10 / 21
ξ
with the approximation error on the level
F
i
=
∂
∂ξ
+
Δ
8
44100
+
O
(Δ
10
)
.
In this case the computation should be done as follows (on the basis of of
the equation system solution with pentadiagonal matrix [7]:
Q
i
=
E
+
5Δ
2
42
˜
f
i
,
e
F
i
=
(
E
+
2Δ
2
7
+
Δ
2
2
70
−
1
Q
i
)
i
,
F
i
=
sign
(
Y
i
+1
−
Y
i
−
1
) min ˜
F
i
+1
,
˜
F
i
,
˜
F
i
−
1
,
F
i
=
F
i
+
sign
(
Y
i
+1
−
Y
i
−
1
)
∙
[1
−
Ind
(
Y
)
i
]
∙
e
F
i
−
F
i
.
In piecewise-polynomial distributions
Y
(
ξ
)
there are space derivatives
of the second order
∂
2
Y
∂ξ
2
i
=
s
i
, which we further conventionally denote
by
s
i
, and compute with the eighth order of accuracy [5]:
9
38
(
s
i
+1
+
s
i
−
1
) +
s
i
=
−
751
342Δ
2
Y
i
+
147
152Δ
2
(
Y
i
+1
+
Y
i
−
1
) +
+
51
380Δ
2
(
Y
i
+2
+
Y
i
−
2
)
−
23
6840Δ
2
(
Y
i
+3
+
Y
i
−
3
)
.
The space derivatives of the fourth
∂
4
Y
∂ξ
4
i
=
S
4
i
and sixth
∂
6
Y
∂ξ
6
i,j
=
S
6
i
orders can be obtained from the relations (i.e., by solving
the equations systems with the tridiagonal matrix) [7]:
Q
i
=
Δ
2
6
˜
f
i
,
f
S
4
i
=
("
E
−
E
+
Δ
2
6
−
1
#
Q
)
i
;
S
4
i
= Δ
4
∂
4
Y
∂ξ
4
i
−
Δ
8
720
+
O
Δ
10
;
Q
i
=
Δ
2
12
˜
S
4
i
,
f
S
6
i
=
("
E
−
E
+
Δ
2
12
−
1
#
Q
)
i
;
S
i
= Δ
6
∂
6
Y
∂ξ
6
i
−
Δ
10
180
+
O
Δ
12
.
The other derivatives included in the piecewise-polynomial distributions
are determined by using variables
∂Y
∂ξ
i
and
∂
2
Y
∂ξ
2
i
.
12 ISSN 0236-3941. HERALD of the BMSTU. Series “Mechanical Engineering”. 2014. No. 1