),r rN T M−= ˆ

rM (34)

2( )

1

r g pp

δ=+

. (38)

g(0) < r, (38) ( , )p rδ . [6] ,

2 2ˆ ( , ) .

1 ( , )

rw rp r

δδ

=+

(39)

, 1T (25) (32), T (36). .

2. g(p) (37) p0 0 , p p0


. ., . .

2013, 5, 2 77

1ˆ ( ) ( ),K p g p−

≤

, 0 20

( )1

rg pp

δ <+

1 2ˆ ˆ( , ) ( , ).w r w rδ δ≤ 2 0{ : 0 }Rδ δ δ< ≤ .

(24) , p 2 1

23ˆ ( ) .2

pK p e

π−

≤ (40)

, (8), (30), (31), (39), (40) 2 ,

023 5re π

δ−

=

0{ : 0 }Rδ δ δ< ≤

22

2( ) .1 21 ln

3

rRr

δ δ

δπ

Δ ≤+

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

2. , . . / . . - // . . . . . – 1966. – . 6, 6. – . 1089–1094. 3. , . . -

/ . . // . . . . . – 1966. – . 6, 1. – . 170–175. 4. , . . -

/ . . // . . – 1954. – . 26, . 5. – . 551–556.

5. , . . / . . , . . // . . . – . – 1968. – . 6. – . 2. – . 27–37.

6. , . . - / . . , . . // . . . . – 2006. – . 9, 4. – . 353–368.

7. , . . / . . . – . . – 1975. – . 220, 5. – . 1035–1037.

8. , . . / . . , . . . – .: , 1978. – . 2. – . 468.


. « . . » 78

ABOUT THE EVALUATION OF INACCURACY OF APPROXIMATE SOLUTION OF THE INVERSE PROBLEM OF SOLID STATE PHYSICS

V.P. Tanana1, A.A. Erygina2

The problem of determining of the phonon spectrum of the crystal from its thermal capacity was studied. The accuracy evaluation for regularization method of A.N. Tikhonov chosen from the residual principle was obtained.

Keywords: regularization, module of continuity, evaluation of inaccuracy, ill-posed problem.

References 1. Tikhonov A.N. Doklady AN SSSR. 1963. Vol. 151, no. 3. pp. 501–504. (in Russ.). 2. Ivanov V.K. Zhurnal vychislitel'noy matematiki i matematicheskoy fiziki. 1966. Vol. 6, no. 6.

pp. 1089–1094. 3. Morozov V.A. Zhurnal vychislitel'noy matematiki i matematicheskoy fiziki. 1966. Vol. 6, no. 1.

pp. 170–175. (in Russ.). 4. Lifshits I.M. Zhurnal eksperimental'noy i teoreticheskoy fiziki. 1954. Vol. 26, Issue 5. pp. 551–

556. (in Russ.). 5. Vasin V.V., Tanana V.P. Matematicheskie zapiski Ural'skogo universiteta. 1968. Vol. 6, Issue 2.

pp. 27–37. (in Russ.). 6. Tanana V.P., Yaparova N.M. Sib. Zh. Vychisl. Mat. 2006. Vol. 9, no. 4. pp. 353–368. (in Russ.). 7. Tanana V.P. Doklady AN SSSR. 1975. Vol. 220, no. 5. pp. 1035–1037. (in Russ.). 8. Uitteker E.T., Vatson Dzh.N. Kurs sovremennogo analiza (A Course of Modern Analysis). Mos-

cow: Nauka, 1978. Part 2. 468 p. (in Russ.). [Whittaker E.T., Watson G.H. A Course of Modern Analy-sis. Cambridge University Press; 1927. 612 p.]

15 2013 .

1 Tanana Vitaliy Pavlovich is Dr. Sc. (Physics and Mathematics), Professor, Department of Computational Mathematics, South Ural State Uni-versity. E-mail: [email protected] 2 Erygina Anna Aleksandrovna is Master Student, Department of Theory of Management and Optimization, Chelyabinsk State University. E-mail: [email protected]


2013, 5, 2 79

519.857

. . 1, . . 2

-. -

. : , ,

.

1. ,

-, [1, . 85–87].

. -

[2–6] , , [7, 8]. [7] , -

, . . [8] , -

, . , , -

, , .

2. [1]

1 1 1 2 1( )( )

m tx a x a x g um t

= + + + .

1nx ∈ , ia ∈ , ng ∈ ; u – ,

constu c= = ; 1

0

( )( ) 1( )

m tm tm t

= + , 1( )m t – t, 0m – -

. ,

2 1 2 2 2 2, , , .n nx a x a x g v x v v b= + + − ∈ ∈ ≤,

1 2( ) ( )x t x t ε− ≤ . 0ε ≥ – .

1(0) 0m ≥ , .

3. 1 2x x x= − , 1 2y x x= − , ( ) ln( ( ))t m tϕ = ,

1 2, ( ) ,x y y a x a y t u vϕ= = + + + (1) ( ) .x t ε≤ (2)

1 – - , , -

, . E-mail: [email protected] 2 – , , -



. « . . » 80

( )m t , ( ) 0tϕ ≤ . -

1(0) 0m ≥ . ( ) 1m t ≥0t ≥ ,

( ) 0tϕ ≥ 0t ≥ . (3) , ( )m t

τ 0 ( ) 1m m τ≤ ≤ −( )u τ . [1]

( ) ( )( 0) ( ) ( ) ( 0) ( ) , ( 0) ln ( )y y u m mτ τ τ ϕ τ ϕ τ ϕ τ τ+ = + − + = − . (4)

: n n nv × × → , ( , , )v t x y b≤ .

( ) 0tϕ ≥ 0t ≥ - : n n nu × × → , ( , , )u t x y c= .

( )tϕ , .

0 10 ... qτ τ τ= < < < . iτ , ( )ix τ , ( )iy τ , ( )iϕ τ , 1i itτ τ +< ≤ ( )tϕ ,

, 1( ) 0, ( ) 0, ( 0) ( ),i i i it t tϕ ϕ ϕ τ ϕ τ τ τ +≥ ≤ + ≤ < ≤ . (5)

(4) iτ τ= ( ), ( ), ( )i i iu u x yτ τ τ=( 0)iy τ + .

, 1( , ]i iτ τ + , . :

(0) (1) ( 1)1: ... k

i it t tω τ τ++= < < < =

( ) ( ) ( )( )1

0max j j

j kd t tω +

≤ ≤= −

( )x tω , ( )y tω , 1i itτ τ +< ≤ . (1) ( ), ( ), ( )i i iu u x yτ τ τ= , ( ), ( ), ( )i i iv v x yτ τ τ= ( )x tω , ( )y tω

(0) (1)t t t< ≤ (0)( ) ( )ix t xω τ= , (0)( ) ( 0)y t yω τ= + . , (0) ( )jt t t< < . (1) ( )ju u= , ( )jv v= ,

( ) ( ) ( )( )( ) ( ) ( ), ,j j jw j w x t y tω ωτ= , ,w u v= (6)

( ) ( ),x t y tω ω( ) ( )1j jt t t +< < . ,

1i itτ τ +< ≤ . , ( ) ( ),x t y tω ω

. , [9, . 236].

( ) ( ),k k

x t y tω ω , ( ) 0kd ω → .

( )x t ( )y t .

[ ]1,i iτ τ + . , .

1. 22 1 0a aλ λ− − = (1)

, . , 2

1 24 0a a+ > ,


. ., . .

2013, 5, 2 81

2 22 1 2 2 1 2

1 24 4

0, .2 2

a a a a a aλ λ

+ + − += ≥ = (7)

2. 1 0.b a ε+ ≥ (8)

4.

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

1 2 1 2

1 1,t t t tf t e e f t e eλ λ λ λλ λλ λ λ λ

= − + = −− −

. (9)

1 2λ λ> , ( )2 0f t > 0t > . -

1 0λ ≥ , ( )20

t

f r dr → +∞ t → +∞ . 0ε ≥ -

:

( )20

t

f r drbε = (10)

( )t t ε= . 0t ≥ , 0ε ≥

( ) ( )20

,t

g t b f r drε ε= − ( )0 t t ε≤ ≤ ,

( ) ( )( ) ( )2,g t t t bf tε ε= − − ( )t tε ≤ . (11)

( ) ( ) ( ) ( )1 221 2

12 2

t taf t f t f t e eλ λ= + = + (12)

( ) ( ) ( )( )0 0 , 0ct t Tb

ε ϕ ε ϕ≤ ≤ + = (13)

( )( ) ( )( )

( ) ( ) ( )( )

2 22

0

0 ,, , 0 , ,

2r t

f r cf r g raK x y t x y Sf r f r

ϕ εϕ ε

≤ ≤

+= − + . (14)

{ }: 1nS z z= ∈ ≤ .

1. , ( ) ( ) ( ) ( ) ( )( )( )0 0 , 0 , 0 , , , 0x K x y Tϕ ε ε ϕ∉ . (15)

, ( )x t ε> 0t ≥ . (16)

(15) . , (14), ( )( )00 , 0T T ε ϕ≤ ≤ , , 0t T=

( ) ( ) ( ) ( )( )0 0 , 0 , 0 , ,x K x y tϕ ε∈ , (17)

00 t T≤ < (11) . 2. , -

00 t T≤ ≤ (2) . 3. 00 T T≤ <

, (16) 0 t T≤ ≤ -.


. « . . » 82

5. ( ) ( ) ( )0 , 0 , 0 0x y ϕ ≥ , ( )0x ε> (15)

. p t= (17) -

( ) ( ) ( ) ( ) ( ) ( ) ( )1 2 20 0 0 , .f p x f p y cf p g pϕ ε+ = + (18)

( ) ( )1 2 .z f p t x f p t y= − + − (19) (4) ,

( ) ( ) ( ) ( ) ( ) ( )( ) ( )( )20 , , 0z z u x y f pτ τ τ τ τ ϕ τ ϕ τ τ+ = + + − − . (20) (19) (1) (9), ,

( ) ( ) ( ) ( ) ( )12 2 , , 0,j jz f p t t u f p t v t t t j kϕ += − + − < ≤ = . (21)

, ( ) ( )x p z p= .

( ), , ; zu t x y cw wz

= = 0z ≠ 1w = 0z = . (22)

( )0 p t ε< ≤ . (18)

( ) ( ) ( ) ( )2 20

0 0p

z cf p b f r drϕ ε= + − . (23)

( ) 0tϕ = 0 t p< ≤ . (20) ,

( ) ( ) ( )( ) ( ) ( )2(0 ) 0 0, 0 , 0 0z z u x y f pϕ+ = − .

(22), ( ) ( ) ( )2(0 ) 0 0z z c f pϕ+ = − . (23) ,

( )20

(0 )p

z b f r drε+ = − .

( )x t ( )y t . (19). ( )z t . (21) , ( )2( )z t b f p t≤ − .

( ) ( ) ( ) ( )20

0p

x p z p z b f r dr ε= ≤ + + = .

( ) ( )( ), 0t p Tε ε ϕ< ≤ . (18)

( ) ( ) ( ) ( )( ) ( )2 20 0 .z c f p p t b f pϕ ε= − − (24)

( ) ( ) ( )( )2

00

z bt tcf p c

ϕ ϕ= − − ( )0 t p t ε< ≤ − (25)

( ) 0tϕ = ( )p t t pε− < ≤ . (24) ,

( ) ( )( ) ( )( )0 0, 0bp t p tc

ϕ ε ϕ ε+ = − > − = .

(20), (22), (25) , ( )0 0z + = . ,

( )( ) 0z p t ε− = .


. ., . .

2013, 5, 2 83

, ( )( ) 0z p t ε− > . , , ( )0 0z + = ,

( )00 t p t ε≤ < − , ( ) 0z t > ( )ot t p t ε< ≤ − ( )0 0z t = . ( )z t

( )0 ,t p t ε− . ( ) 0z t > ( )0t t p t ε< ≤ − , , (21), (22) (25),

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

z tz t bf p t f p t v v b

z t= − − + − ≤ (26)

( )( 0 ,t t p t ε∈ − . z ,

( )z t . [10, . 118]

( ) ( ) ( ) ( )0

limh

d z t z t h z t z tdt h→ +

+ −= .

(26) , ( )

0d z t

dt≤ . ,

( )( ) ( )z p t z tε− < ( )( )0 ,t t p t ε∈ − . 0t t→ , -

( )( ) 0z p t ε− ≤ .

( )p t t pε− ≤ ≤ (21) (25) , ( ) ( )2 ,z t f p t v v b= − ≤ . ,

( )t ε , ( )z t ε≤ .

6. (14).

1. (14) . . (14)

( ) ( )2 ,f t f t ( ),g t ε t .

( )( )

2 , 0f t

tf t

τ = ≥ .

( )( )

( )( )

2 2

1 2

0 20, lim0 t

f f tf f t λ λ→+∞

= =−

( )( ) ( )

222 0a tf td e

dt f t f t= > ,

1 2

20 τλ λ

≤ ≤−

( )t ψ τ= ,

( ) ( ) 22 a tdf t e

dψ τ

τ−= ( )t ψ τ= (27)

τ (14)

( )( )

2

0 2t

a x y G Sτ τ

τ τ≤ ≤

− + . (28)

( ) ( )( )( )( ) ( )

,0

gG c

fψ τ ε

τ ϕ τψ τ

= + . (29)

, ( )G τ . - (28).


. « . . » 84

(11) , ( ),dg tdt

ε0 t≤ .

(29)

1 2

20 τλ λ

≤ <−

. (29) (27), ,

( )( ) ( )0dG Q cd

ψ τ ϕτ

= + . (30)

( ) ( ) ( ) ( ) ( )2

,, a tdg t df t

Q t f t g t edt dt

εε −= − . (31)

, (30) . , ( ) 0

dQ tdt

≤ 0t ≥ .

(12) ( ) ( ) ( )

2

2 12d f t df t

a a f tdtdt

= + . (32)

( ) ( ) ( ) ( ) ( ) 2

2

2 12, ,

, a tdQ t d g t dg ta a g t f t e

dt dtdtε ε

ε −= − − . (33)

( )0 t t ε≤ < . , (11) (33), ( ) 0dQ t

dt≤

( ) ( ) ( )22 2 1 2 1

0

0tdf t

b a f t a f r dr adt

ε− − + ≥ . (34)

( )2f t (32). (34) -. 0t = 1b a ε+ . 2 -

(34) . ( )t tε < (33)

( ) ( ) ( ) ( ) ( )2 22 22 2 0

dQ t df ta f t b f t b f t

dt dt= − + = − < .

2. (15). 1ψ = ,

( ) ( ) ( ) ( ) ( ) ( ) ( )1 2 20 0 , 0 ,f s x f s y cf s g sψ ϕ ε+ > + (35) 0s ≥ .

, n . . (14) , ( )0x (28), -

( )( )( ) 00 , 0Tτ τ ε ϕ τ≤ ≤ = . (11), (13) (29) ,

( )0 0G τ = . (9), (10) (12) , ( )t tε < (31)

( ) ( ) ( ) ( )( )22

a tQ t b f t f t e t t bε−= − − − . (30) ,

( ) ( ) ( ) ( )( )202 0, , 0a TdG

b f T f T e T Td

τε ϕ

τ−= − < = .

[7], ψ .


. ., . .

2013, 5, 2 85

(35) , , -ψ , ( )0x K , -

(14) ( ) ( ) ( )( )0 , 0 , , 0x x y y t T ε ϕ= = = ( . ). 3. 1 2 0ε ε> > 0t >

( ) ( )1 2 1 2, ,g t g tε ε ε ε> + − . (36)

4. 0ε ≥ , ( )0 0ϕ ≥ , ( )( )0 , 0s T ε ϕ≤ ≤ , 0τ >

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

s

g s g s b f d s t f s f s bσ

σ ε ε τ τ σ ε σ+

+ − + ≥ − + − + − . (37)

(36) (37) (11). - (37) ( )2f t .

1. , (15). 0η > , , (15) ε ( )1 η ε+ .

( ) ( )( )1 , 0T T η ε ϕ> + . [ ]0,T

it iσ= . σ

( )0 tσ ε< < , ( ) ( ) ( )( )2 20

1 max 0T

s T

ee f s f sσ

β σ ηε σσ

−−

≤ ≤

−= − + − > . (38)

( )1 ii e σε η ε−= + . (39)

, -it

( ) ( ) ( ) ( ) ( )( )( ), , , , ,i i i i i i ix t K x t y t t T tϕ ε ε ϕ∉ . (40)

, 0i = . 2

iψ ,

( ) ( ) ( ) ( ) ( ) ( ) ( )1 2 2, ,i i i i if s x t f s y t cf s t g sψ ϕ ε+ > + (41) 0s ≥ .

1i it t t +≤ ≤( ) iv t ψ= .

