Issue 48
A. Fesenko et alii, Frattura ed Integrità Strutturale, 48 (2019) 768-792; DOI: 10.3221/IGF-ESIS.48.70
T HE CONSTRUCTION OF THE MATRIX FUNDAMENTAL SOLUTION SYSTEM , THE MATRIX BASIC SOLUTION SYSTEM , THE FUNDAMENTAL MATRIX AND G REEN MATRIX FUNCTION
A
ccording to [39] the basic matrices are the solutions of the boundary matrix problems
( ) 0,
Ψ δ ( )
L z Ψ
U z
(26)
j
i
j
ij
2
ij δ is Kronecker’s symbol. The basic matrices are constructed in the form
z i Ψ Y C Y C 0 1 ( ) , ( ) z ( ) z
0,1
i
i
i
where z z Y Y are the matrix fundamental solutions of the homogeneous equation (26) correspondingly decreasing and increasing in infinity. The algorithm of fundamental solutions ( ) z Y derivation is described in [37], where the method of fundamental matrix equation is also indicated. According to this approach, the solution is searched as the contour integral [39, 40] ( ), ( )
1 2 i
sz
1
( ) z
e
s ds
( )
(27)
Y
M
C
2
1 ( ) s M , where
2
2 s N Q P ,
2 s
2
( ) s s M I
det ( ) s M
N
where contour C covers all poles of the matrix
,
0
s N . Let’s calculate matrix
1
*
s s M M M , so (27) will be transformed ( ) s ( )/ det ( )
sz
e
C
*
( ) z
s ds
( )
1
Y
M
i
2
2
s N 2
2
here * ( ) s M is a union matrix. With the help of the residual theorem, one derives
j
d dz
sz
e
2
( ) j
( ) Re
( ) z
( ), y z y z
s
s
N
,
Y
Γ
2 ) ( s N s N
2
(
)
j
0
3 1 N e Nz ( Nz
Nz
3 1 N e
( ) (4 )
( ) (4 )
y z
Nz
y z
(
1),
1)
1
2
( ) , j
j
0, 2
4
s N N or 2 2 4 2
i i
1
( )
( ) p s s
( ) s
s
Here, to find matrices
one uses the fact, that
,
Γ
M
Γ
4
4 p s
( )
i
0
2
( ) , j
j
0, 2
i i
( )
4 p s
( ) s M Γ
s
( )
. Using that coefficient near the same exponents should be equal, the matrixes
are
Γ
I
i
0
found
2 1
(2)
(1)
(0)
N
,
,
P .
Γ I Γ
Q Γ
0
As a result, the fundamental matrix system is derived
1
Nz
z
Nz e
*
(z)
Y
0
2 N z
1
Nz
(
)
*
776
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