MODELING OF MANUFACTURING OF A FIELDEFFECT TRANSISTOR TO DETERMINE CONDITIONS TO DECREASE LENGTH OF...

Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, June 2015

31

MODELING OF MANUFACTURING OF A FIELD-

EFFECT TRANSISTOR TO DETERMINE CONDITIONS

TO DECREASE LENGTH OF CHANNEL

E.L. Pankratov

1, E.A. Bulaeva

2

1Nizhny Novgorod State University, 23 Gagarin avenue, Nizhny Novgorod, 603950,

Russia 2 Nizhny Novgorod State University of Architecture and Civil Engineering, 65 Il'insky

street, Nizhny Novgorod, 603950, Russia

ABSTRACT

In this paper we introduce an approach to model technological process of manufacture of a field-effect

heterotransistor. The modeling gives us possibility to optimize the technological process to decrease length

of channel by using mechanical stress. As accompanying results of the decreasing one can find decreasing

of thickness of the heterotransistors and increasing of their density, which were comprised in integrated

circuits.

KEYWORDS

Modelling of manufacture of a field-effect heterotransistor, Accounting mechanical stress; Optimization the

technological process, Decreasing length of channel

1. INTRODUCTION

One of intensively solved problems of the solid-state electronics is decreasing of elements of in-

tegrated circuits and their discrete analogs [1-7]. To solve this problem attracted an interest wide-

ly used laser and microwave types of annealing [8-14]. These types of annealing leads to genera-

tion of inhomogeneous distribution of temperature. In this situation one can obtain increasing of

sharpness of p-n-junctions with increasing of homogeneity of dopant distribution in enriched area

[8-14]. The first effects give us possibility to decrease switching time of p-n-junction and value of

local overheats during operating of the device or to manufacture more shallow p-n-junction with

fixed maximal value of local overheat. An alternative approach to laser and microwave types of

annealing one can use native inhomogeneity of heterostructure and optimization of annealing of

dopant and/or radiation defects to manufacture diffusive –junction and implanted-junction rectifi-

ers [12-18]. It is known, that radiation processing of materials is also leads to modification of dis-

tribution of dopant concentration [19]. The radiation processing could be also used to increase of

sharpness of p-n-junctions with increasing of homogeneity of dopant distribution in enriched area

[20, 21]. Distribution of dopant concentration is also depends on mechanical stress in heterostruc-

ture [15].

In this paper we consider a field-effect heterotransistor. Manufacturing of the transistor based on

combination of main ideas of Refs. [5] and [12-18]. Framework the combination we consider a


32

heterostructure with a substrate and an epitaxial layer with several sections. Some dopants have

been infused or implanted into the section to produce required types of conductivity. Farther we

consider optimal annealing of dopant and/or radiation defects. Manufacturing of the transistor has

been considered in details in Ref. [17]. Main aim of the present paper is analysis of influence of

mechanical stress on length of channel of the field-effect transistor.

Fig. 1. Heterostructure with substrate and epitaxial layer with two or three sections

a1

a2 -a

1L

x -a2 -a

1

c1 c1

c2

oxide

source drain

gate

c3

Fig. 2. Heterostructure with substrate and epitaxial layer with two or three sections. Sectional view


33

2. METHOD OF SOLUTION

To solve our aim we determine spatio-temporal distribution of concentration of dopant in the con-

sidered heterostructure and make the analysis. We determine spatio-temporal distribution of con-

centration of dopant with account mechanical stress by solving the second Fick’s law [1-4,22,23]

( ) ( ) ( ) ( )+

∂

∂

∂

∂+

∂

∂

∂

∂+

∂

∂

∂

∂=

∂

∂

z

tzyxCD

zy

tzyxCD

yx

tzyxCD

xt

tzyxC ,,,,,,,,,,,, (1)

( ) ( ) ( ) ( )

∫∇

∂

∂Ω+

∫∇

∂

∂Ω+

zz L

S

SL

S

S WdtWyxCtzyxTk

D

yWdtWyxCtzyx

Tk

D

x 00

,,,,,,,,,,,, µµ

with boundary and initial conditions

( )0

,,,

0

=∂

∂

=xx

tzyxC,

( )0

,,,=

∂

∂

= xLxx

tzyxC,

( )0

,,,

0

=∂

∂

=yy

tzyxC,

( )0

,,,=

∂

∂

= yLxy

tzyxC,

( )0

,,,

0

=∂

∂

=zz

tzyxC,

( )0

,,,=

∂

∂

= zLxz

tzyxC, C (x,y,z,0)=fC(x,y,z).

