PSI - Issue 50
Kirill E. Kazakov et al. / Procedia Structural Integrity 50 (2023) 131–136 Kirill E. Kazakov / Structural Integrity Procedia 00 (2022) 000 – 000
134
4
( ) 2 ( ) 2 f x xf x
( ) f x xf x f x ( ) ( ) 2 2
1 ~ x
( ) k k f x 1
( ) 1
( ) 0.
1 xf x
1 f x
(11)
2
2
Let us compare equation (11) with equation (9). In order, for the resulting equation, to be Kummer ’s differential equation, it is necessary that
( ) 2 ( ) 2 f x xf x
(12)
1
,
1 k x
2
( ) f x xf x f x ( ) ( ) 2 2
1 ~ x
(13)
.
k
2
k
k
2
The solution of ordinary differential equation (12) is the function
(14)
( ) x k f x c x e ( 1)/2 /2 1
,
2
0
where c 0 is an arbitrary constant. Substitute this representation into equation (13):
~
2
2
4 1 ~ ( 1) 1 x k k x k
2 4 k x 1 k x
4 1 ( 1) 1 k x
k
4 1 1 2
.
k
1
2
2
k
Due to the linear independence of the functions 1, x , x – 1 , we obtain
2 1 ~
1, k
2,
k
k
(15)
1
2
and, according to formula (14), the function f 2 ( x ) takes the form
(16)
/2
( ) 2 x f x c e 0
.
Thus equation (11) is transformed to Kummer ’s differential equation
2 1 ~
( ) (1 ) ( ) x f x
( ) 0.
1 xf x
1 f x
(17)
1
The general solution of the resulting differential equation (17) is a linear combination of Kummer ’s function (0.5(1 ~),1, ) x M and Tricomi's function (0.5(1 ~),1, ) x U (confluent hypergeometric functions, see Abramowitz and Stegun (1964), Olde Daalhuis (2010)), that is,
.
(18)
( ) f x cM
(0.5(1 ~),1, )
(0.5(1 ~),1, ) x
x cU
1
1
2
Using formulas (10), (15), (16), (18), it is possible to obtain a general solution of differential equation (8):
(0.5(1 ~),1,2ln( / ))
, (0.5(1 ~),1,2ln( / )) 0 r r
1 CM
r r CU
( )
u r
0
2
(19)
r
1 C and
2 C are new arbitrary constants that are determined from additional conditions (6):
where
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