12 / 21
−
2
5!
∂
4
Y
∂ξ
4
i
Δ
ξ
2
5
+
F
6
,i
˜
ψ
6!
−
Δ
ξ
2
6
∂
5
Y
∂ξ
5
i
−
+
F
7
,i
˜
ψ
7!
−
Δ
ξ
2
7
∂
6
Y
∂ξ
6
i
−
2
7!
∂
6
Y
∂ξ
6
i
Δ
ξ
2
7
,
where
ψ
=
|
V
|
t
,
˜
ψ
= 2
|
V
|
t
/Δ
ξ
= 2
ψ
/Δ
ξ
).
Here, the following functions are introduced (symbol
e
Ψ
m
, used in the
following formulas, means exponentiation of the value
e
Ψ
to the
m
-th
power):
F
2
˜
ψ
=
−
2
e
Ψ
0
+
e
Ψ
1
;
F
3
˜
ψ
=
−
3
e
Ψ
0
+ 3
e
Ψ
1
−
e
Ψ
2
;
F
4
˜
ψ
=
−
4
e
Ψ
0
+ 6
e
Ψ
1
−
4
e
Ψ
2
+
e
Ψ
3
;
F
5
˜
ψ
=
−
5
e
Ψ
0
+ 10
e
Ψ
1
−
10
e
Ψ
2
+ 5
e
Ψ
3
−
e
Ψ
4
;
F
6
˜
ψ
=
−
6
e
Ψ
0
+ 15
e
Ψ
1
−
20
e
Ψ
2
+ 15
e
Ψ
3
−
6
e
Ψ
4
+
e
Ψ
5
;
F
7
˜
ψ
=
−
7
e
Ψ
0
+ 21
e
Ψ
1
−
35
e
Ψ
2
+ 35
e
Ψ
3
−
21
e
Ψ
4
+ 7
e
Ψ
5
−
e
Ψ
6
.
When the solution describing the displaceable part of the equiation
(i.e.
∂c
∂t
+
V
∂c
∂x
= 0)
is obtained, it is also possible to retrive the
distribution by the piecewise-polynomial distributions
Y
(
x
)
,
[
x
=
{
ξ
}
]
,
ξ
∈ −
Δ
ξ
2
,
Δ
ξ
2
,
approximate by parabola
q
(
x
) =
q
L
i
+
ξ
(Δ
q
i
+
q
6
i
(1
−
ξ
))
,
where
ξ
=
x
−
x
i
−
1
/
2
h
−
1
,
Δ
q
i
=
q
R
i
−
q
L
i
,
q
6
i
= 6
q
i
−
(1
/
2)
q
L
i
+
q
R
i
,
q
L
i
= ˜
Y
L
i
+1
/
2
(
|
V
|
t
)
,
q
R
i
= ˜
Y
R
i
+1
/
2
(
|
V
|
t
)
.
Then, the average value
˜
q
of the function
q
(
x
)
can be found in separate
intervals:
•
for interval
x
i
+1
/
2
− |
V
|
t, x
i
+1
/
2
, if
V >
0
:
˜
q
L
i
+1
/
2
(
y
) =
q
R
i
−
(1
/
2)
yh
−
1
Δ
q
i
−
q
6
i
1
−
(2
/
3)
yh
−
1
, y
=
V t
;
•
for interval
x
i
+1
/
2
, x
i
+1
/
2
+
|
V
|
t
, if
V <
0
:
˜
q
R
i
+1
/
2
(
y
) =
q
L
i
+1
+ (1
/
2)
yh
−
1
Δ
q
i
+1
+
q
6
i
+1
1
−
(2
/
3)
yh
−
1
, y
=
−
V t.
Note, that distribution
q
(
x
)
is connected with the mentioned above
distribution
Y
by formulae:
Δ
q
i
=
∂Y
∂ξ
i
,
q
6
i
=
−
1
2
∂
2
Y
∂ξ
2
i
.
14 ISSN 0236-3941. HERALD of the BMSTU. Series “Mechanical Engineering”. 2014. No. 1