Here C(x,y,z,t) is the spatio-temporal distribution of concentration of dopant; Ω is the atomic vo-

lume; ∇S is the operators of surficial gradient; ( )∫zL

zdtzyxC0

,,, is the surficial concentration of

dopant on interface between layers of heterostructure; µ (x,y,z,t) is the chemical potential; D and

DS are coefficients of volumetric and surficial (due to mechanical stress) of diffusion. Values of

the diffusion coefficients depend on properties of materials of heterostructure, speed of heating

and cooling of heterostructure, spatio-temporal distribution of concentration of dopant. Concen-

traional dependence of dopant diffusion coefficients have been approximated by the following

relations [24]

( ) ( )( )

+=

TzyxP

tzyxCTzyxDD

L,,,

,,,1,,,

γ

γ

ξ , ( ) ( )( )

+=

TzyxP

tzyxCTzyxDD

SLSS,,,

,,,1,,,

γ

γ

ξ . (2)

Here DL (x,y,z,T) and DLS (x,y,z,T) are spatial (due to inhomogeneity of heterostructure) and tem-

perature (due to Arrhenius law) dependences of dopant diffusion coefficients; T is the temperature

of annealing; P (x,y,z,T) is the limit of solubility of dopant; parameter γ could be integer in the

following interval γ ∈[1,3] [24]; V (x,y,z,t) is the spatio-temporal distribution of concentration of

radiation of vacancies; V* is the equilibrium distribution of vacancies. Concentrational depen-

dence of dopant diffusion coefficients has been discussed in details in [24]. Chemical potential

could be determine by the following relation [22]

µ =E(z)Ωσij[uij(x,y,z,t)+uji(x,y,z,t)]/2, (3)

where E is the Young modulus; σij is the stress tensor;

∂

∂+

∂

∂=

i

j

j

i

ijx

u

x

uu

2

1 is the deformation

tensor; ui, uj are the components ux(x,y,z,t), uy(x,y,z,t) and uz(x,y,z,t) of the displacement vector

( )tzyxu ,,,r

; xi, xj are the coordinates x, y, z. Relation (3) could be transform to the following form


34

( ) ( ) ( ) ( ) ( )

×

∂

∂+

∂

∂

∂

∂+

∂

∂=

i

j

j

i

i

j

j

i

x

tzyxu

x

tzyxu

x

tzyxu

x

tzyxutzyx

,,,,,,

2

1,,,,,,,,,µ

( )( )

( )( ) ( ) ( ) ( )[ ]

−−

−

∂

∂

−+−

Ω×

ij

k

kij

ijTtzyxTzzK

x

tzyxu

z

zzE δβε

σ

δσδε 000 ,,,3

,,,

212,

where σ is the Poisson coefficient; ε0=(as-aEL)/aEL is the mismatch strain; as, aEL are the lattice

spacings for substrate and epitaxial layer, respectively; K is the modulus of uniform compression;

β is the coefficient of thermal expansion; Tr is the equilibrium temperature, which coincide (for

our case) with room temperature. Components of displacement vector could be obtained by solv-

ing of the following equations [23]

( )( ) ( ) ( ) ( )

( )( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

∂

∂+

∂

∂+

∂

∂=

∂

∂

∂

∂+

∂

∂+

∂

∂=

∂

∂

∂

∂+

∂

∂+

∂

∂=

∂

∂

z

tzyx

y

tzyx

x

tzyx

t

tzyxuz

z

tzyx

y

tzyx

x

tzyx

t

tzyxuz

z

tzyx

y

tzyx

x

tzyx

t

tzyxuz

zzzyzxz

yzyyyxy

xzxyxxx

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

2

2

2

2

2

2

σσσρ

σσσρ

σσσρ

where ( )

( )[ ]( ) ( ) ( ) ( )

×∂

∂+

∂

∂−

∂

∂+

∂

∂

+=

k

k

ij

k

kij

i

j

j

i

ijx

tzyxu

x

tzyxu

x

tzyxu

x

tzyxu

z

zE ,,,,,,

3

,,,,,,

12δ

δ

σσ

( ) ( ) ( ) ( )[ ]r

TtzyxTzKzzK −−× ,,,β , ρ (z) is the density of materials, δij is the Kronecker sym-

bol. With account of the relation the last system of equation takes the form

( )( )