( ) ( ),x t y t . (19) -

ip t≥ . (20) (21) ,

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( )

1 2

1 2 2 2

,

, .i

i

p t

i i i i i i ip t

f p t x t f p t y t

f p t x t f p t y t t t c f p t b f d

ψ

ψ ϕ ϕ τ τ−

−

− + − ≥

≥ − + − − − − +

(41) is p t= − ,

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1 2 2 2, ,ip t

i i i ip t

f p t x t f p t y t cf p t t g p t b f dψ ϕ ε τ τ−

−

− + − > − + − + . (42)

p t= . , ( ) ( ) , i ix t x t ψ ε ε= > > 1i it t t +≤ ≤ . ,

( ) ( )1 10 i ics t tb

ε ϕ+ +≤ ≤ + (43)

y

20,5a x y−

ψ

K

x

,K -, ψ


. « . . » 86

( ) 0A s > .

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1 1 2 1 2 1 1, ,i i i i iA s f s x t f s y t c f s t g sψ ϕ ε+ + + += + − − . (40) 1i + . s ,

(43), (42) 1 1,i ip s t t t+ += + = .

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

i i is

A s c f s f s t g s g s b f dσ

σ ϕ σ ε ε τ τ+

+ +> + − + + − + .

, (36), (37), ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( )1 1 2 2 2 2 1i i i iA s c t bs bt f s f s f s f sϕ ε σ σ σ ε ε+ + +> − + + − − + − + − .

, ( )2f t , (43) (39), ,

( ) ( ) ( ) ( )2 21T eA s e f s f s

σσ ηε σ σβ σ

σ

−− −> − + + ≥ .

(38) ( ) 0A s > .

[ ],2T T . .

3. ( )( )0 , 0T T ε ϕ< < , (17) t T= . 0η >

ψ , 0 s T≤ ≤ (35) ε ( )1 η ε+ . -

[ ]0,T it iσ= . σ (38). , -

it (41) iψ0 is T t≤ ≤ − .

( ) iv t ψ= 1i it t t +≤ ≤ . 1i it t t +≤ ≤ -

ip t≥ (42). , , ( ) ix t ε ε> >

1i it t t +≤ ≤ . , 10 is T t +≤ ≤ − ( ) 0A s > . ,

( )( )1 1 1,i i iT t T tε ϕ+ + +− < , (41) 1i + .

( )( )1 1 1,i i iT t T tε ϕ+ + +≤ − , 1it + - (40).

1. , . . / . . – .: , 1970. –420 .

2. , . . / . . // . . . – 1968. – . 32. – . 2. – . 177–184.

3. , . . - / . . // . . . – 1966. – . 30. – . 5 – . 897–907.

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

. – 1975. – . 39. – . 3. – . 397–406. 5. , . . -

/ . . // , . – 2009. – 2. – . 38–44.

6. , . . : / . . . – :

. – 2005. – 124 . 7. , . . .

/ . . // : . – : - -

, 1988. – . 123–130.


. ., . .

2013, 5, 2 87

8. , . . - / . . , . . // . « ,

, ». – 2010. – . 11. – 2 (178). – . 29–32. 9. , . . / . . , . . . –

.: , 1965. – 520 . 10. , . . / . . -

. – . , 1985. – 224 .

ONE PROBLEM OF PULSE PERSUIT

V.I. Ukhobotov1, A.A. Troitsky2

Optimum time is found in second order linear differential game with pulse control. Optimum con-trol has been developed for players.

Keywords: differential game, pulse control, pursuit time.

References 1. Krasovskii N.N. Teoriia upravleniia dvizheniem (The Theory of Motion Control). Moscow: Nau-

ka, 1970. 420 p. (in Russ.). 2. Krasovskii N.N. Prikladnaya matematika i mekhanika. 1968. Vol. 32. Issue 2. pp. 177–184. (in

Russ.). 3. Pozharitskiy G.K. Prikladnaya matematika i mekhanika. 1966. Vol. 30. Issue 5. pp. 897–907. (in

Russ.). 4. Subbotina N.N., Subbotin A.N. Prikladnaya matematika i mekhanika. 1975. Vol. 39. Issue 3.

pp. 397–406. (in Russ.). 5. Petrov, N.N. Izvestiya RAN, Teoriya i sistemy upravleniya. 2009. no. 2. pp. 38–44. (in Russ.). 6. Ukhobotov V.I. Metod odnomernogo proektirovaniya v lineynykh differentsial'nykh igrakh s in-

tegral'nymi ogranicheniyami: uchebnoe posobie (Method of one-dimensional design in linear differen-tial games with integral constraints: study guide). Chelyabinsk: Chelyabinskiy gosudarstvennyy univer-sitet. 2005. 124 p. (in Russ.).

7. Ukhobotov V.I. Modifikatsiya igry izotropnye rakety. Mnogokriterial'nye sistemy pri neoprede-lyennosti i ikh prilozheniyakh (Modification of the isotropic rockets game. Multi-criterion systems in indeterminedness and its applications) // Mezhvuzovskiy sbornik nauchnykh trudov: Chelyabinskiy go-sudarstvennyy universitet (Interuniversity collection of scientific papers: Chelyabinsk State University). – Chelyabinsk: izd-vo Bashkirskogo universiteta, 1988. pp. 123–130. (in Russ.).

8. Ukhobotov V.I., Zaytseva O.V. Odna zadacha impul'snogo presledovaniya pri ogranichennoy skorosti ubegayushchego (About one Problem of Impulse Pursuit at the Limited Velocity of the Escap-ing) Vestnik YuUrGU. Seriya «Komp'yuternye tekhnologii, upravlenie, radioelektronika». 2010. Issue 11. no. 2(178). pp. 29–32. (in Russ.).

9. Ljusternik L.A., Sobolev V.I. Jelementy funkcional'nogo analiza (Elements of functional analy-sis). Moscow, Nauka, 1965. 520 p. (in Russ.).

10. Filippov A.F. Differentsial'nye uravneniya s razryvnoy pravoy chast'yu (Differential equations with diffuse right member). Moscow: Nauka, 1985. 224 p. (in Russ.).

9 2013 .

1 Ukhobotov Viktor Ivanovich is Dr.Sc. (Physics and Mathematics), Professor, Head of the Theory of Control and Optimization Department, Chelyabinsk State University. E-mail: [email protected] 2 Troitsky Anton Aleksandorvich is Post-graduate Student, Theory of Control and Optimization Department, Chelyabinsk State University. E-mail: [email protected]


. « . . » 88

519.63

. . 1

. -

, .

: , .

-. ,

. , , , -

. -

[1]. , - [2].

[2, 3].

[4].

: ( , ) ( ) ,u W u v l v v Wα α α α∈ Α = ∀ ∈ ,l Wα ′∈ 1, 2,α = (1)

{ }1

12( ) ( ) : 0W W v W v Γ= Ω = ∈ Ω =

1 2(0; ) (0; ),b bΩ = × { } { }1 1 2 1 2[0; ] [0; ]b b b bΓ = × × ,

( , ) ( ( 1) )x x y yu v u v u v cuv dα αΩ

Α = + + − Ω

1 2, 0, 0.b b c> ≥, [5-7], (1) .

( ) , ,l v f vdα αΩ

= Ω Γ = ∂Ω 2 1\ ,Γ = Γ Γ

fα – , (1)

( 1) ,u cu fα α αα−Δ + − =1

0uα Γ = , 2

0unα

Γ∂

=∂

, 1, 2α = . (2)

1 – , , -



. .

2013, 5, 2 89

, (2) -, 1α = , , 2α = .

, - (1), (2)

Nuα

∈ : u fα ααΑ = , Nf

α∈ , 1, 2,α = (3)

Nvα

∈ : ,1 ,( ,..., )Nv v vα α α ′= , N m n= ⋅ , m , n ∈ ,

, , ( 1) , ,m j i i jv vα α− + = , 1,...,i m= , 1,...,j n= ,

, ,i jvα – ,

( )1 2( , ) ( 0,5) , ( 0,5)i jx y i h j h= − − , ,i j ∈ ,

1 1 /( 0,5),h b m= + 2 2 /( 0,5)h b n= + , , Aα N N× -

: 2

, 1, , , , 1, , , 11 1

, (( )( )m n

i j i j i j i ji j

u v u u v v hα αα α α α α

−+ +

= =Α = − − +

2, , 1 , , , , 1 , , 2 , , , , 1 2( )( ) ( 1) ) ,i j i j i j i j i j i ju u v v h cu v h hα α α α α αα−

+ ++ − − + −

, , 1 , , 1 0,i n i nu vα α+ += = 1,..., ,i m= , 1, , 1, 0,m j m ju vα α+ += = 1,..., .j n=

.,. –

, , 1 21

,N

k kk

u v u v h hα α α α

== , .Nu v

α α∀ ∈

fα Ω ,

, , ( , ),i j i jf f x yα = 1,..., ,i m= 1,..., .j n= (3) , . . 0,αΑ > 1, 2.α =

(3) 2Nu ∈ : Du f= , 2Nf ∈ , 3 0if − = , 1, 2,α = (4)

2Nv ∈ : 1 2

( , )v v v′ ′ ′= , , D 2 2N N× ,

11 1D A= = Α , 12D θ= , 21 0D = , 22 2D = Α = Α ,

x y y xθ ′ ′= ∇ ∇ − ∇ ∇ , x x y yA ′ ′= ∇ ∇ + ∇ ∇ , ,x y∇ ∇ N N×

1, 1, , , 1 , , 1 2

1 1, ( ( ) ) ,

m n

x i j i j i ji j

u v u u h v h hα α α α α

−+

= =∇ = − − , 1, , 1, 0,m j m ju vα α+ += = 1,..., ,j n=

1, , 1 , , 2 , , 1 2

1 1, ( ( ) ) ,

m n

y i j i j i ji j

u v u u h v h hα α α α α

−+

= =∇ = − − , , 1 , , 1 0,i n i nu vα α+ += = 1,..., .i m=

2N :


. « . . » 90

{ }1 2 21 ( , ) : 0 ,V v v v v′ ′ ′= = = { }1 2 1 22 ( , ) : 0 .V v v v Av vθ′ ′ ′= = − =

1. (4) u Vα∈ , 1, 2,α = .

C 2 2N N× , 11 22C C A= = , 12C θ= − , 21C θ= .

(4) : 2k Nu ∈ : 1 1( ) ( )k k k

kC u u Du fτ− −− = − − , k ∈ , 0kτ > 0u Vα∀ ∈ , 1, 2α = . (5) , (5)

NU ∈ : LL U F∗ = , NF ∈ ,

NW ∈ : LW F= , NF ∈ , NU ∈ : L U W∗ = , NW ∈ ,

x yL i′ ′= ∇ − ∇ , x yL L i∗ ′= = ∇ + ∇ ,

( )( )x y x yLL i i A iθ∗ ′ ′= ∇ − ∇ ∇ + ∇ = + ,

1 2 1 2( )( )A i u iu f ifθ+ + = + , :

1 2

1 2 2

,

,

Au u f

u Au f

θθ

− =

+ =1 2

1 2

,u iu U

f if F

+ =

+ =, , (5)

Cu f= , 1 2( , )u u u′ ′ ′= , 1 2( , )f f f′ ′ ′= . 2. (5) 1ku u− = , ku u= k∀ ∈ .

k ku u ψ= + {0}k∀ ∈ . 3. (5)

( )1 11 2 1 2(1 )k k k k

kA Aψ θψ τ ψ θψ− −− = − − ,

1α = , 1 1τ = , 2k Vψ ∈ k∀ ∈ , 2α = , 2

k Vψ ∈ {0}k∀ ∈ .

,v v vα ααα αΑ = Α , 1, 2α = .

1. [1; )δ∃ ∈ +∞ : A Aδ≤ Α ≤ ,

11 cδ λ−= + , 2

2 21 2 1,1 1,1 1,1,

2,25( ) (1,1) ( , ) lim ( , )9m n

b b m n m n πλ λ λ λ λ− −

→∞= + = ≤ < = ,

A [1]: 2 2

2 2,

1 2

2 1 (2 1) 2 1 (2 1)( , ) sin sin2(2 1) 2(2 1)i j

m i n jm nb m b n

π πλ + − + −= ++ +

.

4.2 2 1 1, , ,C A Aψ ψ ψ ψ ψ ψ= − 2Vψ∀ ∈ .

. , 1 2 0Aψ θψ− = , θ θ′ = − ,


. .

2013, 5, 2 91

1 2 2 2 1 2 2 2, , ,C A Aψ ψ θψ ψ ψ ψ ψ θ ψ ψ ψ′= + = + =

2 2 2 1 2 2 1 1 .A A Aψ ψ θψ ψ ψ ψ ψ ψ= − = − 1. ( )

1 (0;1)α∃ ∈ : 1 1 1 2 2, ,Av v Av vα≤ 2v V∀ ∈ .

( 11(1 )γ α −= − 1

1 1α γ −= − ), (1; )γ∃ ∈ +∞ : ( )2 2 2 2 1 1, , , ,Av v Cv v Av v Av vγ γ≤ = − 2v V∀ ∈ ,

. . 0A > , [8] 0C > . , 2 2

1 2 1 2( , ) ( ) ( ) 0x y y xCv v v v v v= ∇ − ∇ + ∇ + ∇ > 0v∀ ≠ , , . . 1 2( )( ) 0x yi v iv∇ + ∇ + ≠ 0v∀ ≠ . ,

1 (0; )λ−∃ ∈ +∞ : 1 11 1 2 1 2 2 2 20 , , , , 0Av v v v A v v v vθ θ θ λ θ θ− −≤ = = ≤ → , 1 2, 0h h → ,

2 21,1 1 2(1,1) 2,25( )b bλ λ − −= = + , . .

1 1, 0Av v → , 1 0v → , . . ( )2 2 0x y y xv vθ ′ ′= ∇ ∇ − ∇ ∇ → , 1 2, 0h h → 2v V∀ ∈ .

5.1

1 1 1 2 21

1 , (1 ) , ,A Aα ψ ψ α ψ ψ ψ ψα−

≤ − ≤ 2Vψ∀ ∈ .

. , 1 1 1 2 2, ,Aψ ψ α ψ ψ≤ 2Vψ∀ ∈

11 1 1 2 2 2 2 1 2 2

1

1 , (1 ) , , ,A A A Aα ψ ψ α ψ ψ ψ ψ α ψ ψα−

≤ − = − ≤

2 2 1 1, , ,A Aψ ψ ψ ψ ψ ψ≤ − = 2Vψ∀ ∈ . 6. (5) 1α = , 1k = , 1 1τ = ,

1 1 0 0 0 011 1

1, , ( 1) ,

1C Aαψ ψ ψ ψ γ ψ ψ

α≤ = −

−, 1 0 01

1 1 11

( 1)1A A A

αψ ψ γ ψα

≤ = −−

.

.1 0 0( )C Dψ ψ ψ− = − , 1 0 1 1 1 0 1 0 1 0 1

1 1 1 1( ), , , ,A A

C C D Aψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ− = = − = − ≤ ,

1 11 11 1 0 0 0 0 0 01 1

1 1 1 11 11 1

,, , , ,

1 1,

AC A A

C

ψ ψ α αψ ψ ψ ψ ψ ψ ψ ψα αψ ψ

≤ ≤ =− −

,

, 1 1 1 11 1 1 2 2, ,Aψ ψ α ψ ψ≤ , 1 1 1 1 1 1 1 1

2 2 1 1 1 2 2, , , (1 ) ,C A A Aψ ψ ψ ψ ψ ψ α ψ ψ= − ≥ − .

5. 1 1 1 1 0 01 11 1 1 1

1 1

1 , , ,1

A Cα αψ ψ ψ ψ ψ ψα α−

≤ =−

,

, . 7.

, , ,C Dψ ψ ψ ψ δγ ψ ψ≤ ≤ 2Vψ∀ ∈ . . ,

1 12 2 1 2 2

1 1, , (1 ) , ,D A Cα αψ ψ ψ ψ α ψ ψ ψ ψδ δ

− −≤ Α ≤ − ≤ =

2 2 1 1 2 2, , , ,A A Dψ ψ ψ ψ ψ ψ ψ ψ= − ≤ Α = ,


. « . . » 92

. 8. (5) 1α ≠ 1k ≠ ,

2 201kτ τ

γδ γδ< = = <

+, 1

1

1 1 11 1

q δ α γδδ α γδ

− + −= = <+ − +

,

2 1 1, ,k k k kC qψ ψ ψ ψ− −≤ , k ∈ .

