# Векторные операторы для функций, гармонических в шаре

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• 2

. .

-

13 (17) 2009

i z v e S t i apenzenSkogo goSudarStvennogo

pedagogicheSkogo univerSitetaimeni v. g. belinSkogo

phySical, MatheMatical and technical ScienceS

13 (17) 2009

517. 444

,

. . . . . ,

e-mail: julia5507@mail.ru

. . , // . , // . , // . .. . 2009. 13 (17). . 2834. L

1L , - - . , , . : , , .

parfenova y. a. vector operators for harmonic functions in ball // izv. penz. gos. pedagog. univ. im.i v.g. belinskogo. 2009. 13 (17). p. 2834. In the investigation enters operator L and inverse to it operator, which are used at a finding of operators of transformation and the solution of concrete boundary value problems in homogeneous spherically symmetric areas. In the given work the operational solution method of vector boundary value problems is offered, in particular, solution of the mixed boundary value problem in ball for the Laplas euation and the mixed boundary value problem in ball for the Poisson euation are found.keywords: operator of transformation, vector boundary value problems, harmonic functions.

1. - - ---

( )( )

( )

1 1 2 3

1 2 3

1 2 3

, ,, , ...

, ,n

u x x xu x x x

u x x x

=

( ){ }2 2 21 2 3 1 2 3, , /RB x x x x x x R= + + , R - RB . ( )1 2 3, ,iu x x x - RB , ( )1 2 3, , 0iu x x x = ( )1 2 3, ,x x x

RB . L :

( ) ( ) ( )3

1 2 31 2 3 1 2 3

1

, ,, , , , i

i i

u x x xL u x x x u x x x x

x =

= + , (1)

( )11 1

1

n

ij n n

n nn

= =

-

,

( )

( )

( )

1 1 2 3

1 2 3

1 2 3

, ,

, ,

, ,

i

in

i

u x x xx

u x x xx

u x x xx

=

, 1,3i = .

• 2

1. - ( )1 2 3, ,u u x x x= - RB , - ( ) ( )1 2 3 1 2 3, , , ,v x x x L u x x x= RB .

. :

( ) ( ) ( )1 2 3 1 2 3 1 2 3, , , , , ,v x x x L u x x x L u x x x = = . - ( )1 2 3, ,u u x x x= RB , ( )1 2 3, , 0u x x x = .

( )1 2 3, , 0v x x x = , - ( )1 2 3, ,v v x x x= RB . .

2 (.[2]). - ( )1 2 3, ,u u x x x= RB , :

( ) ( )1 2 3 1 2 30

, , , ,kk

u x x x P x x x

=

= ,

( )1 2 3, ,kP x x x - k ; - RB .

3. - ( )1 2 3, ,u u x x x= -:

( )( )

( )( )

1 1 2 3

1 2 3 1 2 30

1 2 3

, ,, , ... , ,

, ,k

kn

u x x xu x x x P x x x

u x x x

=

= =

,

( )1 2 3, ,L u x x x :

( ) ( ) ( )1 2 3 1 2 30

, , , ,kk

L u x x x kE P x x x

=

= + , E - n .

[3].

4. , :

1

2

3

sin cos , 0 2 , 0 ,sin sin , 0 2 , 0 ,cos , 0 .

x rx rx r

= = =

( )1 2 3, ,L u x x x :

( ) ( ) ( ), ,

, , , ,u r

L u r u r rr

= + . (2)

( )1 2 3, ,L u x x x . 1L , L . .

1. , ( )det 0E = . 5. ( ) ( )1 2 3 1 2 3, , , ,L u x x x v x x x = ( )1 2 3, ,v x x x

( ) ( )1 2 3 1 2 30

, , , ,kk

v x x x P x x x

=

= , ( )1 2 3, ,kP x x x , ( )1 2 3, ,v x x x - -, ( )kE + - - k N ,

( ) ( ) ( ) ( )111 2 3 1 2 3 1 2 30

, , , , , ,kk

u x x x L v x x x kE P x x x

=

= = + (3) 3.

1. ,

Re 0i > , 1,3i = ,

( ) ( )1

11 2 3 1 2 3

0

, , , ,EL v x x x v x x x d = ,

( )( )exp lnE E =

• 30

- 13 (17) 2009 .

2. 1 ,..., n ( )minRe 1i m = + , m - ,

( ) ( ) ( )11 1 2 3 1 2 30

, , , ,m

kk

L v x x x kE P x x x=

= + + ( ) ( )1

1 2 3 1 2 300

, , , ,m

E kk

kv x x x P x x x d

=

+

1. k , ( )kE + , -

. -

.

6. ( )kE + pk = , 1 p n , ( ) ( )1 2 3 1 2 30

, , , ,kk

v x x x P x x x

=

= , ( )1 2 3, , 0klP x x x = , 1,l p=

( ) ( ) ( )11 1 2 3 1 2 30

, , , ,kk

L v x x x kE P x x x

=

= + , () , 1,...,k kp .

