Issue 45

O. Reut et alii, Frattura ed Integrità Strutturale, 45 (2018) 183-190; DOI: 10.3221/IGF-ESIS.45.16



  , r       



sin

2 2

n





sin n

  r           , r    , 2 2 r q r ' , '

  , r     n n

2



r

0

(4)

n

n n

n

2





sin

x r q

 was done in the Eqn. (4). It was rewritten with the new variables in the following form

A change of variables

   

   

  

  

  

  

  

  

x q

x q

x q

2

2









    

x

x

'

,

'

,

,

0

(5)

n

n n

n

The Kantorovich-Lebedev integral transformation with regard to variable x is applied to the equality (5)

 

0  

 

, x dx q    

i K x 

    

 

(6)



n

n

x

In the transformations (6) domain the Eqn. (5) can be reformulated as

                  1 4 n n n  

2



0

(7)

There is no possibility to apply the integral Legendre transformation by the usual scheme to the Eqn. (7) because there are discontinuities of the function   n    and its derivative when    . The jumps have the following form

  

  

  

   

  

x q

x q

x q



, 0,         ,   

 



, ,

0,

  

  

  

  

x q 

x q 

(8)





 

 

, ,

, ,

  

  

x q









 

, ,





 

 

 

 

0

0

The integral Legendre’s transformation is applied to the Eqn. (7) by the generalized scheme [15]



   P 

 cos sin n 

n k     

d  

(9)



n

k

0

It leads to the linear algebraic equation in the transformations (3), (6), (9) domain

   

   





n

cos

dP











  





  

2

n

k

2    k  n k 

1 / 2 sin 

cos

n 

P



 





k

n





d



  









n

n





dP

dP

cos

cos

k

k



We will accept the designation

in future. Here





d

d

  

185