. , 1 1( )k k kC Dψ ψ τ ψ− −− = − , 1k kTψ ψ −= , 1T E C Dτ −= − , 0T T ′= > ,

2

1 1 1 1,, , sup ,

,k k k k k k

V

CT TC T T C

Cψ

ψ ψψ ψ ψ ψ ψ ψ

ψ ψ− − − −

∈= ≤ =

2 2

2 21 1 1 1, ( ) ,

sup , sup ,, ,

k k k k

V V

CT C DTC C

C Cψ ψ

ψ ψ τ ψ ψψ ψ ψ ψ

ψ ψ ψ ψ− − − −

∈ ∈

−= = =

{ }2

22 21 1 1 1 2 1 1,

sup 1 , max 1 1 , ,,

k k k k k k

V

DC C q C

Cψ

ψ ψτ ψ ψ τ τγδ ψ ψ ψ ψ

ψ ψ− − − − − −

∈= − = − − = .

1. (5) 0ku u u u

α αα α α α αε

Α Α− ≤ − ,

1 (2 ) ( 1)τ α α τ= − + − , kτ τ= , { }\ 1k ∈ , ,

( ) 1(2 )( 1) ( 1) kq qαε α γ α γδ −≤ − − + − .

. 1α = , 5 1

1 11

1 , ,Aα ψ ψ ψ ψα−

≤ 2Vψ∀ ∈ ,

6 1 1 0 0 0 011 1 1 1

1, , ( 1) ,

1C A Aαψ ψ ψ ψ γ ψ ψ

α≤ = −

−.

2α = , 7 1

2 21 , ,Cα ψ ψ ψ ψ

δ− Α ≤ , 2 2, ,Cψ ψ ψ ψ≤ Α 2Vψ∀ ∈ .

. ,L L∗ , , (3) N -, 1,

(5) αε , 1( ln )O N αε −

.

1. , . . - / . . // . « , , ». –

2006. – . 7. – 7(62). – . 64–70. 2. , . . / . . //

. – , 1991. – 40 . ( . 11.11.91, 4232- 91)

3. , . . / . . , . . // -

. – 1993. – . 33, 1. – . 52–68.


. .

2013, 5, 2 93

4. , . . / . . // . – 2007. –

. 1 (35) – . 33–36. 5. , . . - /

. . , . . . – : - , 1979. – 235 . 6. , . .

/ . . . – .: - , 1950. – 256 . 7. , .- . / .- . . – .: , 1977. – 383 . 8. , . . / . . , . . . – .: ,

1984. – 320 .

UPDATING ITERATIVE FACTORIZATION FOR THE NUMERICAL SOLUTION OF TWO ELLIPTIC EQUATIONS OF THE SECOND ORDER IN RECTANGULAR AREA

A.L. Ushakov1

Two elliptic equations of the second order in rectangular area under the mixed regional conditions are considered. Their numerical decision is reduced by means of iterative factorization and fictitious continuations to the solution of systems of the linear algebraic equations with triangular matrixes, in which the quantity of nonzero elements in each line do not exceed three.

Keywords: iterative factorization, fictitious continuations.

References 1. Ushakov A.L. Modelirovanie iteratsionnoy faktorizatsii dlya ellipticheskoy kraevoy zadachi

vtorogo poryadka (Simulation of iterative factorization for the elliptic second-order boundary value problem). Vestnik YuUrGU. Seriya «Matematika, fizika, khimiya». 2006. Issue 7. no. 7(62). pp. 64–70. (in Russ.).

2. Ushakov A.L. Modifikatsiya metoda fiktivnykh komponent (Updating of the method of fictitious entries). Chelyabinsk: Chelyabinskiy gosudarstvennyy tekhnicheskiy universitet, 1991. 40 p. (Dep. v VINITI 11.11.91, no. 4232-V91). (in Russ.).

3. Matsokin A.M., Nepomnyaschikh S.V. The fictitious-domain method and explicit continuation operators. Computational Mathematics and Mathematical Physics. 1993. Vol. 33, no. 1. pp. 45–59.

4. Ushakov A.L. Izvestiya Chelyabinskogo nauchnogo tsentra. 2007. Issue 1(35). pp. 33–36. (in Russ.).

5. Oganesyan L.A., Rukhovets L.A. Variatsionno-raznostnye metody resheniya ellipticheskikh uravneniy (Variational differential solution method of elliptic equations). Erevan: Izd-vo AN ArmSSR, 1979. 235 p. (in Russ.).

6. Sobolev S.L. Nekotorye primeneniya funktsional'nogo analiza v matematicheskoy fizike (Some applications of functional analysis in mathematical physics). Leningrad: Izd-vo LGU, 1950. 256 p. (in Russ.).

7. Oben Zh.-P. Priblizhyennoe reshenie ellipticheskikh kraevykh zadach (Approximate answer to elliptic boundary value problems). Moscow: Mir, 1977. 383 p. (in Russ.). [Aubin J.-P. Approximation of elliptic boundary-value problems. New York: Wiley-Interscience, 1972. 360 p.]

8. Voevodin V.V., Kuznetsov Yu.A. Matritsy i vychisleniya (Matrices and calculations). Moscow: Nauka, 1984. 320 p. (in Russ.).

11 2013 .

1 Ushakov Andrei Leonidovich is Associate Professor, Differential and Stochastic Equations Department, South Ural State University. E-mail: [email protected]


. « . . » 94

538.91+544.77.022.823

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. , -, .

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. , -, 2–4 Å. -

-.

1. , . . / . . // . « .

. ». – 2009. – . 1. – 22(155). – . 39–44. 2. Inorganic nanoparticles: synthesis, applications, and perspectives / ed. by C.Altavilla, E.Ciliberto.

– Boca Raton; London: CRC, 2011. – 558 p. 3. Baletto, Fr. Structural properties of nanoclusters: Energetic, thermodynamic, and kinetic effects /

Fr. Baletto, R. Ferrando // Reviews of Modern Physics. – 2005. – Vol. 77. – P. 371–423. 4. , . . / . . , . . ,

. . , . . . – .: . – 1978. – 223 . 5. , . .

/ . . , . . , . . // . – 2012. – . 142. – . 5(11). – . 897–907.

6. , . . - / . . , . . , . . //

. « . . ». – 2011. – . 4. – 10(227). – . 61–66. 7. Vorontsov, A.G. Use of interatomic space for studying structure transition in melts / A.G. Vo-

rontsov, G.P. Vyatkin, A.A. Mirzoev // Journal of Non-Crystalline Solids. – 2007. – Vol. 353, 32–40. – P. 3510–3514.

8. Vorontsov, A.G. Structural changes of simple expanded liquids at high temperatures / A.G. Vo-rontsov, D.A. Kuts // Journal of Physics: Conference Series. – 2008. – Vol. 98. – P. 012004.

STRUCTURE OF HEATED METAL CLUSTERS DURING ITS PREPARATION

A.G. Vorontsov1

Computer simulation study of non-equilibrium metal clusters are performed. It is found that cluster contains three layers: core layer, gas-like outer layer and intermediate well ordered layer. The changes of the layers with cluster size and its internal energy are discussed.

Keywords: structure of metal clusters, condensation, molecular dynamics.

References 1. Vorontsov A.G. Modelirovanie zarozhdeniya i rosta metallicheskikh nanochastits v protsesse

kondensatsii iz peresyshchennogo para (Modeling of Nucleation and Grouth of Metal Nanoparticles During the Condensation from Vopour Phase). Vestnik YuUrGU. Seriya «Matematika. Mekhanika. Fizika». 2009. Issue 1. no. 22(155). pp. 39–44. (in Russ.).

2. Altavilla C. (ed.), Ciliberto E. (ed.) Inorganic nanoparticles: synthesis, applications, and per-spectives. Boca Raton; London: CRC, 2011. 558 p.

3. Baletto Fr., Ferrando R. Structural properties of nanoclusters: Energetic, thermodynamic, and ki-netic effects. Reviews of Modern Physics. 2005. Vol. 77. pp. 371–423. 1 Vorontsov Alexander Gennadevich is Cand.Sc. (Physics and Mathematics), Associate Professor, General and Theoretical Physics Department, South Ural State University. E-mail: [email protected]


. « . . » 98

4. Frishberg I.V., Kvater L.I., Kuz'min B.P., Gribovskiy S.V. Gazofaznyy metod polucheniya poroshkov (The gas-phase method for producing powders). oscow: Nauka. 1978. 223 p. (in Russ.).

5. Vorontsov A.G., Gel’chinskii B.R., Korenchenko A.E. Kinetics and energy states of nanoclusters in the initial stage of homogeneous condensation at high supersaturation degrees. JETP. Vol. 115. Issue 5. pp. 789–797.

6. Vorontsov A.G., Gel'chinskiy B.R., Korenchenko A.E. Statisticheskiy analiz stolknoveniy metal-licheskikh nanochastits pri vysokikh stepenyakh peresyshcheniya (Statistical analysis of metal nanop rticle collisions at high supersaturation rates). Vestnik YuUrGU. Seriya «Matematika. Mekhanika. Fizika». 2011. Issue 4. no. 10(227). pp. 61–66. (in Russ.).

7. Vorontsov A.G., Vyatkin G.P., Mirzoev A.A. Use of interatomic space for studying structure transition in melts. Journal of Non-Crystalline Solids. 2007. Vol. 353, no. 32–40. pp. 3510–3514.

8. Vorontsov A.G., Kuts D.A. Structural changes of simple expanded liquids at high temperatures. Journal of Physics: Conference Series. 2008. Vol. 98. pp. 012004.

22 2013 .


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– 1950. – . 14, 3. – . 129–133. 2. , . . / . . , . . . – .: , 1991. – 304 . 3. -

/ . . , . . , . . , . . // . . – 1983. – . 29, 4. – . 561–562.

4. , . . -: . … - . /

. . . – : - . . . - , 1995. – 43 . 5. , . .

/ . . , . . , . . // . – 2004. – . 97, 2. – . 8–12. 6. , . . -

/ . . , . . , . . // . . – 2011. – . 57, 5. – . 600–606.

7. , . . . : 10- . / . . , . . . – .: , 1987. – . VII. – 248 .

8. , . . / . . , . . , . . // . – 2010. – 3. – . 45–49.

9. / . . , . . , . . , . . // . – 1987. –

2. – . 47–52. 10. , . . : 3 . / . . , . . , . . -

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/ . . , . . , . . , . . // . « -

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. ., . ., : . .

2013, 5, 2 107

LAMB WAVES IN FERROMAGNETIC METALS: LASER EXITATIONAND ELECTROMAGNETIC REGISTRATION

S.Yu. Gurevich1, E.V. Golubev2, Yu.V. Petrov3

The processes of laser excitation and receiving of the Lamb waves in ferromagnetic metals in the range of from room temperature to magnetic phase transition (the Curie point). The basic parameters of the relationship between these processes were investigated.

Keywords: Lamb waves, laser excitation, the electromagnetic receiving, ferromagnetic metals.

References 1. Danilovskaya V.I. Temperaturnye napryazheniya v uprugom poluprostranstve, voznikayushchee

vsledstvie vnezapnogo nagrevaniya poverkhnosti (Thermal stresses in an elastic half-space due to sud-den heating surface). Prikladnaya matematika i mekhanika. 1950. Vol. 14, no. 3. pp. 129–133. (in Russ.).

2. Gusev V.E., Karabutov A.A. Lazernaya optoakustika (Laser Optical Acoustics). Moscow: Nauka, 1991. 304 p. (in Russ.).

3. Budenkov G.A., Gurevich S.Yu., Kaunov A.D., Maskaev A.F. Akusticheskiy zhurnal. 1983. Vol. 29, no. 4. pp. 561–562. (in Russ.).

4. Gurevich S.Yu. Osnovy teorii i prakticheskogo primeneniya vysokotemperaturnogo ul'trazvuko-vogo kontrolya ferromagnitnykh metalloizdeliy: avtoreferat dis. … d-ra tekhn. nauk (Basic theory and practical applications of high-temperature ultrasonic testing of ferromagnetic metal: thesis abstract of doctor of science). Chelyabinsk: Izd-vo Chelyab. gos. tekhn. un-ta, 1995. 43 p. (in Russ.).

5. Golubev E.V., Gurevich S.Yu., Petrov Yu.V. Laser Generation of Surface Acoustic Waves in a Ferromagnetic Metal. The Physics of Metals and Metallography. 2004. Vol. 97, no. 2. pp. 124–128.

6. Golubev E.V., Gurevich S.Yu., Petrov Yu.V. On the Theory of Lamb wave excitation in metals by pulsed laser radiation. Acoustical Physics. 2011. Vol. 57, no. 5, pp. 620–626.

7. Landau L.D., Lifshits E.M. Teoriya uprugosti. Teoreticheskaya fizika: v 10-ti t. (The theory of elasticity. Theoretical physics: in 10 volumes.). Moscow: Nauka, 1987. Vol. 7. 248 p.

8. Petrov Yu.V., Gurevich S.Yu., Golubev E.V. Experimental determination of parameters of laser-generated Lamb waves. Russian Journal of Nondestructive Testing. 2010. Vol. 46, no. 3. pp.185–188.

9. Gurevich S.Yu., Gal'tsev Yu.G., Kaunov A.V., Karimov R.S. Defektoskopiya. 1987. no. 2. pp. 47–52. (in Russ.).

10. Komarov V.A., Muzhitskiy V.F., Gurevich S.Yu. Teoriya fizicheskikh poley: v 3 t. (Field The-ory: in 3 volumes). Chelyabinsk–Izhevsk: Izdatel'stvo YuUrGU, 2000. Vol. 3. 627 p.

11. Gurevich S.Yu., Petrov Yu.V., Golubev E.V., Shulginov A.A. Eksperimental'naya ustanovka dlya registratsii impul'snykh elektromagnitnykh poley, vozbuzhdaemykh volnami Lemba v magnitopol-yarizovannoy metallicheskoy plastine (Experimental setup for registration pulsed electromagnetic fields, excited by Lamb waves in magnetically polarized metal plate). Vestnik YuUrGU. Seriya «Matematika. Mekhanika. Fizika». 2012. Issue 7. no. 34(293). pp. 146–149. (in Russ.).

30 2013 .

1 Gurevich Sergei Yurevich is Dr. Sc. (Engineering), Professor, General and Experimental Physics Department, South Ural State University. E-mail: [email protected] 2 Golubev Evgeniy Valerievich is Cand. Sc. (Physics and Mathematics), Associate Professor, General and Experimental Physics Department, South Ural State University. E-mail: [email protected] 3 Petrov Yuriy Vladimirovich is Cand. Sc. (Engineering), Associate Professor, General and Experimental Physics Department, South Ural State University.


. « . . » 108

669.112.227.1:538.915

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1. , / . . , . . , . . , . . . – .: . – 1982. – 632 c.

2. , . . / . . , . . , . . . – .: . – 1977. – C. 236.


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7. , . . / . . , . . // . – 1949. – 6. – . 755–760.

8. Kaufman, L. Thermodynamics of bainite reaction / L. Kaufman, S.V. Radcliffe, M. Cohen / De-composition of Austenite by Diffusional Processes: . . . – AIME, New York: Interscience Pub-lishers, 1962. – P. 313–352.

9. McLellan, R.B. A quassi-chemical treatment of interstitial solid solutions: it application to carbon austenite / R.B. McLellan, W.W. Dunn // J.Phys.Chem.Solids. – 1969. – V. 30, 11. – P. 2631–2637.

10. , . . / . . // -. . – 1965. – 2. – . 10–16. 11. Dunn, W.W. The Application of Quassi-chemical Solid Solution Model to Carbon Austenite /

W.W. Dunn, R.B. McLellan // Metall Trans. – 1970. – Vol. 1, 5. – P. 1263–1265. 12. , . . Fe-C -

- / . . , . . . // -. – 2004. – . 98, 6. – P. 3–7.

13. , . . / . . . – : , 1987. – C. 208.

14. , . . -- : . - .- . / . . .

– : . . . , 1997. – 55 . 15. Jiang, D.E. Carbon dissolution and diffusion in ferrite and austenite from first principles /

D.E. Jiang, E.A. Carter // Physical Review B. – 2003. – Vol. 67. – P. 214103. 16. Slane, J.A. Experimental and Theoretical Evidence for Carbon-Vacancy Binding in Austenite /

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18. . . Mn, Al, Si, - / . . , . . . // ,

. – 2012. – 10. – P. 24–28. 19. Timoshevskii, A.N. Ab-initio modeling of the short range order in Fe-N and Fe-C austenitic al-

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20. Gavriljuk, V.G. Change in the electron structure caused by C, N and H atoms in iron and its ef-fect on their interaction with dislocations / V.G. Gavriljuk, V.N. Shivanyuk, B.D. Shanina // Acta Mate-rialia. – 2005. – Vol. 53. – P. 5017–5024.

21. Schwarz, . Electronic structure calculations of solids using the WIEN2k package for material science / K. Schwarz, P. Blaha, G.K.H. Madsen // Computer Physics Communications. – 2002. – Vol. 147. – P. 71–76.