( )( )

( )[ ]( )

( )( )

( )[ ]( )

+∂∂

∂

+−+

∂

∂

++=

∂

∂

yx

tzyxu

z

zEzK

x

tzyxu

z

zEzK

t

tzyxuz

yxx,,,

13

,,,

16

5,,,2

2

2

2

2

σσρ

( )( )[ ]

( ) ( )( )

( )( )[ ]

( )( ) ×−

∂∂

∂

+++

∂

∂+

∂

∂

++ zK

zx

tzyxu

z

zEzK

z

tzyxu

y

tzyxu

z

zE zzy ,,,

13

,,,,,,

12

2

2

2

2

2

σσ

( ) ( )x

tzyxTz

∂

∂×

,,,β

( )( ) ( )

( )[ ]( ) ( )

( ) ( )( )

+∂

∂−

∂∂

∂+

∂

∂

+=

∂

∂

y

tzyxTzKz

yx

tzyxu

x

tzyxu

z

zE

t

tzyxuz xyy ,,,,,,,,,

12

,,, 2

2

2

2

2

βσ

ρ

( )( )[ ]

( ) ( ) ( )( )[ ]

( )( )

+∂

∂

++

+

∂

∂+

∂

∂

+∂

∂+

2

2 ,,,

112

5,,,,,,

12 y

tzyxuzK

z

zE

y

tzyxu

z

tzyxu

z

zE

z

yzy

σσ

( )( )

( )[ ]( )

( )( )

yx

tzyxuzK

zy

tzyxu

z

zEzK

yy

∂∂

∂+

∂∂

∂

+−+

,,,,,,

16

22

σ (4)

( )( ) ( ) ( ) ( ) ( )

×

∂∂

∂+

∂∂

∂+

∂

∂+

∂

∂=

∂

∂

zy

tzyxu

zx

tzyxu

y

tzyxu

x

tzyxu

t

tzyxuz

yxzzz,,,,,,,,,,,,,,,

22

2

2

2

2

2

2

ρ


35

( )( )[ ]

( )( ) ( ) ( ) ( )

×∂

∂−

∂

∂+

∂

∂+

∂

∂

∂

∂+

+×

z

tzyxT

z

tzyxu

y

tzyxu

x

tzyxuzK

zz

zE xyx ,,,,,,,,,,,,

12 σ

( ) ( ) ( )( )

( ) ( ) ( ) ( )

∂

∂−

∂

∂−

∂

∂−

∂

∂

+∂

∂+×

z

tzyxu

y

tzyxu

x

tzyxu

z

tzyxu

z

zE

zzzK zyxz

,,,,,,,,,,,,6

16

1

σβ .

Conditions for the displacement vector could be written as

( )0

,,,0=

∂

∂

x

tzyur

; ( )

0,,,

=∂

∂

x

tzyLux

r

; ( )

0,,0,

=∂

∂

y

tzxur

; ( )

0,,,

=∂

∂

y

tzLxuy

r

;

( )0

,0,,=

∂

∂

z

tyxur

; ( )

0,,,

=∂

∂

z

tLyxuz

r

; ( )00,,, uzyxurr

= ; ( )0,,, uzyxurr

=∞ .

Farther let us analyze spatio-temporal distribution of temperature during annealing of dopant. We

determine spatio-temporal distribution of temperature as solution of the second law of Fourier [25]

( ) ( ) ( ) ( ) ( )+

∂

∂

∂

∂+

∂

∂

∂

∂+

∂

∂

∂

∂=

∂

∂

z

tzyxT

zy

tzyxT

yx

tzyxT

xt

tzyxTTc

,,,,,,,,,,,,λλλ

( )tzyxp ,,,+ (5)

with boundary and initial conditions

( )0

,,,

0

=∂

∂

=xx

tzyxT,

( )0

,,,=

∂

∂

= xLxx

tzyxT,

( )0

,,,

0

=∂

∂

=yy

tzyxT,

( )0

,,,=

∂

∂

= yLxy

tzyxT,

( )0

,,,

0

=∂

∂

=zz

tzyxT,

( )0

,,,=

∂

∂

= zLxz

tzyxT, T (x,y,z,0)=fT (x,y,z).

