Issue 49

V. Matveenko et alii, Frattura ed Integrità Strutturale, 49 (2019) 225-242; DOI: 10.3221/IGF-ESIS.49.23

1 2



 

 

 x x 2

 1 4  2



 

2

2



18 1 x      k 

 



2 f x

x

13



  f x x x   3 6

  3 1 2 1 x  

  f x x x 



4

4  



1

(39)

3

4

 1 1 x       1 1



 

  

   x x x x 2 1

 2   



2

2



x x 



k

;

1

0

2

4



1 4

  

 1 1 x      2 k

  x

x x 

 

0 

 

1



k

2







x

1

  x x x   1 4



 



 

2

2

1 



4 k       1 x

1



k

2







2

3



k 

x

x

1

1

  x

 x x 2 5 2 1 

 

 

3

2 

3 

 

2 x x 

k

u to the functions

Moreover, the results of these transformations is a set of equations, which relate the function k

w v

, k k

and their derivatives

ì ïï

2

( x x - 1

)

d w x

dw x

( )

( )

( x x 2

k

k

)

( 4 2 1 x x )

=

-

+ -

-

u x

) íï - + + ïî 4 2 2

1

( )

k

2

( ) ( x x k S S a 2 1

dx

dx

ü ùïï ú ý ï ú ûïþ

é ê ë

dv x

( )

( x x é ù - - + + + + - - ê )( 1 1 ) ( k S ) 1 1 ( ) w x kSx x ( 2 4 2 a a

k

( + - + ) ( 2 1 2 k x S v x ) ( )

(40)

)

1

ê ë

ú û

k

k

dx

Eqn. (38) is independent of (39) and represents a linear differential equation of the fourth order with respect to the function k w . Eqn. (39) can be considered as a differential equation of the second order with respect to the function k v with right –hand side depending on k w . Such a peculiarity of the differential Eqns. (28) and the obtained relation (30) allows us to determine the sequence of constructing partial solutions for functions , , k k k w v u . The hang of this sequence can be outlines as follows. First, from the solution of Eqn. (38) we obtain four partiсular solutions (1) (2) (3) (4) , , , k k k k w w w w , written as

   

   

   

   





k

k

1

1

  

  

  

  

0      m

 





m

m

 

 

(1) w x

(1) A x m

(2) w x

(2) A x m

2

2





,

,

k

k



m

0

(41)

   

   

   

   





k

k

1

1

  

  

  

  

 





m

m









 

  A B x x  (3) (3) ln

 

  A B x x  (4) (4) ln

(3) w x

(4) w x

2

2





,

k

m

m

k

m

m





m

m

0

0

(1)

(2)

(3)

(4)

(3)

(4)

, m m m A A A A B B can be determined and calculated E-resource (www. , , , , m m m

where the coefficients

icmm.ru/compcoeff). Then substituting the obtained particular solutions (1) (2) (3) (4) , , , k k k k w w w w

into the right-hand side of Eqn. (28. b) and v v v v , which can be written as

solving it as an inhomogeneous equation we arrive at four partial solutions (1) (2) (3) (4) , , , k k k k

é ê ê ê ë

ù ú ú ú û

é ê ê ê ë

ù ú ú ú û

æ ç ç ç è

ö ÷ ÷ ÷ ø

æ ç ç ç è

ö ÷ ÷ ÷ ø

+

-

k

k

1

1

¥ å

¥ å

+

+

m

m

(1) v x

(2) v x

(1) P x m

(2) P x m

2

2

=

=

,

,

( )

( )

k

k

=

=

m

m

0

0

ì ï ï ï ï ï ï î

ü ï ï ï ý ï ï ï þ

( ) P D x x æ ù + í (3) (3) ln m m

ö- ÷

k

1

¥ å

2 ç - ÷ ç ÷ çè ø m

é ê ë

(3) v x

=

,

(42)

( )

ú û

k

=

m

0

234