22. The Role of the Nature of Magnetic Coupling on the Martensitic Transformation in Fe-Ni / M. Acet, E.F. Wassermann, K. Andersen et al. // Journal de Physique IV France. – 1997. – Vol. 7,

C5. – P. 401–404. 23. Medvedeva, N.I. Magnetism in bcc and fcc Fe with carbon and manganese / N.I. Medvedeva,

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troduction/ S. Cottenier, 2004. [http://www.wien2k.at/reg_user/textbooks/DFT_and_LAPW-2_cottenier.pdf]


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25. , . . -- / . . , . . , . . . // . « -

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/ M.C. , . . , . . // . « -». – 2010. – . 14. – 13(189). – P. 67–71.

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8. – P. 2239–2245. 29. Lobo, J.A. Thermodynamics of carbon in austenite and Fe-Mo austenite / J.A. Lobo,

G.H. Geiger // Metallurgical Transactions A. – 1976. – Vol. 7, 8. – P. 1359–1364.

AB-INITIO SIMULATION OF INFLUENCE OF SHORT-RANGE ORDERING CARBON IMPURITIES ON THE ENERGY OF THEIR DISSOLUTION IN THE FCC-IRON

Ya.M. Ridnyi1, A.A. Mirzoev2, D.A. Mirzaev3

The first principle simulation of equilibrium structure and FCC-iron characteristics was carried out by the software package WIEN2k. The optimal parameters which allow building the most accurate model were generated. Energies of dissolution of carbon atoms, their relative positions and the contri-bution of the elastic effects to the energy of system were calculated for non-magnetic (NM) and double-layer antiferromagnetic states (AFMD) of FCC-iron.

Keywords: first principle simulation, FCC-iron, carbon impurity, WIEN2k.

References 1. Umanskiy Ya.S., Skakov Yu.A., Ivanov A.N., Rastorguev L.N. Kristallografiya, rentgenografiya

i elektronnaya mikroskopiya (Crystallography, radiographic imaging and electronic microscopy). Moscow: Metallurgiya. 1982. 632 p. (in Russ.).

2. Kurdyumov G.V., Utevskiy L.M., Entin R.I. Prevrashcheniya v zheleze i stali (Transformations in iron and steel). Moscow: Nauka. 1977. 236 p. (in Russ.).

3. Khachaturyan A.G. Uglerod v martensite stali. Nesovershenstva kristallicheskogo stroeniya i martensitnye prevrashcheniya (Carbon in martensite of steel. Imperfections of crystalline construction and martensitic transformations). Moscow: Nauka. 1971. pp. 34–45. (in Russ.).

4. Mogutnov B.M., Tomilin N.A., Shvartsman L.A. Termodinamika zhelezo-uglerodistykh splavov (Thermodynamics of iron-carbon alloys). Moscow: Metallurgiya. 1972. 328 p. (in Russ.).

5. Dünwald H., Wagner C. Thermodynamische Untersuchungen zum System Eisen–Kohlenstoff–Sauerstoff. Z. Anorg. Allgem Chem. 1931. Vol. 199. pp. 321–346. DOI: 10.1002/zaac.19311990132

6. Darken L.S., Smith R.P. Appendix to the paper by Smith R.P. Equilibrium of iron – carbon alloys / // J.Amer.Chem. Soc. 1946. no. 7. pp. 1163–1175.

7. Temkin M.I., Shvartsman L.A. Zhurnal fizicheskoy khimii. 1949. no. 6. pp. 755–760. (in Russ.). 8. Kaufman L.R., Radcliffe S.V., Cohen M. Thermodynamics of the Bainite Reaction. Decomposi-

tion of Austenite by Diffusional Processes. AIME, New York: Interscience Publishers, 1962. pp. 313–352.

1 Ridnyi Yaroslav Maksimovich is Student, South Ural State University. E-mail: [email protected] 2 Mirzoev Aleksander Aminulaevich is Dr. Sc. (Physics and Mathematics), Professor, General and Theoretical Physics Department, South Ural State University. E-mail: [email protected] 3 Mirzaev Dzhalal Aminulovich is Dr. Sc.(Physics and Mathematics), Professor, Department of Physical Metallurgy and Material Science, Southern Ural State University. E-mail: [email protected]


. « . . » 116

9. McLellan R.B., Dunn W.W. A quassi-chemical treatment of interstitial solid solutions: it applica-tion to carbon austenite. J.Phys.Chem.Solids. 1969. Vol. 30. no. 11. pp. 2631–2637.

10. Kozheurov V.A. Izvestiya VUZov. Chernaya metallurgiya. 1965. no. 2. pp. 10–16. (in Russ.). 11. Dunn W.W., McLellan R.B. The Application of Quassi-chemical Solid Solution Model to Car-

bon Austenite. Metall Trans. 1970. Vol. 1, no. 5. pp. 1263–1265. 12. Bol'shov L.A., Suslov V.N. Fizika metallov i metallovedenie. 2004. Vol. 98, no. 6. pp. 3–7. (in

Russ.). 13. Gavrilyuk V.G. Raspredelenie ugleroda v stali (Carbon distribution in steel). Kiev: Naukova

Dumka. 1987. p. 208. 14. Nadutov V.M. Mezhatomnoe vzaimodeystvie i raspredelenie atomov vnedreniya v zhelezo-

azotistykh i zhelezo-uglerodistykh splavakh. Avtoreferat na soiskanie uchyenoy stepeni doktora fiziko-matematicheskikh nauk (Interatomic interaction and distribution of interstitial atoms in iron-nitrogenuous and iron-carbon alloys. Dr. phys. and math. sci. synopsis of diss.). Kiev: IMF NAN Ukrainy im. G.V. Kurdyumova, 1997. 55 p.

15. Jiang D.E., Carter E.A. Carbon dissolution and diffusion in ferrite and austenite from first prin-ciples. Physical Review B. 2003. Vol. 67. pp. 214103.

16. Slane J.A., Wolverton C., Gibala R. Experimental and Theoretical Evidence for Carbon-Vacancy Binding in Austenite. Metallurgical and Materials Transactions A. 2004. Vol. 35, no. 8. pp. 2239–2245.

17. Boukhvalov D.W., Gornostyrev Y.N., Katsnelson M.I., Lichtenstein A.I. Magnetism and Local Distortions near Carbon Impurity in g-Iron. Physical Review Letters. 2007. Vol. 99. p. 247205.

18. Ivanovskiy L.I., Medvedeva N.I. Fazovye perekhody, uporyadochennye sostoyaniya i novye ma-terialy. 2012. no. 10. pp. 24–28. (in Russ.).

19. Timoshevskii A.N., Yablonovskii S.O. Functional Materials. 2011. Vol. 18, no. 4. pp. 517–522. 20. Gavriljuk V.G., Shivanyuk V.N., Shanina B.D. Acta Materialia. 2005. Vol. 53. pp. 5017–5024. 21. Schwarz K., Blaha P., Madsen G.K.H. Computer Physics Communications. 2002. Vol. 147. pp.

71–76. 22. Acet M., Wassermann E.F., Andersen K., Murani A., Scharpff O. Journal de Physique IV

France. 1997. Vol. 7, no. C5. p. 401–404. 23. Medvedeva N.I., Aken D.V., Medvedeva J.E. Journal of Physics: Condensed Matter. 2010.

Vol. 22. pp. 316002. 24. Cottenier S. Density Functional Theory and the family of (L)APW-methods: a step-by-step in-

troduction. 2004. Available at: http://www.wien2k.at/reg_user/textbooks/DFT_and_LAPW-2_cottenier.pdf

25. Ursaeva A.V., Ruzanova G.E., Mirzoev A.A. Vybor optimal'nykh parametrov dlya postroeniya maksimal'no tochnoy modeli OTsK-zheleza (Selection of optimal parameters for formation the most ac-curate model of BCC iron). Vestnik YuUrGU. Seriya «Matematika. Mekhanika. Fizika». 2010. Issue 2. no. 9(185). pp. 97–101. (in Russ.).

26. Rakitin M.C., Mirzoev A.A., Mirzaev D.A. Izmenenie elektronnoy struktury -zheleza, soderz-hashchego vnedrennye atomy vodoroda (Change of electronic structure in iron containing interstitial atoms of hydrogen) // Vestnik YuUrGU. Seriya: Metalurgiya. 2010. Issue 14. no. 13(189). pp. 67–71. (in Russ.).

27. Onink M., Brakman C.M., Tichelaar F.D., Mittemeijer E.J., Van der Zwaag S., Root J.H., Kon-yer N.B. Scripta Metallurgica Et Materialia. 1993. Vol. 29, no. 8. pp. 1011–1016.

28. Slane J.A., Wolverton C., Gibala R. Metallurgical and Materials Transactions A. 2004. Vol. 35, no. 8. pp. 2239–2245.

29. Lobo J.A., Geiger G.H. Metallurgical Transactions A. 1976. Vol. 7, no. 8. pp. 1359–1364. 11 2013 .


2013, 5, 2 117

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.

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

« . . ». – 2009. – . 1. – 22(155). – . 91–96. 2. ,

/ . . , . . , . . , . . // . « . . ». – 2010. – . 3. – 30(206). – . 95–100.

3. , . . / . . , . . , . . // . « . .

». – 2011. – . 4. – 10(227). – . 72–76.

A METHOD FOR MONITORING SMALL SURFACE MOVEMENTS OF A SESSILE DROP DURING EVAPORATION

V.L. Ushakov1, G.P. Pyzin2, V.P. Beskachko3

Interference methods allow researchers to obtain the information about normal displacements of a phase surface with high resolution and in real time. This opportunity was realized to observe time changes of the height of evaporating drop accurate to a fraction of the wavelength of visible light. We consider the possibilities of studying processes on a drop surface, that accompany the evaporation and are small-scale in terms of space and time.

Keywords: interfacial phenomena, evaporation, drop shape, sessile drop method, interferometric methods.

References 1. Ushacov V.L., Pyzin G.P., Rechkalov V.G., Beskachko V.P. Opredelenie radiusa krivizny v ver-

shine lezhashchey kapli po nablyudeniyam kartin interferentsii (The determination of the curvature ra-dius at the top of a sessile drop based on observations of interference patterns). Vestnik YuUrGU. Seriya «Matematika. Mekhanika. Fizika». 2009. Issue 1. no. 22(155). pp. 91–96. (in Russ.).

2. Pyzin G.P., Ushacov V.L., Rechkalov V.G., Beskachko V.P. Otsenka parametrov interfero-gramm, neobkhodimykh dlya opredeleniya radiusa krivizny v vershine lezhashchey kapli (The estima-tion of parameters of the interference patterns needed to determine the radius of curvature at the top of a sessile drop). Vestnik YuUrGU. Seriya «Matematika. Mekhanika. Fizika». 2010. Issue 3. no. 30(206). pp. 95–100. (in Russ.).

3. Korenchenko A.E., Ilimbaeva A.G., Beskachko V.P. Chislennoe modelirovanie svobodnykh kolebaniy lezhashchey kapli. (The numerical simulation of sessile drop free oscillations). Vestnik YuUrGU. Seriya «Matematika. Mekhanika. Fizika». 2011. Issue 4. no. 10(227). pp. 72–76. (in Russ.).

1 2013 .

1 Ushakov Vladimir Leonidovich is Assistant, General and Theoretical Physics Department, South Ural State University. 2 Pyzin Georgii Petrovich is Cand. Sci. (Engineering), Associate Professor, General and Theoretical Physics Department, South Ural State University. E-mail: [email protected] 3 Beskachko Valeriy Petrovich is Dr. Sc. (Physics and Mathematics), Professor, General and Theoretical Physics Department, South Ural State University.


2013, 5, 2 123

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2013, 5, 2 125

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. ., . .

2013, 5, 2 127

PROBLEMS OF VIBRATION SAFETY OF OPERATOR OF INDUSTRIAL TRACTOR

I.Ya. Berezin1, Yu.O. Petrenko2

Registration of dynamic processes in the context of real operation gave the researches opportunity to find the source of low-frequency vibration impact driven by rotation of caterpillar track in supporting frame of by-pass which is, according to biomedical researches, the most unfavorable for the body of the operator. The linear discrete model of the “Antivibration seat – Pelvis – Body – Head” system is studied as well as the impact of the dynamic parameters of antivibration seat on the level of vibrational load of the operator.

Keywords: progressive wave, haphazard vibrational process, discrete linear model, transfer func-tion, spectral density of vibrational acceleration, sanitary standards.

References 1. Chelomey V.N. (ed.) Vibratsii v tekhnike: spravochnik. T. 6. Zashchita ot vibratsii i udarov (Vi-

bration technology: a handbook. Vol. 6. Protection from vibration and shocks). Moscow: Mashinostroe-nie, 1981. 456 p. (in Russ.).

2. GOST 27259-87 (ISO 7096-82) Mashiny zemleroynye. Siden'e operatora. Peredavaemaya vi-bratsiya. (Earth-moving machinery. The operator's seat. Transmitted vibration.) (in Russ.).

3. GOST 12.1.012-90. Vibratsionnaya bezopasnost'. Obshchie trebovaniya. (GOST 12.1.012-90. Vibration safety. General requirements.) (in Russ.).

4. Khripunov D.V. Metody otsenki vibronagruzhennosti promyshlennogo traktora so storony gusen-ichnogo dvizhitelya: avtoreferat dis. … kand. tekhn. nauk (Methods for assessing vibration loading the tractor from the caterpillar drive: synopsis of the cand. tech. sci. diss.). Chelyabinsk, 2002. 22 p. (in Russ.).

5. Panovko G.Ya. Postroenie dinamicheskikh modeley tela cheloveka-operatora pri vibratsion-nykh vozdeystviyakh: avtoreferat dis. … kand. tekhn. nauk (Building a dynamic model of the body of the hu-man operator with vibration impacts: synopsis of the cand. tech. sci. diss.). Mashinovedenie, 1973. 28 p. (in Russ.).

6. Panovko G.Ya., Traktovenko B.G. Diskretnaya kolebatel'naya model' tela cheloveka i opredele-nie ee parametrov (Discrete vibrational model of the human body and the determination of its parame-ters). Mashinovedenie. 1974. no. 4. pp. 16–20. (in Russ.).

11 2013 .

1 Berezin Igor Yakovlevich is Dr. Sc. (Engineering), Professor, Applied Mechanics, Dynamics and Strength of Machines Department, South Ural State University. 2 Petrenko Yulia Olegovna is Post-graduate Student, Applied Mechanics, Dynamics and Strength of Machines Department, South Ural State University. E-mail: [email protected]


. « . . » 128

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− . -:

1 , 14.132.21.1396, 14. 37.21.1633 , 12-02-31448 _ . 2 – - , , , -

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2013, 5, 2 129

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ϕ

ϕ

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1, 1,2,1, 1,

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N NN

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N NN

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+,

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− +.

(1) :

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i iϕ ϕϕΣ + −= +

− +,

( ) 1 1 cos, , 2 .

sinxi i

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ϕϕ

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:


. « . . » 130

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2Retg 2 ,

1

χθ

χ=

−ε :

( ) ( )2

2Imsin 2 .

1

χε

χ=

+ (3)

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1 tgϕθϕ

=−

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, . -

, , -.

, π . -:

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− +, :

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cosxi i

y

Ee e

Ei iϕ π ϕ ϕ

ϕ+ + −+ = =

− (4)

(4) , sin cos tgχ ϕ ϕ ϕ= = , , :

( )tg 2 tg 2( ) ,2πθ ϕ= − +

, ( 2)θ ϕ π= − + , . . -. , ,

. , , -

, . . . , -

. , -

, , , -, , -

.

1. Biss, D.P. Polarization-vortex-driven second-harmonic generation / D.P. Biss, T.G. Brown // Op-tics Letters. – 2003. – Vol. 28, 11. – P. 923–925.

2. Second- and third-harmonic generation with vector Gaussian beams / S. Carrasco, B.E. Saleh, M.C. Teich, J.T. Fourkas // J. Opt. Soc. Am. B. – 2006. – Vol. 23, 10. – P. 2134–2141.

3. Biss, D.P. Dark-field imaging with cylindrical-vector beams / D.P. Biss, K.S. Youngworth, T.G. Brown // Appl. Opt. – 2006. – Vol. 45, 3. – P. 470–479.

4. Zhan, Q. Radiation forces on a dielectric sphere produced by highly focused cylindrical vector beams / Q. Zhan // J. Opt. A: Pure Appl. Opt. – 2003. – Vol. 5, 3. – P. 229–232.

5. Zhan, Q. Trapping metallic Rayleigh particles with radial polarization / Q. Zhan // Opt. Express. – 2004. – Vol. 12, 15. – P. 3377–3382.