Here λ is the heat conduction coefficient. Value of the coefficient depends on materials of hetero-

structure and temperature. Temperature dependence of heat conduction coefficient in most inter-

est area could be approximated by the following function: λ(x,y,z,T)=λass(x,y,z)[1+µ Tdϕ/

Tϕ(x,y,z,t)] (see, for example, [25]). c(T)=cass[1-ϑ exp(-T(x,y,z,t)/Td)] is the heat capacitance; Td is

the Debye temperature [25]. The temperature T(x,y,z,t) is approximately equal or larger, than

Debye temperature Td for most interesting for us temperature interval. In this situation one can

write the following approximate relation: c(T)≈cass. p(x,y,z,t) is the volumetric density of heat,

which escapes in the heterostructure. Framework the paper it is attracted an interest microwave annealing of dopant. This approach leads to generation inhomogenous distribution of temperature

[12-14,26]. Such distribution of temperature gives us possibility to increase sharpness of p-n-

junctions with increasing of homogeneity of dopant distribution in enriched area. In this situation

it is practicably to choose frequency of electro-magnetic field. After this choosing thickness of

scin-layer should be approximately equal to thickness of epitaxial layer.

First of all we calculate spatio-temporal distribution of temperature. We calculate spatio-temporal

distribution of temperature by using recently introduce approach [15, 17,18]. Framework the ap-

proach we transform approximation of thermal diffusivity α ass(x,y,z) =λass(x,y,z)/cass =α0ass[1+εT

gT(x,y,z)]. Farther we determine solution of Eqs. (5) as the following power series

( ) ( )∑ ∑=∞

=

∞

=0 0

,,,,,,i j

ij

ji

TtzyxTtzyxT µε . (6)


36

Substitution of the series into Eq.(6) gives us possibility to obtain system of equations for the ini-

tial-order approximation of temperature T00(x,y,z,t) and corrections for them Tij(x,y,z,t) (i≥1, j≥1).

The equations are presented in the Appendix. Substitution of the series (6) in boundary and initial

conditions for spatio-temporal distribution of temperature gives us possibility to obtain boundary

and initial conditions for the functions Tij(x,y,z,t) (i≥0, j≥0). The conditions are presented in the

Appendix. Solutions of the equations for the functions Tij(x,y,z,t) (i≥0, j≥0) have been obtain as

the second-order approximation on the parameters ε and µ by standard approaches [28,29] and

presented in the Appendix. Recently we obtain, that the second-order approximation is enough

good approximation to make qualitative analysis and to obtain some quantitative results (see, for

example, [15,17,18]). Analytical results have allowed to identify and to illustrate the main depen-

dence. To check our results obtained we used numerical approaches.

Farther let us estimate components of the displacement vector. The components could be obtained

by using the same approach as for calculation distribution of temperature. However, relations for

components could by calculated in shorter form by using method of averaging of function correc-tions [12-14,16,27]. It is practicably to transform differential equations of system (4) to integro-

differential form. The integral equations are presented in the Appendix. We determine the first-

order approximations of components of the displacement vector by replacement the required

functions uβ(x,y,z,t) on their average values αuβ1. The average values αuβ1 have been calculated by

the following relations

αus1=Ms1/4LΘ, (7)

where ( )∫ ∫ ∫ ∫=Θ

− −0 0

,,,x

x

y

y

zL

L

L

L

L

isistdxdydzdtzyxuM . The replacement leads to the following results

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

−∫ ∫

∂

∂−−∫ ∫

∂

∂=

∞ t zz

xdwd

x

wyxTwwKtdwd

x

wyxTwwKttzyxu

0 00 01

,,,,,,,,, τ

τβττ

τα

( )]101

,,,uxxux

twyx αφα +Φ− ,

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

−∫ ∫

∂

∂−−∫ ∫

∂

∂=

∞ t zz

y dwdy

wyxTwwKtdwd

y

wyxTwwKttzyxu

0 00 01

,,,,,,,,, τ

τβττ

τα

( )]101

,,,uyyuy

twyx αφα +Φ− ,

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

−∫ ∫

∂

∂−∫ ∫

∂

∂=

∞ t zz

zdwd

w

wyxTwwKtdwd

w

wyxTwwKttzyxu

0 00 01

,,,,,,,,, τ

τβττ

τα

( )]101

,,,zzuz

twyx αφα +Φ− .

