Issue 52

O. Kryvyi et alii, Frattura ed Integrità Strutturale, 52 (2020) 33-50; DOI: 10.3221/IGF-ESIS.52.04



1

  ρ , 

  

  V , V   3,

  

  







3 n v       

3,

3,



n j        v 



, e d j    in

j

,



V

.



v

V

1, 3, 5,



n

n

n

n

j



2





 , V ( ρ ) j n

     ( 0, 1, 2, 3, ) n

For determination of coefficient в

in expansions (12), we move on in system (9) to polar

coordinates and apply the finite Fourier transform. Then, taking into account formulas [19]





1

(   ) ik e   



( ) ( ) , dt  

J t J t



k

k

2

2





2 cos( 

)  

 

 

,



k

0









i m 



e

d

0  of representation (12), and the convolution theorem, after simple transformations, we obtain the following system of integral equations 0 1, m m  *   *  * 0 W ( , ), W ( , )   ( ) ( ) , t dt  kn k n i J t J e            





1, 1 4 , W [U ] G , 0 n n n n    4

, a n      

W [U] nn

0, 1, 2, , 

(13)

B

G

n

Here we use the notation





j

1,

*

*

1 3 3 U {U , U , U }, U ( ), U 0.5(U ( 1) U ), n n n n n jn n n V       2 3 1



j

2, 3,

1     n

1     n

  

5, V ( )]   

n 

n 

1 3, V ( )] , U [ n   

n

* 3

1 3, V ( )] , U ( )   

* 3





U ( ) 





[

[



n

n

n

n

n

n

4



c

0.5



  







n n 

n 

5,

1 G 0.5 с   n









c  



24 q F

q

c

c

W [V ],G n n n

,

,



n

n

n

n

n

n

25 ,

3

0 0,



q 

23





 

[ ],  

5, W [V ], n n n

G ( )  

F q F 

( ) 2  

3 0  



12 1  



( ) 

 

q

F

n

n

n

n

n

n

n

2

44

45 ,

a

2



q d

 

0

0 00

G { 



1, ))     n



0 







* 

* * *     d

[ ( d

id

, n n )]}, W [ ] f

f

(

( )W ( , )

,



n

n

n

n

n

kn

4

10 1,

01 1,

1,

q

2

4

65

0



    

   

41 q q q q  2

0

21 23



41 q q q q  2

21 43 41 23 23 q q q q q q q          23 23 , (



3

1

* 3

21 23





 





{ } jm b

{ } jm b

0 , 

2

),

B

B

43

2

43

0 0 1

, j n   Kronecker symbol; n c  constants that are determined from conditions

a

0 

1 d     n

* n U 3



(14)

( )

0

 3 ( ) R

For the operator

there is an inverse [19], which is an isomorphism in

and allows the representation

, W l l

a

t



1 2 

l

1

l



( ) 

f

t

dt

d

2



0 

* , l l W [ ( )] f 

   



d

(15)



l

1 

1 2

1 2

2

2

2

2

dt









t

t

(

)

(

)



37