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2013, 5, 2 131

6. Kozawa, Y. Optical trapping of micrometer-sized dielectric particles by cylindrical vector beams / Y. Kozawa, S. Sato // Opt. Express. – 2010. – Vol. 18, 10. – P. 10828–10833.

7. Zhang, Y. Trapping two types of particles using a double-ring-shaped radially polarized beam / Y. Zhang, D. Biaofeng, S. Taikei // Phys. Rev. A. – 2010. – Vol. 81, 2. – P. 023831.

8. Rang, H. Enhanced photothermal therapy assisted with gold nanorods using a radially polarized beam / H. Rang, . Jia, J. Li // Applied Physics Letters. – 2010. – Vol. 96, 6. – P. 063702.

9. Meier, M. Material processing with pulsed radially and azimuthally polarized laser radiation / M. Meier, V. Romano, . Feurer // Applied Physics A: Materials Science & Processing. – 2007. – Vol. 86, 3. – P. 329–334.

10. Kraus, M. Microdrilling in steel using ultrashort pulsed laser beams with radial and azimuthal polarization / M. Kraus, M.A. Ahmed, A. Michalowski // Opt. Express. – 2010. – Vol. 18, 21. – P. 22305.

11. Niziev, V.G. Influence of beam polarization on laser cutting efficiency / V.G. Niziev, A.V. Nesterov // Journal of Physics D: Applied Physics. – 1999. – Vol. 32, 13. – P. 1455.

12. Steinhauer, L.C. A new approach for laser particle acceleration in vacuum / L.C. Steinhauer, W. D. Kimura // J. Appl. Phys. – 1992. – Vol. 72, 8. – P. 3237.

13. Wong, L.J. Direct acceleration of an electron in infinite vacuum by a pulsed radially-polarized laser beam / L.J. Wong, F.X. Kartner // Opt. Express. – 2010. – Vol. 18, 24. – P. 25035.

14. Bochkareva, S.G. Vacuum electron acceleration by a tightly focused, radially polarized, relativ-istically strong laser pulse/ S.G. Bochkareva, K.I. Popov, V.Yu. Bychenkova // Plasma Physics Reports. – 2011. – Vol. 37, 7. – P. 603.

15. Demonstration of an elliptical plasmonic lens illuminated with radially-like polarized field / G.M. Lerman, A. Yanai, N. Ben-Yosef, U. Levy // Opt. Express. – 2010. – Vol. 18, 10. – P. 10871.

16. Zhan, Q. Cylindrical vector beams: from mathematical concepts to applications / Q. Zhan // Ad-vances in Optics and Photonics. – 2009. – Vol. 1. – Issue 1. – P. 1–57.

17. / . . , . . , . . , . . // . – 1995.

– . 107, 5. – . 1464–1472. 18. ., . . . .: , 1978. – 341 c. 19. , A. / A. , . . – .: , 1987. – 656 .

FIBER AND INTERFERENTIAL METHOD OF OBTAINING NON-HOMOGENEOUS POLARIZED BEAM

M.V. Bolshakov1, A.V. Guseva2, N.D. Kundikova3, I.I. Popkov4

The process of distribution of circularly polarized radiation in small-mode optical fiber is repre-sented in this paper. The interferential method of obtaining non-homogeneous polarized beams is de-scribed.

Keywords: polarization, optical fiber, interference.

References 1. Biss D.P., Brown T.G. Polarization-vortex-driven second-harmonic generation. Optics Letters.

2003. Vol. 28, no. 11. pp. 923–925.

1 Bolshakov Maxim Vyacheslavovich is Cand. Sc. (Physics and Mathematics), associate professor, Optics and Spectroscopy Department, South Ural State University. E-mail: [email protected] 2 Guseva Anna Valentinovna is Undergraduate Student, Optics and Spectroscopy Department, South Ural State University. E-mail: [email protected] 3 Kundikova Nataliya Dmitrievna is Dr. Sc. (Physics and Mathematics), Professor, Dean of Physics Faculty, Join Nonlinear Optics Laboratory of IEF RAS, Optics and Spectroscopy Department, South Ural State University. E-mail: [email protected] 4 Popkov Ivan Igorevich is Assistant, Optics and Spectroscopy Department, South Ural State University. E-mail: [email protected]


. « . . » 132

2. Carrasco S., Saleh B.E., Teich M.C., Fourkas J.T. Second- and third-harmonic generation with vector Gaussian beams. J. Opt. Soc. Am. B. 2006. Vol. 23, no. 10. pp. 2134–2141.

3. Biss D.P., Youngworth K.S., Brown T.G. Dark-field imaging with cylindrical-vector beams / Appl. Opt. 2006. Vol. 45, no. 3. pp. 470–479. [http://dx.doi.org/10.1364/AO.45.000470]

4. Zhan Q. Radiation forces on a dielectric sphere produced by highly focused cylindrical vector beams. J. Opt. A: Pure Appl. Opt. 2003. Vol. 5, no. 3. pp. 229–232.

5. Zhan Q. Trapping metallic Rayleigh particles with radial polarization. Opt. Express. 2004. Vol. 12, no. 15. pp. 3377–3382.

6. Kozawa Y., Sato S. Optical trapping of micrometer-sized dielectric particles by cylindrical vector beams. Opt. Express. 2010. Vol. 18, no. 10. pp.10828–10833.

7. Zhang Y., Biaofeng D., Taikei S. Trapping two types of particles using a double-ring-shaped ra-dially polarized beam. Phys. Rev. A. 2010. Vol. 81, no. 2. p.023831.

8. Rang H., Jia ., Li J. Enhanced photothermal therapy assisted with gold nanorods using a radially polarized beam. Applied Physics Letters. 2010. Vol. 96, no. 6. p. 063702.

9. Meier M., Romano V., Feurer . Material processing with pulsed radially and azimuthally polar-ized laser radiation. Applied Physics A: Materials Science & Processing. 2007. Vol. 86, no. 3. pp. 329–334.

10. Kraus M., Ahmed M.A., Michalowski A. Microdrilling in steel using ultrashort pulsed laser beams with radial and azimuthal polarization. Opt. Express. 2010. Vol. 18, no. 21. p. 22305.

11. Niziev V.G., Nesterov A.V. Influence of beam polarization on laser cutting efficiency. Journal of Physics D: Applied Physics. 1999. Vol. 32, no. 13. p. 1455.

12. Steinhauer L.C., Kimura W.D. A new approach for laser particle acceleration in vacuum. J. Appl. Phys. 1992. Vol. 72, no. 8. p. 3237.

13. Wong L.J., Kartner F.X. Direct acceleration of an electron in infinite vacuum by a pulsed ra-dially-polarized laser beam. Opt. Express. 2010. Vol. 18, no. 24. p.25035.

14. Bochkareva S.G., Popov K.I., Bychenkova V.Yu. Vacuum electron acceleration by a tightly fo-cused, radially polarized, relativistically strong laser pulse. Plasma Physics Reports. 2011. Vol. 37, no. 7. p. 603.

15. Lerman G.M., Yanai A., Ben-Yosef N., Levy U. Demonstration of an elliptical plasmonic lens illuminated with radially-like polarized field. Opt. Express. 2010. Vol. 18, no. 10. pp. 10871.

16. Zhan Q. Cylindrical vector beams: from mathematical concepts to applications. Advances in Optics and Photonics. 2009. Vol. 1. Issue 1. pp. 1–57. [http://dx.doi.org/10.1364/AOP.1.000001]

17. Darsht M.Ya., Zel'dovich B.Ya., Kataevskaya I.V., Kundikova N.D. Formation of an isolated wavefront dislocation. JETP. Vol. 80, no. 5. p. 817.

18. Dzherard A., Berch D.M. Vvedenie v matrichnuyu optiku (Introduction into matrix optics). Moskva, Mir, 1978. 341 p. (in Russ.). [Gerrard A., Burch J.M. Introduction to Matrix Methods in Op-tics. John Wiley & Sons, New York, 1975. 356 p.]

19. Snayder A., Lav D. Teoriya opticheskikh volnovodov (Theory of optical waveguide). Moscow: Radio i svyaz', 1987. 656 p. (in Russ.). [Snyder A.W., Love J.D. Optical Waveguide Theory. Springer, 1983. 734 p.]

21 2013 .


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1. DiVincenzo, D.P. Quantum computation / D.P. DiVincenzo // Science. – 1995. – Vol. 270, 5234. – P. 255–261.

2. Kilin, S.Ya. Quantum information / S.Ya. Kilin // Physics – Uspekhi. – 1999. – Vol. 42, 5. – P. 435–452.

3. Experimental Realization of Teleporting an Unknown Pure Quantum State via Dual Classical and Einstein-Podolsky-Rosen Channels / D. Boschi, S. Branca, F. De Martini et al. // Physical Review Let-ters. – 1998. – Vol. 80, 6. – P. 1121–1125.

4. Molina-Terriza, G. Management of the Angular Momentum of Light: Preparation of Photons in Multidimensional Vector States of Angular Momentum / G. Molina-Terriza, J.P. Torres, L. Torner // Physical Review Letters. – 2001. – Vol. 88, 1. – P. 013601.

5. Spin-to-orbital momentum conversion via electro-optic Pockels effect in crystals / I. Skab, Yu. Vasylkiv, I. Smaga, R. Vlokh // Physical Review A. – 2011. – Vol. 84, 4. – P. 043815.

6. Brasselet, E. Electrically controlled topological defects in liquid crystals as tunable spin-orbit en-coders for photons / E. Brasselet, C. Loussert // Optics Letters. – 2011. – Vol. 36, 5. – P. 719–721.

7. Quantum interference by coherence transfer from spin to orbital angular momentum of photons / E. Nagali, F. Sciarrino, L. Sansoni et al. // SPIE Proceedings-Quantum Cryptography and Quantum In-formation Processing . – 2009. – Vol. 7355. – P. 735507

8. / . . , . . , . . , . . // . – 1995.

– . 107, 5. – .1464–1472. 9. , A. / A. , . . – .: , 1987.

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. ., . ., . ., . .

2013, 5, 2 137

TRANSFORMATION OF THE SPIN MOMENT INTO ORBITAL MOMENT IN LASER BEAM

M.V. Bolshakov1, A.V. Guseva2, N.D. Kundikova3, I.I. Popkov4

The process of distribution of circularly polarized radiation in small-mode optical fiber is studied in this paper. It was experimentally proven that the transformation of spin moment of ±1 beam into orbital moment with topological charge ±1.

Keywords: spin moment, orbital moment, optical fiber, topological charge.

References 1. DiVincenzo D.P. Quantum computation. Science. 1995. Vol. 270, no. 5234. pp. 255–261. 2. Kilin S.Ya. Quantum information. Physics – Uspekhi. 1999. Vol. 42, no. 5. pp. 435–452. 3. Boschi D., Branca S., De Martini F., Hardy L., Popescu S. Experimental Realization of Teleport-

ing an Unknown Pure Quantum State via Dual Classical and Einstein-Podolsky-Rosen Channels. Physi-cal Review Letters. 1998. Vol. 80, no. 6. pp. 1121–1125.

4. Molina-Terriza G., Torres J.P., Torner L. Management of the Angular Momentum of Light: Preparation of Photons in Multidimensional Vector States of Angular Momentum. Physical Review Let-ters. 2001. Vol. 88, no. 1. p. 013601.

5. Skab I., Vasylkiv Yu., Smaga I., Vlokh R. Spin-to-orbital momentum conversion via electro-optic Pockels effect in crystals. Physical Review A. 2011. Vol. 84, no. 4. p. 043815.

6. Brasselet E., Loussert C. Electrically controlled topological defects in liquid crystals as tunable spin-orbit encoders for photons. Optics Letters. 2011. Vol. 36, no. 5. p. 719–721.

7. Nagali E., Sciarrino F., Sansoni L., De Martini F., Marrucci L., Piccirillo B., Karimi E., Santamato E. Quantum interference by coherence transfer from spin to orbital angular mo-mentum of photons. SPIE Proceedings-Quantum Cryptography and Quantum Information Processing. 2009. Vol. 7355. p. 735507

8. Darsht M.Ya., Zel'dovich B.Ya., Kataevskaya I.V., Kundikova N.D. Formation of an isolated wavefront dislocation. JETP. Vol. 80, no. 5. p. 817.

9. Snayder A., Lav D. Teoriya opticheskikh volnovodov (Theory of optical waveguide). Moscow: Radio i svyaz', 1987. 656 p. (in Russ.). [Snyder A.W., Love J.D. Optical Waveguide Theory. Springer, 1983. 734 p.]

21 2013 .

1 Bolshakov Maxim Vyacheslavovich is Cand. Sc. (Physics and Mathematics), associate professor, Optics and Spectroscopy Department, South Ural State University. E-mail: [email protected] 2 Guseva Anna Valentinovna is Undergraduate Student, Optics and Spectroscopy Department, South Ural State University. E-mail: [email protected] 3 Kundikova Nataliya Dmitrievna is Dr. Sc. (Physics and Mathematics), Professor, Dean of Physics Faculty, Join Nonlinear Optics Laboratory of IEF RAS, Optics and Spectroscopy Department, South Ural State University. E-mail: [email protected] 4 Popkov Ivan Igorevich is Assistant, Optics and Spectroscopy Department, South Ural State University. E-mail: [email protected]


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1. Saffman, M. Mode multiplexing and holographic demultiplexing communication channels on a multimode fiber / M. Saffman, D.Z. Anderson // Optics Letters. – 1991. – Vol. 16, 5. – . 300–302.

2. Stuart, H.R. Dispersive multiplexing in multimode optical fiber / H.R Stuart // Science. – 2000. – Vol. 289, 5477. – . 281–283.

3. Tai, A.M. Transmission of two-dimensional images though a single fiber by wavelength-time en-coding / A.M. Tai, A.A. Friesem // Optics Letters. – 1983. – Vol. 8, 1. – . 57–59.

4. Complete modal decomposition for optical fibers using CGH-based correlation filters / T. Kaiser, D. Flamm, S. Schroter, M. Duparre // Optics Express. – 2009. – Vol. 17, 11. – . 9347–9356.

5. Complete modal decomposition for optical waveguides / O. Shapira, A.F. Abouraddy, J.D. Joannopoulos, Y. Fink // Physical Review Letters. – 2005. – Vol. 94, 14. – . 143902.

6. , . . , - / . . , . . , . . //

. « . . ». – 2012. – . 7. – 34(293). – . 138-141. 7. Bolshakov, M. Optical effects connected with coherent polarized light propagation through a

step-index fiber / M. Bolshakov, A. Ershov, N. Kundikova // Fiber Optic Sensors: . . . – In-Tech, 2012. – P. 249–274.

8. , . / . , . . – .- .: , 1951. – . 1. – 476 .

9. Fienup, J.R. Phase retrieval algorithms: comparison / J.R. Fienap // Applied Optics. – 1982. – Vol. 21, 15. – . 2758–2769.

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. « . . » 142

EXPERIMENTAL DETERMINATION OF MULTIMODE OPTICAL FIBER RADIATION MODE COMPOSITION

M.V. Bolshakov1, M.A. Komarova2, N.D. Kundikova3

We proposed the method of the optical fiber mode composition determination of the radiation propagating in a multimode optical fiber with a stepwise refractive index profile. The method is based on the field decomposition by non-orthogonal modes. Complex coefficients of optical fiber radiation modes were calculated for experimental data.

Keywords: optical fiber, optical fiber modes.

References 1. Saffman M., Anderson D.Z. Mode multiplexing and holographic demultiplexing communication

channels on a multimode fiber. Optics Letters. 1991. Vol. 16, no. 5. pp. 300–302. 2. Stuart H.R. Dispersive multiplexing in multimode optical fiber. Science. 2000. Vol. 289,

no. 5477. pp. 281–283. 3. Tai A.M., Friesem A.A. Transmission of two-dimensional images though a single fiber by wave-

length-time encoding. Optics Letters. 1983. Vol. 8, no. 1. pp. 57–59. 4. Kaiser T., Flamm D., Schroter S., Duparre M. Complete modal decomposition for optical fibers

using CGH-based correlation filters. Optics Express. 2009. Vol. 17, no. 11. pp. 9347–9356. 5. Shapira O., Abouraddy A.F., Joannopoulos J.D., Fink Y. Complete modal decomposition for op-

tical waveguides. Physical Review Letters. 2005. Vol. 94, no. 14. p. 143902. 6. Bolshakov M.V., Komarova M.A., Kundikova N.D. Opredelenie modovogo sostava izluchenia,

rasprostraniaushegosia v malomodovom opticheskom volokne (Determination of the mode composition propagating in a few-mode optical fiber). Vestnik YuUrGU. Seriia «Matematika. Mekhanika. Fizika».2012. Issue. 7. no. 34(293). pp. 138–141. (in Russ.).

