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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

ξ

= 2

ψ

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

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