Issue 49

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

3. Substituting relation (30) into the initial differential Eqn. (21) and equating to zero the expressions with the same powers of x , we obtain the recurrence relations for determination of the coefficients (2) m R , (2) m T           2 (2) (2) (2) 1 2 1 2 3 2 3 2 2 5 2 1 4 1 m m m m m m m T T T m m m m                                             2 (2) 2 (2) (2) 1 1 (2) (2) 2 2 1 2 3 1 2 4 5 1 1 1 2 3 2 2 5 2 2 2 4 1 1 m m m m m m m m m m R T R T m m m m m m m m m R T m m m m                                   (31)   (2) (2) (2) (2) 0 1 0 1 ( 1, 1, 1 , 1, 1) m T T R R          The proposed algorithm is valid for the linear ordinary differential equations of any order. Hence, the proposed method allows us to define step by step the types of the generalized power series for all partial solutions of the initial differential equation and in all partial solutions to set apart the regular from irregular solutions (in our case for 0 x  ). Such a capability of the method is rather essential for constructing solutions to particular problems, for example, for hollow and composite cones. The form of the obtained solutions (1) (2) 0 0 , w w suggests that (1) 0 w is a regular solution, and (2) 0 w is irregular solution at 0 x  . A general solution to differential Eqn. (12) takes the following form:

 

 

 

(1)

(2)

0 2 0 w x C w x C w x     1 0

(32)

where 2 C are the constants defined by the prescribed combination of the boundary conditions (3) - (5). To construct partial solutions to Eqns. (10), (11), corresponding to the axisymmetric deformation, we need to solve this system for 0 v 1 C ,

 









  

   



3 d u x

2 d u x

1 x x



S

   0 du x

1

  

 



 



2 x x 

0

0



 

 

(33)

0 v x

4 2 x

2 1 S

   

 







3

2

2    S 

dx

1

dx

dx

2

 



and with respect to the function 0

u , which yields a differential equation of the fourth order

4

3

2

d u x

d u x

d u x

( )

( )

( )

2

[

]

( x x 2

0

0

0

)

( x x - - -

)( 1 4 8

)

( + - + - - + )( a a )( 2 1 2 2 3 x

)

-

-

x

x

1

4

3

2

dx

dx

dx

(34)

du x

( ) (

) (

2

0

( a a - + - )( 2 2 2 1 x )

)( 1 2 a a a a - + - + )

=

u x

0

( )

0

dx

This equation is the differential equation with a regular singular point. Therefore, the linearly independent partial solutions can be represented in the form of convergent generalized power series. Using the above procedure for constructing such series, we obtain four partial solutions (1) (2) (3) (4) 0 0 0 0 , , , u u u u written as

0      m

    





 

 

  

 



m

1

m

(1) u x 0

(1) A x m

(2) u x 0

(2) A x m

 



,

,



m

0

(35)

 







    1 m  

 

 

  A B x x   (4) (4) ln m m

 

 

m

(3) u x

(3) A B x x  (3) ln m m

(4) u x





,

,





0

0





m

m

0

0

232