3. , Re 0k > k , 1 ,..., pk k , ( )kE + , ( )1 2 3, , 0klP x x x = , 1,l p=

( ) ( )1

11 2 3 1 2 3

0

, , , ,EL v x x x v x x x d = 4. 1 ,...,p n +

( )minRe 1i m = + , m - ,

( ) ( ) ( )11 1 2 3 1 2 30

, , , ,m

kk

L v x x x kE P x x x=

= + + ( ) ( )1

1 2 3 1 2 300

, , , ,m

E kk

kv x x x P x x x d

=

+

L

1L ,

- .

2. ( ),u u r = - ( ):

( ), 0u r =

( ) ( )1

,r

u r f == .

, , ( ),u u r = , ( ),u u r = , - :

( ), 0u r =

( ) ( ) ( )1

,,

r

u rhu r r f

r

=

+ =

, (4)

h - , . :

( )0

, k ikkk

u r c r e

=

= .

( ),u u r = ( )0

, k ikkk

u r a r e

=

= , ( ) ikkf e = , ( ), k ikku r r e = .

(4), k - :

( ) ( ) ( )1

,, kk k

r

u rhu r r f

r

=

+ =

,

• 31

:1

1

k ik k ik ikk k r

hc r e rkc r e e =

+ = ,

ik ik ikk khc re kc re e

+ = ,

1kc h k=

+.

( ) 1, k ikku r r eh k =

+ :

1k ik k ikr e r eh k

+

,

kk rr

h k

+.

:1 1 1

0 0 0

kk k k k rf r f r F d r F d r F d

h k,

1

0

1k F dh k

.

( )F [4]. ( ) 1hF = . ,

11 11 1

0 0 0

1k hk h k hd dk h h k

+ + = = =

+ + :

( ) ( )1

1

0

hf r f r d = 1 :

1k

kr rh k

+

( ) ( ) ( ) ( )1 2 3

1 / / /1 2 3x x xf h f x x f x x f x x f x

= + + + .

, .

, -

.

.

3

- ( )( )

( )

1 , ,, , ...

, ,n

u ru r

u r

=

, RB ,

( ), , 0u r =

( ) ( ) ( ), ,

, , ,r R

u ru r R f

n

=

+ =

, (5)

( ),f - RS ,R - , - n ,

n

- .

• 32

- 13 (17) 2009 .

2. RS

( ) ( ), , , ,r R r R

u r u rr

n r

= =

=

(6)

R - .. , -

, .

.

7. ,

3 :2 2 22 1

32 2 2 20 0 0

sin ,, ,

4 2 cosE

R r fRu r d d dR Rr r

, (7)

( )cos sin sin cos cos cos = + .. :

( ) ( ) ( ), ,

, , , ,u r

u r r v rr

+ =

,

( ) ( ), , , ,v r L u r = 1 ( ), ,v r RB . , 2 , , ,

r Rv r f . , ( ), ,v r -

:

( ), , 0v r =

, , ,r R

v r f . :

2 22

32 2 20 0

sin ,, ,

4 2 cos

R r fRv r d dR Rr r

, (8)

( )cos sin sin cos cos cos = + . 1 ,

, :

( ) ( ) ( )1

1

0

, , , , , ,Eu r L v r v r d = = .

(8) :2 2 22 1

32 2 2 20 0 0

sin ,, ,

4 2 cos

ER r fRu r d d dR Rr r

,

( )cos sin sin cos cos cos = + . .

5. , -

:

2 2 221

32 2 2 20 0

sin ,, ,

4 2 cos

R r fRu r L d dR Rr r

.

, 3 .

3. 3

• 33

- ( )( )

( )

1 , ,, , ...

, ,n

u ru r

u r

=

,

( ) ( ), , , ,u r f r = , RS :

( ) ( ), ,

, , 0r R

u ru r R

n

=

+ =

, (9)

( ), ,f r - RB , R - , - n ,

n

- .

3. ( ), ,u u r = RB , :

( ) ( )1

1

0

, , , ,Eu v r d L v r = , ( ), ,v v r = - .

. :

( )( ) ( ) ( ), ,, , , ,E u ru r u r

= + =

( ) ( ) ( ) ( ), , , ,

, , , ,E Eu r u r

u r r u r rr r

= + = + =

( )( ) ( ), , , ,E EL u r v r = = .

1 11

00 0

, , , , , , , ,Ev r d u r d u r u r .

.

6. L 1L

.

8. ,

:

( )( ) ( )( )

2 2 21 2 1

32 2 2 20 0 0 0

sin , ,, ,

4 2 cos

ER r L f RRu r d d d dR

R Rr r

=

+ ,

( )cos sin sin cos cos cos = + ,( ), ,f r - .

. :

( ) ( ) ( ), ,

, , , ,u r

u r r v rr

+ =

,

( ) ( ), , , ,v r L u r = 3 :

( ) ( )1, , , ,u r L u r = ., ( ), ,u u r = ( ) ( ), , , ,u r f r = , :

( ) ( ) ( ), , , , , ,v r L u r L f r = = .

• 34

- 13 (17) 2009 .