Issue 38

N. Vaysfeld et alii, Frattura ed Integrità Strutturale, 38 (2016) 1-11; DOI: 10.3221/IGF-ESIS.38.01

























    

1 

1 

2 



  













c

c

x c

c

x

1 

1 

1 

1 

c

с

3,1

3,2

3,1

3,2

    

2

















c

d

1

2

2

2

2

 2 2

















x     x  



x

2

 

 

 

 

x

x

x

x

2

2

2





1

1

(13)

 

1 



 1 1 

 x c



 1 1 







 

 

c

x

  

   K x d r x ,    

4

4

  









x    1

d

,

1

3

3









  

x    2 

x

2



1



 1   



a

here     

,     K x r x , ,   are the known regular functions, i

c i , 1, 4   are shown in the Application B.

  



 

2



The Eq. (13) is the partial case of the equation with two fixed singularities considered [21]



m k 

  m k

  y



 

  y



1

1

n









c

x y ,

x

x

1

1

c

1

k    2 

  A x c  

  x

1









dy

dy

0





i 

i 

y x 

m k 

m k k 



k

1



















y

y

xy

1

1

1

k



k

0





1

1

1 



   

 





y dy f x 

, 0 Re 

m k 

K x y ,

k



1

which can be rewritten as

1 



   

  A x c  

 

 

 

 

 

 



x c S x N x N x K x K x y ,         

y dy f x 





0

1 1

1

1

1



1

where



m

  y





1



n



x

1



k

c

k   2 

 



  c x 1, 1 lim ,





N x   1



   



dy c ,

c

,

k k

k

m kk 

i 

 

x

1 1



k

1







   



 

y

x y

1

2





k

0



1





The symbol of the singular integral Eq. (13) was constructed, which has the following form, where all designations correspond to the designations in [21]

  m k 

      

2



  

  

c n 







c S 1

R

,



k k 

2 ,



     

A A

  



   ,  



k

0



,  





A

(14)

  m k 

2





  

  

c n 





,    



c S 1

R

,



k k 

2 ,





k

0

    z , , ,     

  p S z cth i 

  









   

1 / , 

z   , 

,



  m k 

k

   k 1   

 1 ,





iz  

  





n

m

,

k

k

k

 



k

,

k



sh i

z

6