7. Bolshakov M., Ershov A., Kundikova N. Optical effects connected with coherent polarized light propagation through a step-index fiber. In book: Fiber Optic Sensors. InTech, 2012. pp. 249–274.

8. Kurant R., Gil'bert D. Metody matematicheskoy fiziki (Methods of Mathematical Physics). Mos-cow, Leningrad: GITTL, 1951. Vol. 1. 476 p. (in Russ.).

9. Fienup J.R. Phase retrieval algorithms: comparison. Applied Optics. 1982. Vol. 21, no. 15. pp. 2758–2769.

19 2013 .

1 Bolshakov Maxim Vyacheslavovich is Cand. Sc. (Physics and Mathematics), Associate Professor, Optics and Spectroscopy Department, South Ural State University. 2 Komarova Marianna Alekseevna is Post-graduate student, Optics and Spectroscopy Department, South Ural State University. 3 Kundikova Nataliya Dmitrievna is Dr. Sc. (Physics and Mathematics), Professor, Dean of Physics Faculty, Join Nonlinear Optics Laboratory of IEF RAS, Optics and Spectroscopy Department, South Ural State University. E-mail: [email protected]


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. « . . » 146

1. Br ckner, R. Silicon Dioxide / R. Br ckner // Encyclopedia of Applied Physics. – 1997. –Vol. 18. – P. 95–131

2. , . . / . . , . . . – .: , 1985. – 168 .

3. , . . : / . . // . – 2001. – . 7, 3. – . 95–102.

4. , . . -1 / . . , . . , . . // .

2 « ». – 2010. – . 51, 1. – . 43–47. 5. , . . . / . . . – .:

, 1976. – 328 . 6. -

41 / . . , . . , . . , . . // . « ». – 2008. – 4. – . 376–388.

INFLUENCE OF HEAT-TREATING ON MICROHARDNESS OF SILICA GLASS -1

A.N. Bryzgalov1, P.V. Volkov2

The influence of heat-treating on the microhardness of the silica glass K -1 is investigated. It is found out that the heat-treatment at 1100 °C increase in microhardness and the uniformity improves. This fact is explained by the change of the phase composition of the surface, namely by the appearance of re-flexes, related to the phase of cristobalite.

Keywords: silica glass K -1, microhardness, heat-treatment, X-ray analysis, cristobalite.

References 1. Br ckner R. Silicon Dioxide. Encyclopedia of Applied Physics. 1997. Vol. 18. pp. 95–131. 2. Leko V.K., Mazurin O.V. Svoystva kvartsevogo stekla (Properties of silica glass). Moscow:

Nauka, 1985. 168 p. (in Russ.). 3. Khonik V.A. Sorosovskiy obrazovatel'nyy zhurnal. 2001. Vol. 7, no. 3. pp. 95–102. (in Russ.). 4. Lunin B.S., Kharlanov A.N., Kozlov S.E. Vestnik Moskovskogo universiteta. Seriya 2 «Khimiya».

2010. Vol. 51, no. 1. pp. 43–47. (in Russ.). 5. Mirkin L.I. Rentgenostrukturnyy analiz. Spravochnoe posobie. (X-ray structural analysis. Study

guide). Moscow: Nauka, 1976. 328 p. (in Russ.). 6. Kozlova S.A., Parfenov V.A., Tarasova L.S., Kirik S.D. Zhurnal Sibirskogo Federal'nogo uni-

versiteta. Seriya «Khimiya». 2008. no. 4. p. 376–388. (in Russ.). 11 2013 .

1 Bryzgalov Aleksandr Nikolaevich is Dr.Sc. (Physics and Mathematics), Professor, General and Theoretical Physics Department, Chelyabinsk State Pedagogical University. E-mail: [email protected] 2 Volkov Petr Vyacheslavovich is Post-Graduate Student, General and Theoretical Physics Department, Chelyabinsk State Pedagogical Univer-sity. E-mail: [email protected]


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1. Zaporozhchenko, R.G. Relation between Efficiency of Second Harmonic Generation and Spectral Properties of a One-Dimensional Photonic Crystal / R.G. Zaporozhchenko // Optics and Spectroscopy. – 2003. – Vol. 95, 6. – P. 976–982.

2. Multiple wavelength second-harmonic generation in one-dimensional nonlinear photonic crystals / L.M. Zhao, C. Li, Y.S. Zhou, F.H. Wang // J. Opt. Soc. Am. B. – 2008. – Vol. 25, 12. – P. 2010–2014.

3. Sasaki, Y. Phase-matched sum-frequency light generation in optical fibers / Y. Sasaki, Y. Ohmori // Appl. Phys. Lett. – 1981. – Vol. 39, 6. – P. 466–468.

4. Ohmori, Y. Two-Wave Sum-Frequency Light Generation in Optical Fibers / Y. Ohmori, Y. Sasaki // IEEE Journal of Quantum Elektronics. – 1982. – Vol. 18, 4. – P. 758–762.

5. , . . / . . , . . // . – 1987. – . 45, 12. – . 562–565.

6. Stolen, R.H. Self-organized phase-matched harmonic generation in optical fibers / R.H. Stolen, H.W.K. Tom // Optics Letters. – 1987. – Vol. 12, 8. – . 585–587.

7. http://ab-initio.mit.edu/wiki/index.php/Meep

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. « . . » 150

EFFICIENCY OF SECOND HARMONIC GENERATION IN ONE-DIMENTIONAL PHOTONIC CRYSTAL FROM ISOTROPIC MATERIAL

A.M. Gerasimov1, N.D. Kundikova2, Yu.V. Miklyaev3, D.G.Pikhulya4, M.V.Terpugov5

Investigation of second-harmonic generation in one-dimensional photonic crystal was carried out. The calculation of the recording of the grate (2) in a periodic system consisted of layers of air and glass and is followed by second-harmonic generation was performed. The effective factor of transforma-tion of radiation into the second harmonic at different wavelengths of the first harmonic was estimated.

Keywords: photonic crystal, second harmonic generation (SHG), optical nonlinearity.

References 1. Zaporozhchenko R.G. Relation between Efficiency of Second Harmonic Generation and Spectral

Properties of a One-Dimensional Photonic Crystal. Optics and Spectroscopy. 2003. Vol. 95, no. 6. pp. 976–982.

2. Zhao L.M., Li C., Zhou Y.S., Wang F.H.. Multiple wavelength second-harmonic generation in one-dimensional nonlinear photonic crystals. J. Opt. Soc. Am. B. 2008. Vol. 25. Issue 12, pp. 2010–2014. [http://dx.doi.org/10.1364/JOSAB.25.002010]

3. Sasaki Y., Ohmori Y. Phase-matched sum-frequency light generation in optical fibers. Appl. Phys. Lett. 1981. Vol. 39, no. 6. pp. 466–468.

4. Ohmori Y., Sasaki Y. Two-Wave Sum-Frequency Light Generation in Optical Fibers. IEEE Journal of Quantum Elektronics. 1982. Vol. 18, no. 4. pp. 758–762.

5. Baranova N.B., Zel'dovich B.Ya. Extension of holography to multifrequency fields. JETP Let-ters. Vol. 45. Issue 12. p. 717.

6. Stolen R.H., Tom H.W.K. Self-organized phase-matched harmonic generation in optical fibers. Optics Letters. 1987. Vol. 12, no. 8. pp. 585–587.

7. http://ab-initio.mit.edu/wiki/index.php/Meep 21 2013 .

1 Gerasimov Alexander Michaylovich is Post-graduate Student, Optics and Spectroscopy Department, School of Physics, South Ural State University. E-mail: [email protected] 2 Kundikova Nataliya Dmitrievna is Dr. Sc. (Physics and Mathematics), Professor, Head of the Nonlinear Optics Laboratory of IEF RAS, Dean of Physics Faculty, South Ural State University. E-mail: [email protected] 3 Miklyaev Yuriy Vladimirovich is Cand. Sc. (Physics and Mathematics), Associate Professor, Senior Staff Scientist of the Nonlinear Optics Laboratory of IEF RAS, Optics and Spectroscopy Department, South Ural State University. E-mail: [email protected] 4 Pikhulya Denis Grigorievich is Assistant, Junior Research Fellow of the Nonlinear Optics Laboratory of IEF RAS, Department of Optics and Spectroscopy, South Ural State University. E-mail: [email protected] 5 Terpugov Mikhail Vladimirovich is Master Student, South Ural State University. E-mail: [email protected]


2013, 5, 2 151

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2013, 5, 2 153

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1. , . . / . . // . « .

. ». – 2012. – . 7. – 34(293). – . 157–160. 2. , . . /

. . // . – 2012. – 6(62). – . 21–25. 3. Khokhlova, T.N. Stability of a ring and linear neural networks with a large number of neurons /

T.N. Khokhlova, M.M. Kipnis // Applied Mathematics and Computation. – 2012. – P. 1–14. 4. Ivanov, S.A. The stability cone for a difference matrix equation with two delays / S.A. Ivanov,

M.M. Kipnis, V.V. Malygina // ISRN J. Applied Mathematics. – 2011. – P. 1–19. ID 910936.

. 3. (1) -( , )a b 0, 4, 3nγ = = -

k

. 2. (1) -( , )a b 0, 4, 3kγ = =

n


. « . . » 154

5. Ivanov, S.A. Stability analysis of discrete-time neural networks with delayed interactions: torus, ring, grid, line / S.A. Ivanov, M.M. Kipnis // International Journal of Pure and Applied Math. – 2012. – Vol. 78, 5. – P. 691–709.

6. Kipnis, M.M. The stability cone for a matrix delay difference equation / M.M. Kipnis, V.V. Malygina // International Journal of Mathematics and Mathematical Sciences. – 2011. – P. 1–15. ID 860326.

7. Khokhlova, T.N The stability cone for a delay differential matrix equation / T.N. Khokhlova, M.M. Kipnis, V.V. Malygina // Applied Math. Lett. – 2011. – Vol. 24. – P. 742–745.

STABILITY OF TWO-LAYER RECURSIVE NEURAL NETWORKS

S.A. Ivanov1

The stability conditions are described for the discrete neural networks. Stability regions are con-structed in the parameter space for these networks. The problem reduces to the stability problem for the matrix difference equations of higher order with delay. The main tool is the stability cone.

Keywords: neural networks, difference matrix equations, stability of difference equations, two-layer network.

References 1. Ivanov S.A. Oblast' ustoychivosti v prostranstve parametrov rekursivnykh neyronnykh setey s to-

pologiey mnogomernogo kuba (The stability domain in the parameters space of recursive neural net-works with hypercube topology). Vestnik YuUrGU. Seriya “Matematika. Mekhanika. Fizika”. 2012. Issue 7. no. 34(293). pp. 157–160. (in Russ.).

2. Ivanov S.A. Ustoychivost' rekursivnykh neyronnykh setey so zvezdnoy topologiey svyazey (Sta-bility of recursive neural networks with star topology). Estestvennye i tekhnicheskie nauki. 2012. no. 6(62). pp. 21–25. (in Russ.).

3. Khokhlova T.N., Kipnis M.M. Stability of a ring and linear neural networks with a large number of neurons. Applied Mathematics and Computation. 2012. pp. 1–14.

4. Ivanov S.A., Kipnis M.M., Malygina V.V. The stability cone for a difference matrix equation with two delays. ISRN J. Applied Mathematics. 2011. pp. 1–19. ID 910936.

5. Ivanov S.A., Kipnis M.M. Stability analysis of discrete-time neural networks with delayed inter-actions: torus, ring, grid, line. International Journal of Pure and Applied Math. 2012. Vol. 78, no. 5. pp. 691–709.

6. Kipnis M.M., Malygina V.V. The stability cone for a matrix delay difference equation. Interna-tional Journal of Mathematics and Mathematical Sciences. 2011. pp. 1–15. ID 860326

7. Khokhlova T.N., Kipnis M.M., Malygina V.V. The stability cone for a delay differential matrix equation. Applied Math. Lett. 2011. Vol. 24. pp. 742–745.

4 2013 .

1 Ivanov Sergey Alexandrovich is Post-graduate Student, Mathematical Analysis Department, Chelyabinsk State Pedagogical University. E-mail: [email protected]


2013, 5, 2 155

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. . , . . . – .: , 1978. – 246 . 3. - / . , . , . .; . . ;

. . – .: , 1990. – 256 . 4. / . . ,

. . , . . , . . // . – 2007. – . 6. – . 131–133.

5. , . . / . . , . . // « »_22. URL:

http://www.rit.informost.ru/rit/4-2002/28.pdf ( : 01.03.2013) 6. , . . -

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2013, 5, 2 159

FLUORESCENT IMPURITY CENTERS FOR MONITORING THE PHYSICAL STATE OF POLYMERS AND A COMPUTER SIMULATION OF PROCESSES OF STRUCTURING OF THE POLYMER MATRICES

S.G. Karitskaya1, Ya.D. Karitskiy2

The temperature influence and the structure changes of polymer matrix on fluorescence of impurity luminescence centers were investigated. The obtained data broadens the perspectives of using triphe-nylmethane dye as a detector that helps to analyse physical conditions of polymers. Computer simula-tion of polymer matrices structuring process was carried out. The results of the calculations by the methods of MM+ and ZINDO/S prove the mechanism of appearance of hysteresis as a change of envi-roning geometry of luminescent centers.

Keywords: thermal hysteresis of the luminescence, triphenylmethane dyes, structural modification of the polymer.

References 1. Parker C.A. Photoluminescence of Solutions. Elsevier Publishing Company. Amsterdam-

London-New York, 1968. 544 p. 2. Golovina A.P., Levshin L.V. Khimicheskiy lyuminestsentnyy analiz neorganicheskikh veshchestv

(Chemical luminescent analysis of non-organic substances). Moscow: Khimiya, 1978. 246 p. (in Russ.). 3. Okosi T., Okamoto K., Otsu M., Nisikhara Kh., Kyuma K., Khatate K. Volokonno-opticheskie

datchiki (Fiberoptic detectors). Leningrad: Energoatomizdat, 1990. 256 p. 4. Urakseev M.A., Farrakhov R.G., Kireev M.G., Dmitriev D.A. Istoriya nauki i tekhniki. 2007.

Vol. 6. pp. 131–133. (in Russ.). 5. Larin Yu.T., Nesterko V.A. Polimernye opticheskie volokna (Polymeric optical fibers).

INFORMOST «Radio-elektronika i Telekommunikatsii»_22. [ URL: http://www.rit.informost.ru/rit/4-2002/28.pdf (accessed 01.03.2013)].

6. Krichevskiy G.E. Fotokhimicheskoe prevrashchenie krasiteley i svetostabilizatsiya okrashennykh materialov (Photochemical transformation of dyes and light stabilization of dyed materials). M.: Khimiya, 1986. 248 p.

7. Tager A.A. Fiziko-khimiya polimerov (Physics and chemistry of polymers). Moscow: Khimiya, 1978. 544 p. (in Russ.).

8. Tarutina L.I., Pozdnyakova F.O. Spektral'nyy analiz polimerov (Spectral analysis of polymers). Leningrad: Khimiya, 1986. 248 p. (in Russ.).

24 2013 .

1 Karitskaya Svetlana Gennadievha is Cand. Sc. (Engineering), Associate Professor, Department of Applied Mathematics, Ural Federal Univer-sity named after the first President of Russia B.N. Yeltsin, University Branch in the town of Pervouralsk. E-mail: [email protected] 2 Karitskiy Yaroslav Dmitrievich is Student, Ural Federal University named after the first President of Russia B.N.Yeltsin.


. « . . » 160

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α , (1). 0u - . T , X , -

0u - , ;u v

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( )( ); ; 1q t u v < , ;x u v∈

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; ;T u v u v⊂ . . T 0u -

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1 – , - , ,

- . E-mail: [email protected]


. .

2013, 5, 2 161

0 0;n nn nu T u v T v= = .

, 0n n mnv u t u≤ + ,

( ),mn n mt d u v= . { }mnt . , lim 0mnm

n

t t→∞→∞

= ≠ .

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, , ( )1 1 0 0m n mnv u qtu t t u+ +≤ + + − .

, ( ) ( )1 1;n n mnd u v qt t t+ + ≤ + − ,

. { }nu { }nv .

lim limnn nu x∗→∞ →∞

= = .

T – , x∗ – . .

1. , . . / . . -, . . . – .: , 1975. – 512 . 2. . . / . .

// . – 1978. – . 36. – . 237–380.

EXISTENCE OF THE FIXED POINT IN THE CASE OF EVENLY CONTRACTIVE MONOTONIC OPERATOR

M.L. Katkov1

The article proves the existence of the fixed point in the case of evenly contractive monotonic op-erator in the Banach K-space. It proved to be right that the iterations converge to the fixed point in the metric of the even convergence. Compactness of the invariant set and the total continuity of the operator are not assumed.

Keywords: positive operator, monotonic concave operator, heterotonic operator.