Substitution of the above relations into relations (7) gives us possibility to obtain values of the

parameters αuβ1. Appropriate relations could be written as

( ) ( )[ ]( ) ( )∫ −

ΘΧ−∞ΧΘ=

zL

z

xxz

ux

zdzzLL

L

0

3

20

1

8 ρα ,

( ) ( )[ ]( ) ( )∫ −

ΘΧ−∞ΧΘ=

zL

z

yyz

uy

zdzzLL

L

0

3

20

1

8 ρα ,

( ) ( )[ ]( ) ( )∫ −

ΘΧ−∞ΧΘ=

zL

z

zzz

uz

zdzzLL

L

0

3

20

1

8 ρα ,

where ( ) ( ) ( ) ( ) ( )∫ ∫ ∫ ∫

∂

∂−

Θ+=ΘΧ

Θ

− −0 0

,,,1

x

x

y

y

zL

L

L

L

L

z

i

isdxdydzd

s

zyxTzzKzL

tτ

τχ .


37

The second-order approximations of components of the displacement vector by replacement of

the required functions uβ(x,y,z,t) on the following sums αus2+us1(x,y,z,t), where αus2=(Mus2-Mus1)/

4L3Θ. Results of calculations of the second-order approximations us2(x,y,z,t) and their average

values αus2 are presented in the Appendix, because the relations are bulky.

Spatio-temporal distribution of concentration of dopant we obtain by solving the Eq.(1). To solve

the equations we used recently introduce approach [15,17,18] and transform approximations of

dopant diffusion coefficients DL (x,y,z,T) and DSL (x,y,z, T) to the following form:

DL(x,y,z,T)=D0L[1+ε L gL(x,y,z,T)] and DSL(x,y,z,T)=D0SL[1+ εSLgSL(x,y,z,T)]. We also introduce the

following dimensionless parameter: ω = D0SL/ D0L. In this situation the Eq.(1) takes the form

( ) ( )[ ] ( )( )

( ) ( )

×∂

∂

∂

∂+

∂

∂

++

∂

∂=

∂

∂

y

tzyxC

yx

tzyxC

TzP

tzyxCTzg

xD

t

tzyxCLLL

,,,,,,

,

,,,1,1

,,,0 γ

γ

ξε

( )[ ] ( )( )

( )[ ] ( )( )

×

++

∂

∂+

++×

TzyxP

tzyxCTzyxg

zDD

TzyxP

tzyxCTzg

LLLLLL,,,

,,,1,,,1

,,,

,,,1,1

00 γ

γ

γ

γ

ξεξε

( ) ( )[ ] ( )( )

( )

×∫

++

∂

∂Ω+

∂

∂×

zL

SLSLSLWdtWyxC

TzP

tzyxCTzg

xD

z

tzyxC

00 ,,,

,

,,,1,1

,,,γ

γ

ξεω (8)

( ) ( )( )

( )( )

∫∇

+

∂

∂Ω+

∇

×zL

S

SLL

S WdtWyxCTk

tzyx

TzP

tzyxC

yDD

Tk

tzyx

000 ,,,

,,,

,

,,,1

,,, µξω

µγ

γ

.

We determine solution of Eq.(1) as the following power series

( ) ( )∑ ∑ ∑=∞

=

∞

=

∞

=0 0 0

,,,,,,i j k

ijk

kji

LtzyxCtzyxC ωξε . (9)

Substitution of the series into Eq.(8) gives us possibility to obtain systems of equations for initial-

order approximation of concentration of dopant C000(x,y,z,t) and corrections for them Cijk(x, y,z,t)

(i ≥1, j ≥1, k ≥1).The equations are presented in the Appendix. Substitution of the series into ap-

propriate boundary and initial conditions gives us possibility to obtain boundary and initial condi-

tions for all functions Cijk(x,y,z,t) (i ≥0, j ≥0, k ≥0). Solutions of equations for the functions

Cijk(x,y,z,t) (i ≥0, j ≥0, k ≥0) have been calculated by standard approaches [28,29] and presented in

the Appendix.

Analysis of spatio-temporal distributions of concentrations of dopant and radiation defects has

been done analytically by using the second-order approximations framework recently introduced

power series. The approximation is enough good approximation to make qualitative analysis and

to obtain some quantitative results. All obtained analytical results have been checked by compari-

son with results, calculated by numerical simulation.

