Issue 48
A. Fesenko et alii, Frattura ed Integrità Strutturale, 48 (2019) 768-792; DOI: 10.3221/IGF-ESIS.48.70
, ( , , ) (1 )w ( , , ) Z x y h x y h
1 ( ) ( ) 2 G x a y b 2
1
T x y h
( , , )
.
0
0
In the domain of Fourier’s transforms (the cosine transform with regard to the variable x and full Fourier transform with regard to the variable y ) one can derive two boundary problems. First, it is formulated for the unknown function ( , , ) S x y z
S z N z 2 ( )
( ) 0,
(0) 0
z h S h ,
S
( ) 0, 0
S
( ) 0 S z , and, hence,
( , , ) 0 S x y z .
It’s obvious, that this problem has a trivial solution
( ), w ( ) Z z z the one dimensional boundary problem is stated
For the functions
( ) ( ), 0
L z z y f
z h
2
(24)
( ) y
( ) y
U z
U z
,
γ
γ
0
0 1
1
2 L is the differential operator of second order
here
2
" ( ) z
z N z z Py f '( )
( ) ( ), 0
2 L z
z h
( ) y I y
Qy
0
N z T T 2 ( ) z
1
0
w ( ) ( ) ( ) z Z z z y
1
0
*
*
( ) z
I is unit matrix,
,
,
,
is the unknown vector of
Q
* f
Р
2
N
( )
0
0
transforms. The boundary functionals
, 0,1
i U i
are written in the corresponding forms
( ) z z y I y I y γ [ ] ( )
U
0
1
2
0
( ) z z y Ay By γ [ ] ( )
U
1
1
1 0 0 0
0 0 0 1
0 1
2
0 N
0
,
,
,
I
I
A
B
1
2
1 0
0
0
,
.
γ
γ
)
)
1
1
(2 G
1
a e
cos
(2
0
i b
T h
( )
0
0
The solution of the problem (24) is searched in the form [37]
βα w ( ) z
y
h
0
( , ) ( ) z d G f
( ) z Ψ γ Ψ γ ( ) z
( ) z
(25)
βα Z z
0
0
1
1
( )
( ), 0,1 i
z i Ψ
( , ) z G іs Green matrix function,
where
are the basis matrices.
775
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