References 1. Krasnoselskiy M.A., Zabreiko P.P. Geometricheskie metody nelineynogo analiza (Geometrical

methods of nonlinear analysis). Moscow: Nauka, 1975. 512 p. (in Russ.). 2. Opoitsev V.I. Trudy moskovskogo matematicheskogo obshchestva. 1978. Vol. 36. pp. 237–380.

(in Russ.). 28 2013 .

1 Katkov Mikhail Lvovich is Associate Professor, Mathematical and Functional Analysis Department, South Ural State University. E-mail: [email protected]


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102–105 / 2.

1 - « » ( ), -

( 2012043- 04) 2 – - , , , -



. .

2013, 5, 2 163

, . « -

– ». -

. -H(x,t). Z(x,t). Z(x,t) H(x,t) –

. , Z(x,t) H(x,t), ( ) – Ω1. , Z(x,t) -

( ) – Ω2. «1», «2» -, ρ1, ρ2 – .

, υ ϕ= ∇ , ,

, yeυ ψ= ∇ .

, 1ka (k – , a – -). (1)–(4) -

. 2( , , ) ( ) ( , )G x Z t g t Z x t= − ,

1( , , ) ( ) ( , )G x H t g t H x t= − , 1 2( ), ( )g t g t – , -, . 2 1

2 1A ρ ρ

ρ ρ−

=+

.

. , , :

21( , , ) 0,x z tϕ∇ = 2

2 ( , , ) 0,x z tϕ∇ = (1)

12

1

( , , ) ( , , )( ) ( , ) ,x Z t P x Z tAg t Z x tt

ϕρ

∂= −

∂

11

( , , ) ( ) ( , ),x H t g t H x tt

ϕ∂ =∂

22

2

( , , ) ( , , )( ) ( , ) ,x Z t P x Z tg t Z x tt

ϕρ

∂= −

∂ (2)

11

( , , )( , , ) ,zx Z tZ x Z t

t zϕυ ∂∂ = =

∂ ∂ 1

1( , , )( , , ) ,zx H tH x H t

t zϕυ ∂∂ = =

∂ ∂2

2( , , )( , , )zx Z tZ x Z t

t zϕυ ∂∂ = =

∂ ∂. (3)

, : 1 2ϕ ϕ= − (4). -

: 0( , ) ( )cos( )Z x t Z t kx= , 0( , ) ( )cos( ),H x t H t kx=

Z0(t) – , H0(t) – .

, ,

1( , )sh( ( )) ( , )sh( )( , , ) cos( ),

sh( )z t k H z H t kzx z t kx

kHϕ ϕϕ − += (5)

(1). (5) -.

(5) (1)–(3), -,

. 2 0 1 0

0

0 0 1 0

( )ch( )( )sh( )

( )ch( ) ( ) sh( )

Ag kZ t kH kg HZ tkH

H t kH Z t g kH kH

− +=

− = (6)


. « . . » 164

: kH→ –∞, ( H λ>> , λ - ) . (6) :

0 2 0

0 1 0

( ) ( ).

( ) ( )

Z t Ag kZ t

H t g kH t

=

= −, H λ>> ,

, ( -– ), , -

( ). – . -

: ( ) ( )g t U tδ= , U – -. (6) :

2 0 1 00

0 0 1 0

( )ch( )( )sh( )

( )ch( ) ( ) sh( )

AkU Z t kH kU HZ tkH

H t kH Z t kU H kH

− +=

− =. (7)

(6) (7) . -:

0 0( ) , ( ) .n nn n

n nZ t a t H t b t= =

(6), (7) an, bn. –

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

1 ( 1)ch( )n n n n

n nBa Db Ba C D ba a Z t b b H t

n n kH+ ++ + +

= = = = = =+ +

,

ch( ) sh( ), sh( ), sh( )B AUk kH kH D kU kH C kU kH= − = = . –

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

( 1)( 2) ( 1)( 2)ch( )n n n n

n nBa Db Ba C D ba a Z t b b H tn n n n kH+ +

+ + += = = = = =

+ + + +ch( ) sh( ) , sh( ), sh( )B Agk kH kH D kg kH C kg kH= − = = .

, -,i i i ia a b aε ε< < , ε ~ 10–4.

Fe. - [3, 4]. .

10 . ( « – ») .

1 ( ). (7) , 20 -

15 / 2 ( ) - 7,8 , [3].

, -

.

( , ). , -

-.

,

.


. .

2013, 5, 2 165

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

. – 2013. – 10. – . 1–8. 2. Mechanisms of Metal Surface Modification under Processing by Compression Plasma Flows /

A.Ya. Leyvi, V.M. Astashynski, M.Yu. Zotova et al. // . -. – 2012. – . 55. – . 12-2. – . 202–205.

3. / . . , . . , . .

. // . – 2001. – . 74, 4. – C. 234–236. 4. -

/ . . , . . , . . , . . // . – 2007. – . 77. – . 4. – . 41–49.

5. , / . . , . . , . . . // . – 2006. –

. 32, 10. – . 20–29. 6. . . -

/ . . , . . , . . // . – 2003. – . 73, 3. – . 1–9. 7. Leyvi A.Ya. The dynamics of metal target surface at irradiation by intense plasma streams.//

A.Ya. Leyvi, K.A. Talala, .P. Yalovets // Proceedings of 10-th International Conference on Modifica-tion of Materials with Particle Beams and Plasma Flows. – Tomsk, 2010. – P. 173–176.

ANALYSIS OF MECHANISMS OF SURFACE PATTERN FORMATION DURING THE HIGH-POWER PLASMA STREAM PROCESSING

A.Ya. Leyvi1

The analysis of mechanisms of surface pattern formation during the high-power plasma stream processing is carried out. It is showed that in the case of plasma processing the surface dynamics is de-termined not only by the surface tension forces and viscosity but also by the forces connected with the plasma stream pressure on the substance. By means of these forces the roughness of the surface in-creases even during the processing in the precritical mode.

Keywords: non-linear dynamics, radiation processing, mass transfer, Rayleigh–Taylor instability, material modification.

References 1. Astashinskiy V.M., Leyvi A.Ya., Talala K.A, Uglov V.V., Cherenda N.N., Yalovets A.P. Po-

verkhnost'. Rentgenovskie, sinkhrotronnye i neytronnye issledovaniya. 2013. no. 10. p. 1–8. (in Russ.). 2. Leyvi A.Ya., Astashynski V.M., Zotova M.Yu., Cherenda N.N., Uglov V.V., Yalovets A.P. Iz-

vestiya vysshikh uchebnykh zavedeniy. Fizika. 2012. Vol. 55. Issue 12-2. P. 202–205. (in Russ.).3. Uglov V.V., Anishchik V.M., Astashinskiy V.V., Astashinskiy V.M., Ananin S.I., Askerko V.V.,

Kostyukevich E.A., Kuz'mitskiy A.M., Kvasov A.M., Danilyuk A.L. JETP Letters. 2001. Vol. 74, no. 4. pp. 234–236.

4. Krasnikov V.S., Leyvi A.Ya., Mayer A.E., Yalovets A.P. ZhTF. 2007. Vol. 77. Issue 4. pp.41–49.

5. Volkov N.B., Mayer A.E., Talala K.A., Yalovets A.P. Pis'ma v ZhTF. 2006. Vol. 32, no. 10. pp. 20–29.

6. Volkov N.B., Mayer A.E., Yalovets A.P. ZhTF. 2003. Vol. 73, no. 3. pp.1–9. 7. Leyvi A.Ya., Talala K.A., Yalovets .P. The dynamics of metal target surface at irradiation by

intense plasma streams. Proceedings of 10-th International Conference on Modification of Materials with Particle Beams and Plasma Flows. Tomsk, 2010. pp. 173–176.

20 2013 . 1 Leyvi Artem Yacheslavovich is Cand. Sc. (Physics and Mathematics), General and Experimental Physics Department, South Ural State Uni-versity. E-mail: [email protected]


. « . . » 166

517.521.8

. . 1

, « -» .

: , , .

( )f z .

( )f z0

kk

ka z

∞

=. m- -

0

mf k

m kk

s a z=

= , fS ( )f z fA –

, ( )f z . 11 z− -

( )n zσ . . B , ,( )m nb ,

0

kk

ka z

∞

=( )f z D ⊂ 0m ,

Q D⊂ 0m m≥ ,0

( )fm k k

kb s z

∞

=-

( )f z Q . . fU A⊂ - ( )f z ,

, , -{ ( )}f

ms z ( )f z U . , -

0

kk

ka z

∞

=.

[1,2] , ( ),m nb0

∞

=

k

k

z 1 (1 )z/ − -

D , fD , , 0

kk

ka z

∞

=( )f z . - -

0

∞

=

k

k

z .

1 – - , ,



. .

2013, 5, 2 167

[3] . 1( );

1F w

w− 11

( ) ( 0)kk

kF w c w c

∞

== ≠

11 w−

,

, 11 (1 ( )) ( ) ( , 0)

2 (1 )

nFm n m

F w F wc dw m nw wπ +

−= ≥−

,

γ – -. ( )F w [1] : 1) ( )F w -

K , , , 1w = , K ; 2) ( )F w K , (0) 0F = , (1) 1F = , ( )F K K⊆ ; 3) ( )F w -

1w = .

. ( ),0 0( ) m nF F km m nn kP z c z= == m -

[ ]( );1 (1 )F w w− .

(0) 0F = , , 0Fm nc = n m> , m - -

m . ( )T F , -1( ) { [ ( )] [0 2 ]}ixF z z F e xτ π−= : = , ∈ ; , [ ]( );1 (1 )F w w− .

[3] [4], [ ]( );1 (1 )F w w−0

∞

=

k

k

z 1 (1 )z/ − -

( )T F . ( )FmP z . ( )F K K⊆ ,

[ ]( );1 (1 )F w w−

0

k

kz

∞

=.

[5] ( )T F . [6] ,

( )FmP z ( )T F 1z ≠ . ,

( )FmP z

11mρ + , 0 1ρ< < .

. z , ( )T F , -

0w K∈ 0C ≠ , 1

0lim ( )F mmm

P z w C+

→∞= .

. ( )z T F∈ , 1( ( ))z F K −∈ ( )T F . ( )F W K , 0w K∈ ,

0

1( )

zF w

= , 1 ( )( , )1 ( )

F ww zzF w

−Φ =−

.

γ – 0w . 0w –( , )w zΦ ,

0 11 1 1

0 0 0

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

F w Cdww zi w zF w w wγπ + + +

−− Φ = − =

′,


. « . . » 168

01

0

1 ( ) 0( )

F wCzF w−

= − ≠′

.

μ , γ .

11 1 1

0 | | 1

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

w

Cdw dww z w zi iw w wμπ π+ + +

=

Φ + = Φ .

12

0 1CC

w=

−:

221 1 1

0 00 | | 1

1 1( , ) ( , )2 2

m m

k m kk k w

Cdw dww z w z Ci iw w wμπ π+ + +

= = =

Φ − = Φ − .

M | ( , ) |w zΦ { :| | 1}w w = .

1 10 2 0 21

0

1 ( , ) ( | |)2

mm m

kk

dww w z C w mM Ci wμπ

+ ++

=Φ − ≤ + ,

0 m → ∞ , 0| | 1w < . ,

10 21

0

1lim ( , )2

mm

km k

dww w z Ci wμπ

++→∞ =

Φ = .

[1] , 1z ≠

11 1 (1 ( ))( )

1 1 2 (1 ( ))(1 )F

m mz F w dwP z

z z i zF w w wγπ +−= − ⋅

− − − −.

1 10

1 ( , ) ( , )2 (1 )

m

m kk

dw dww z w zi w w wμ μπ + +

=Φ = Φ

−,

1 20lim ( )

1F m

mm

zCP z w Cz

+

→∞= − =

−, 2

1zCC

z= −

−. .

. z , ( )T F ,

0w K∈ , 0m 1C , 2C , 0m m≥

1 21 1

0 0| ( ) |

| | | |F

mm mC CP z

w w+ +< < .

. 0w K∈ 0C ≠ , 1

0lim ( )F mmm

P z w C+

→∞= . 1C , 2C -

, 1 2| |C C C< < . z [1, 2, 4]

-.

, , ( )f z [1, 7].

1. , . . . / . . // : . . . – , . . - , 1978. – C. 49–64.

2. , . . / . . // -: . . . – : - , 2011. – . 3.– C. 149–152.


. .

2013, 5, 2 169

3. Gronwall, T.H. Summation of series and conformal mapping / T.H. Gronwall // Ann. Math. – 1932. – Vol. 33, 2. – P. 101–117.

4. Birindelli, C. Contributo all’ analisi dei metodi di sommazione di Gronwall / C. Birindelli // Ren-dic. del Circolo Matemat. de Palermo. – 1937. – Vol. 61. – P. 157–176.

5. , . . / . . // : . . . – : - , 2009. – . 2. – C. 153–

156. 6. , . . /

. . // . « , , ». – 2012. – . 7. – 34(293). – . 165–168.

7. , . . II. / . . // . . : . . . – , . . - . – 1979. – C. 30–37.

BEHAVIOR OF GRONWALL POLYNOMIALS OUTSIDE THE BOUNDARY OF SUMMABILITY REGION

L.V. Matveeva1

In this paper we continue the study of Gronwall’s polynomials which was started by the author in “The evaluation of convergence rate of the sequence of Gronwall’s polynomials” and continued in other works.

Keywords: methods of summation, Gronwall polynomials, convergence rate.

References 1. Matveeva L.V. Otsenka skorosti skhodimosti posledovatel'nosti polinomov Gronuolla (Estimates

of the rate of convergence of the sequence of polynomials Gronwall). Issledovaniya po funktsional'nomu analizu: sb. nauch. tr. (Research on functional analysis: a collection of scientific works). Sverdlovsk, Ur. gos. un-t, 1978. pp. 49–64. (in Russ.).

2. Matveeva L.V. Oblasti ravnomernoy skhodimosti i teorema Okada (Areas of uniform conver-gence theorem and Okada). Nauka YuUrGU: sb. nauch. tr. (Science of SUSU: a collection of scientific works). Chelyabinsk: Izd-vo YuUrGU, 2011. Vol. 3. pp. 149–152. (in Russ.).

3. Gronwall T.H. Summation of series and conformal mapping. Ann. Math. 1932. Vol. 33, no. 2. p. 101–117.

4. Birindelli C. Contributo all’ analisi dei metodi di sommazione di Gronwall. Rendic. del Circolo Matemat. de Palermo. 1937. Vol. 61. pp. 157–176. 5. Matveeva L.V. Polinomy Gronuolla s analiticheskoy otobrazhayushchey funktsiey (Gronwall poly-nomials with analytic mapping function). Nauka YuUrGU: sb. nauch. tr. (Science of SUSU: a collection of scientific works). Chelyabinsk: Izd-vo YuUrGU, 2009. Vol. 2. pp. 153–156. (in Russ.).

6. Matveeva L.V. Povedenie polinomov Gronuolla na granitse oblasti summiruemosti (Behavior of Gronwall polynomials on the boundary of summability domain). Vestnik YuUrGU. Seriya “Matematika, mekhanika, fizika”. 2012. Issue 7. no. 34(293). pp. 165–168. (in Russ.).

7. Matveeva L.V. Otsenka skorosti skhodimosti posledovatel'nosti polinomov Gronuolla II (Esti-mates of the rate of convergence of the sequence of polynomials Gronwall II). Issledovaniya po funktsional'nomu analizu: sb. nauch. tr. (Research on functional analysis: a collection of scientific works). Sverdlovsk, Ur. gos. un-t, 1979. pp. 30–37. (in Russ.).

30 2013 .

1 Matveeva Liubov Vasilievna is Cand. Sci. (Physics and Mathematics), Professor, Department of mathematical and functional analysis, South Ural State University. E-mail: [email protected]


. « . . » 170

519.633

. . 1

-, -

. - -,

, .

: , , .

1. . ( . . -

) [1]

( )

( ) ( )

0,

,

divt

div F divTt

ρ ρυ

ρυρυ υ ρ

∂ + =∂∂

+ ⊗ = +∂

(1)

ρ υ – , F – (0F = ), T – ( υ υ⊗ ).

: ( ) 2T p div G eμ υ μ′= − + + , (2)

p – , μ – , μ′ – 2- , G – ( ), e – -

,

( )( )12

Te υ υ= ∇ + ∇ , (3)

∇ – , T . , ( ), ,r zϕ -

:

( ) ( ) ( )1r zdiv r r

r r r zϕυ

υ υ υϕ

∂ ∂ ∂= + +∂ ∂ ∂

, (4)

( ) ( ) ( )1r zdiv r r

r r r zϕρρυ ρυ υ ρυ

ϕ∂ ∂ ∂= + +∂ ∂ ∂

. (5)

( )( ) ( ) ( ) ( )1 ii i r i zi

div div r rr r r z

ϕρυ υρυ υ ρυ υ ρυ υ ρυ υ

ϕ∂ ∂ ∂⊗ = = + +∂ ∂ ∂

, { }, ,i r zϕ∈ (6)

1 1, , ,2 2

1, ,2

.

r r r zrr r rz

zr z

zzz

e e er r r z r

e r ez

ez

ϕ ϕϕ

ϕ ϕϕϕ ϕ

υ υυ υ υ υϕ

υ υ υυϕ ϕ

υ

∂∂ ∂ ∂ ∂= = + − = +∂ ∂ ∂ ∂ ∂

∂ ∂ ∂= + = +∂ ∂ ∂

∂=∂

(7)

1 – , , - . E-mail: [email protected]


. .

2013, 5, 2 171

( ) ( )2divT p div div eμ υ μ′= −∇ + ∇ + , (8)

( )div eμ :

( )( ) ( ) ( )

( )( ) ( ) ( )

( )( ) ( ) ( )

31 ,

1 ,

1 .

rr r zrr

r z

rz z zzz

div e r e e r e er r r z r

div e r e e r er r r z

div e r e e r er r r z

ϕ ϕϕ

ϕ ϕϕ ϕϕ

ϕ

μ μμ μ μϕ

μμ μ μϕ

μμ μ μϕ

∂ ∂ ∂= + + −∂ ∂ ∂

∂ ∂ ∂= + +∂ ∂ ∂

∂ ∂ ∂= + +∂ ∂ ∂

(9)

, . . 0ϕ∂ =

∂, 0ϕυ = , (4)–(9)

:

( ) ( ) ( )1r zdiv r r

r r zυ υ υ∂ ∂= +

∂ ∂, ( ) ( ) ( )1

r zdiv r rr r z

ρυ ρυ ρυ∂ ∂= +∂ ∂

; (10)

( )( ) ( ) ( )1i r i zi

div r rr r z

ρυ υ ρυ υ ρυ υ∂ ∂⊗ = +∂ ∂

, { },i r z∈ ; (11)

1, 0, ,2

, 0,

;

r r zrr r rz

r z

zzz

e e er z r

e r e

ez

ϕ

ϕϕ ϕ

υ υ υ

υυ

∂ ∂ ∂= = = +∂ ∂ ∂

= =

∂=∂

(12)

( )( ) ( ) ( ) 31

rr zrrdiv e r e r e e

r r z r ϕϕμμ μ μ∂ ∂= + −

∂ ∂, (13)

( )( ) ( ) ( )1rz zzz

div e r e r er r z

μ μ μ∂ ∂= +∂ ∂

. (14)

2. , ( ), ,r zϕ -

0ρ0rϕυ υ= = , ( )z u rυ = , 0 r R≤ ≤ , (15)

( ) 0u R = , . . . -R .

, ( ) 0div υ = . e - (12) :

12rz zr

ue er

∂= =∂

,

. , μ e , . .

( )eμ μ= ΙΙ , 21

4eur

∂ΙΙ = −∂

– 2- e [1, 2], -

,

( )1

1 2 22 22

0 04

nn

eur

μ μ λ μ λ

−− ∂= − ΙΙ =

∂, (16)


. « . . » 172

0μ , λ , n – , 1n > ( [3]). -

r z R , V , 0ρ , /R V , -

0μ , P , 0/P μ , 0P V Rμ= . (1)

( ) ( )Re 2div p div etυ υ υ μ∂ + ⊗ = −∇ +

∂, (17)

0

0Re RVρ

μ= – , .

(17) (10)-(11) -. (17)

(12-14), pr z

τ∂ ∂=∂ ∂

, (18)

( )1 rpz r r

τ∂∂ =∂ ∂

, (19)

( )u u r= , 0 1r≤ ≤ , ( )1 0u = , ur

τ μ ∂=∂

, 1nuk

rμ

−∂=∂

, VkR

λ= .

( ) ( )( )

11

11 2

nnzz nk pn sign pu r r

k n

+

= −+

, (20)

zp p L= Δ , pΔ – L , p z .

(20) 1n = ( ) ( )2 14

zpu r r= −

.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

n = 1n = 1.2

n = 1.4n = 1.6

n = 1.8

n → +∞

r

u(r)

( )u r n ( 10k = , 0,8zp = − )

( )u r n ,

n → +∞ – , (20), ( ) ( ) ( )sign1zp

u r rk

= − .

, , ,


. .

2013, 5, 2 173

( )0

2R

Q ru r drπ= ,

( ) 20

RQ u r drπ= .

( ) 0u R = ,

20

R uQ r drr

π ∂= −∂

.

(20) ,

( )1

0

sign2

nzz p rpur

λλ μ

∂ =∂

,

( )1

3

0

sign3 1 2

nzz p Rpn RQn

λπλ μ

= −+

. (21)

( 1n = ) 4

08zR pQ π

μ= − .

1. , . / . ; . . . . – .: , 1963. – 256 .

2. Georgiou G.C. The time-dependent, compressible Poiseuille and extrudate-swell flows of a Carreau fluid with slip at the wall / G.C. Georgiou // J. Non-Newtonian Fluid Mech. – 2003. – Vol. 109, 2. – P. 93–114.

3. , . / . , . . . . – .: , 1964. – 216 .

POISEUILLE FLOW OF A FLUID WITH VARIABLE VISCOSITY

K.Z. Khayrislamov1

The Poiseuille flow in a pipe for non-Newtonian fluid was examined. It is assumed that fluid viscos-ity is dependent on shear rate by power law. By solving Navier-Stokes equations we obtained velocity and volumetric flow rate solutions which summarize Poiseuille law for a Newtonian fluid.

Keywords: Poiseuille flow, non-Newtonian fluid, viscous fluid.

References 1. Serrin Dzh. Matematicheskie osnovy klassicheskoy mekhaniki zhidkosti (Mathematical fundamen-

tals of classical mechanics of fluids). Moscow: Izdatel'stvo inostrannoy literatury, 1963. 256 p. (in Russ.). [Serrin J. Mathematical principles of classical fluid mechanics. Springer-Verlag, Berlin-Göttingen-Heidelberg, 1959. 148 p.]

2. Georgiou G.C. The time-dependent, compressible Poiseuille and extrudate-swell flows of a Carreau fluid with slip at the wall. J. Non-Newtonian Fluid Mech. 2003. Vol. 109, no. 2. pp. 93–114.

3. Uilkinson U. Nen'yutonovskie zhidkosti (Non-Newtonian Fluids). Moscow: Mir, 1964. 216 p. (in Russ.). [Wilkinson W.L. Non-Newtonian fluids; fluid mechanics, mixing and heat transfer. Pergamon Press, New York, 1960. 138 p.]

6 2013 .

1 Khayrislamov Kirill Zinatullaevich is Post-Graduate student, Applied Mathematics Department, South Ural State University. E-mail: [email protected]


. « . . » 174

519.633

. . 1, . . 2

-, . -

- – . -

. : , -

, , .

, [1, 2], -.

1. -

[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 x L≤ ≤ – ; 0 t T≤ ≤ – ; L – ; T – ; ( )u – ;

( )q u – ; ( )xϕ – -; ( )l uλ – ; ( )r uλ – -

; lθ – ; rθ – ; ( )l lQ Q t= – -

; ( )r rQ Q t= – -. ( )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∂ ∂=∂ ∂

, ( )( )( )

q ua uc u

= –

1 – , , - . E-mail: [email protected] 2 – - , , , -

.


. ., . .

2013, 5, 2 175

. ( )G u, 1G−

, , .

. ( , )x t [2]

h hτ τω ω ω= × , 1 , 1, 2, ...,2h ix i h i Nω = = − = , { , 0, 1, ...},jt j jτω τ= = =

h L N= – x , τ – t . . 1. G -

(+1), -

– 12

+ , – 12

− .

2

22 ( )

( 1) ( )ia u

hi iG G B e A B

ττ

−+ = − + + , (2)

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

− −− ++

= + − ,

2 2

2 21 ( ) ( )2 1 11

2

i ia u a ui ih h

i iG GG G e e

τ τ+ − ⋅ − ⋅− ++

= + − .(3)

0 1N + ( . . 1): , ,

, .

1 23( ) ( )( (0, ))2

l ll u t λ λλ −

= , 1 23( (0, ))2

q qq u t −= , 1 0(0, ) u uu tx h

−∂ =∂

, 0 1(0, )2

u uu t += ,

0: 1 21 2

1 1 2

01 21 2

3( ) ( )3 (3( ) ( ) ) 22

3( ) ( )32

l ll l l l

l l

q qu Qhu q q

h

λ λ λ λ θ

λ λ

−− − + − += −− +

. (4)

1ix −0 ix 1ix +

0t

h 2h ih Nh L=x

t

1 2 N 1N +

0t τ+

0t τ−

iG1iG − 1iG +

( 1)iG +

12

1iG+

−

12

1iG+

+

12

1iG−

−

12

1iG−

+

. 1.


. « . . » 176

1N +

1 11

11 1

3 3( ) ( ) (3( ) ( ) ) 22

3 3( ) ( )2

N N r N r NN r N r N r r

NN N r N r N

q qu Qhu q q

h

λ λ λ λ θ

λ λ

− −−

+− −

− −− + − += − −+

. (5)

2. [4]

-. -

, [4, 5], --

:

, 0, 0nu uu t xt x x

∂ ∂ ∂= > >∂ ∂ ∂

, (6)

( , 0) 0, 0u x x= ≥ , (7) 1

00

, 0n nx

x

uu u Qt tx

λ ==

∂ = − >∂

, (8)

0n > – , , 0λ > – -, 0Q > – , -

. , 2n = (6)–(8) :

2 ( ), ,( , )

0, ,t x x t

u x tx t

α α αα

− <=

≥ (9)

2 2 22

Qλ λα − + += .

3. (6)–(8) 2n = , 21 /( × × )λ = , 25 /( × )Q = . -

h 1, τ 0,05. 10, 20, 30t = c . 2.

.

0 5 10 15 20 25 30 35 40 45 500

2

4

6

8

10

12

x,

,

t=30c

t=20c

t=10c

. 2. -

.


. ., . .

2013, 5, 2 177

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

2. , . . - / . . , . . , . // . « -

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

/ . . , . . // . – 2007. – . 47, 1. – . 110–120.

5. , . . , / . . , . . // 70- . . : . . . – .: -, 1950. – . 61–71.

EXPLICIT SCHEME FOR THE SOLUTION OF THIRD BOUNDARY VALUE PROBLEM FOR QUASI-LINEAR HEAT EQUATION

M.Z. Khayrislamov1, A.W. Herreinstein2

In produced paper numerical method for the solution of third boundary value problem for one-dimensional quasi-linear heat equation grounded on the use of explicit finite-difference scheme is of-fered. The coefficients’ dependence on temperature is overcome by introducing the new unknown func-tion – a primitive integral of conduction. Test problem with known exact solution for numerical calcula-tions is proposed.

Keywords: thermal conductivity, quasi-linear heat equation, explicit finite-difference schemes, ap-proximation.

References 1. Gerenshteyn A.V., Mashrabov N. Obozrenie prikladnoy i promyshlennoy matematiki. 2008.

Vol. 15, no. 5. pp. 870–871. 2. Herreinstein A.W., Herreinstein E.A., Mashrabov N. Ustoychivye 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. Samarskiy A.A. Teoriya raznostnykh skhem (The theory of difference schemes). Moscow: Nauka, 1989. 616 p. (in Russ.).

4. Kudryashov N.A., Chmykhov M.A. Approximate solutions to one-dimensional nonlinear heat conduction problems with a given flux. Computational Mathematics and Mathematical Physics. 2007. Vol. 47, no. 1. pp. 107–117

5. Zel'dovich Ya.B., Kompaneets A.S. K 70-letiyu A.F. Ioffe: sb. nauch. tr. (On the 70th anniversary of the A.F. Ioffe: collection of scientific papers). Moscow: Izd-vo AN SSSR, 1950. pp. 61–71. (in Russ.).

6 2013 .

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


. « . . » 178

517.17:517.51

. . 1

, n-,

n+1. : , .

, - [1, 2].

1. n 1n +.

. . . 1, 2 1( ,..., )nP x x x + n 1n +

1

1 2 1 1, 2 11

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

n i ni

P x x x f x x x+

+ +=

= , i 1 2 1( , ,..., )i nf x x x +

ix . . , k n≤

1, 2 1( ,..., )k nP x x x + k 1k + -, , , 1 2 1, ,..., .n k n k nx x x− + − + +

( 0k = ) ( 0P – ). . 1 2 1( , ,..., )nT x x x +

1, 2 1( ,..., )k nP x x x + 1n kx − + 0.

1, 2 1 1 2 1 1 1 1 2 1( ,..., ) ( , ,..., ) ( , ,..., ),k n n n k k nP x x x T x x x x P x x x+ + − + − += +

1 2 1( , ,..., )nT x x x + 1n kx − + ,

1 1, 2 1( ,..., )k nP x x x− + 1k − . k n= .

1. 1 2 1, ,..., nT T T + – ( . . 1 1 2 2 1 1... 0n nk T k T k T+ ++ + + ≠

1 2 1, ,..., ,nk k k + ). : 1 2x x , -

1 2 1, ,..., nk k k +

1 2 1 1 2 2 1 1... .n nx x k T k T k T+ +− = + + +K -

z . x K∈1 1 2 2 1 1... ,n nx z m T m T m T+ += + + + +

1 2 1, ,..., nm m m + – . ( )Q x – n .

1 1 2 2 1 1( ... )n nQ z m T m T m T+ ++ + + + -

1 2 1, ,..., nm m m + . 1

1 1 2 2 1 1 1 2 11

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

n n i ni

Q z m T m T m T f m m m+

+ + +=

+ + + + =

1 – , , , - -



. .

2013, 5, 2 179

i 1 2 1( , ,..., )i nf m m m + im . , -

1 2 1( , ,..., )i nf m m m + x ( im -x ) iT .

2. n - n .

. ( )nP x – n1

( ) ( ),n

n kk

P x f x=

= k

( )kf x kT .

1( ) ( ) ( ).n n n nQ x P x T P x− = + −1,n − 1n − :

1

11

( ) ( ( ) ( )).n

n k n kk

Q x f x T f x−

−=

= + −

, - ( ), – !

. [3] : -, ,

.

1. , . . / . . // -. « , , ». – 2005. – . 5. – 2(42). – . 56–61.

2. , . . f: → , / . . // . « -

. . ». – 2011. – . 4. – 10(227). – . 38–39. 3. , . .

/ . . , . . // . – 2013. – 5. – . 72–74. 30 2013 .

POLYNOMIAL AS A SUM OF PERIODIC FUNCTIONS

A.Yu. Evnin1

It is proved that an arbitrary polynomial of degree n representatives as a sum of periodic functions, the minimum number of terms in this sum is n+1.

Keywords: periodic functions, counterexamples in the analysis.

References 1. Evnin A.Yu. Period summy dvukh periodicheskikh funktsiy (The period of the sum of two peri-

odic functions). Vestnik YuUrGU. Seriya “Matematika, fizika, khimiya”. 2005. Issue 5. no. 2(42). pp. 56–61. (in Russ.).

2. Evnin A.Yu. Primer vsyudu razryvnogo biektivnogo otobrazheniy f: → , obratnoe k ko-toromu nepreryvno v schyetnom mnozhestve tochek (The example of the bijective mapping f: →such that f is everywhere discontinuous, but an inverse of the f is continuous at a countable set of points). Vestnik YuUrGU. Seriya «Matematika. Mekhanika. Fizika». 2011. Issue 4. no. 10(227). pp. 38–39. (in Russ.).

3. Evnin A.Yu., Shved D.A. Predstavimost' funktsiy v vide summy konechnogo chisla periodi-cheskikh funktsiy (Functions’ representability as a sum of a finite number of periodic functions). Mate-matika v shkole. 2013. no. 5. pp. 72–74. (in Russ.).

1 Evnin Alexander Yurievich is Cand. Sc. (Pedagogical), Associate Professor, Applied Mathematics Department, South Ural State University. E-mail: [email protected]


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7. Адрес редакции журнала «Вестник ЮУрГУ» серии «Математика. Механика. Физика»: Россия 454080, г. Челябинск, пр. им. В.И. Ленина, 76, Южно-Уральский государственный универси-

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ВЕСТНИК ЮЖНО-УРАЛЬСКОГО

ГОСУДАРСТВЕННОГО УНИВЕРСИТЕТА

Серия «МАТЕМАТИКА. МЕХАНИКА. ФИЗИКА»

2013, том 5, № 2

Редактор А.Ю. Федерякин

Издательский центр Южно-Уральского государственного университета

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