3. DISCUSSION

In this section we analyzed redistribution of dopant under influence of mechanical stress based on

relations calculated in previous section. Fig. 3 show spatio-temporal distribution of concentration

of dopant in a homogenous sample (curve 1) and in heterostructure with negative and positive

value of the mismatch strain ε0 (curve 2 and 3, respectively). The figure shows, that manufactur-

ing field-effect heterotransistors gives us possibility to make more compact field-effect transistors

in direction, which is parallel to interface between layers of heterostructure. As a consequence of


38

the increasing of compactness one can obtain decreasing of length of channel of field- effect tran-

sistor and increasing of density of the transistors in integrated circuits. It should be noted, that

presents of interface between layers of heterostructure give us possibility to manufacture more

thin transistors [17].

x, y

C(x

,Θ),

C(y

,Θ)

2

3

1

Fig. 3. Distributions of concentration of dopant in directions, which is perpendicular to interface between

layers of heterostructure. Curve 1 is a dopant distribution in a homogenous sample. Curve 2 corresponds to

negative value of the mismatch strain. Curve 3 corresponds to positive value of the mismatch strain

4. CONCLUSION

In this paper we consider possibility to decrease length of channel of field-effect transistor by us-

ing mechanical stress in heterostructure. At the same time with decreasing of the length it could

be increased density of transistors in integrated circuits.

ACKNOWLEDGEMENTS

This work is supported by the contract 11.G34.31.0066 of the Russian Federation Government,

Scientific School of Russia SSR-339.2014.2 and educational fellowship for scientific research.

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[4] V.I. Lachin, N.S. Savelov. Electronics (Phoenix, Rostov-na-Donu, 2001).

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APPENDIX

Equations for functions Tij(x,y,z,t) (i≥0, j≥0)

( ) ( ) ( ) ( ) ( )

ass

ass

tzyxp

z

tzyxT

y

tzyxT

x

tzyxT

t

tzyxT

να

,,,,,,,,,,,,,,,2

00

2

2

00

2

2

00

2

0

00 +

∂

∂+

∂

∂+

∂

∂=

∂

∂

( ) ( ) ( ) ( )+

∂

∂+

∂

∂+

∂

∂=

∂

∂2

0

2

2

0

2

2

0

2

0

0 ,,,,,,,,,,,,

z

tzyxT

y

tzyxT

x

tzyxT

t

tzyxT iii

ass

i α

( ) ( ) ( ) ( ) ( ) ( )

∂

∂

∂

∂+

∂

∂+

∂

∂+ −−−

z

tzyxTzg

zy

tzyxTxg

x

tzyxTzg i

T

i

T

i

Tass

,,,,,,,,,10

2

10

2

2

10

2

0α , i ≥1


40

( ) ( ) ( ) ( )( )

×+

∂

∂+

∂

∂+

∂

∂=

∂

∂

tzyxT

T

z

tzyxT

y

tzyxT

x

tzyxT

t

tzyxTdass

ass,,,

,,,,,,,,,,,,

00

0

2

01

2

2

01

2

2

01

2

0

01

ϕ

ϕαα

( ) ( ) ( )( )

( )

+

∂

∂−

∂

∂+

∂

∂+

∂

∂×

2

00

00

0

2

00

2

2

00

2

2

00

2 ,,,

,,,

,,,,,,,,,

x

tzyxT

tzyxT

T

z

tzyxT

y

tzyxT

x

tzyxTdass

ϕ

ϕα

( ) ( )

∂

∂+

∂

∂+

2

00

2

00,,,,,,

z

tzyxT

y

tzyxT

( ) ( ) ( ) ( )( )

×+

∂

∂+

∂

∂+

∂

∂=

∂

∂

tzyxT

T

z

tzyxT

y

tzyxT

x

tzyxT

t

tzyxTdass

ass,,,

,,,,,,,,,,,,

00

0

2

02

2

2

02

2

2

02

2

0

02

ϕ

ϕαα

( ) ( ) ( )( )

( )

×

∂

∂−

∂

∂+

∂

∂+

∂

∂×

+ x

tzyxT

tzyxT

T

z

tzyxT

y

tzyxT

x

tzyxTdass

,,,

,,,

,,,,,,,,,00

1

00

0

2

01

2

2

01

2

2

01

2

ϕ

ϕαϕ

( ) ( ) ( ) ( ) ( )

∂

∂

∂

∂+

∂

∂

∂

∂+

∂

∂×

z

tzyxT

z

tzyxT

y

tzyxT

y

tzyxT

x

tzyxT ,,,,,,,,,,,,,,, 0100010001

( ) ( ) ( )( )

( )( )

( )

×+∂

∂+

∂

∂=

∂

∂Tzyxg

x

tzyxTTzyxg

tzyxT

tzyxT

x

tzyxT

t

tzyxTTTass

,,,,,,

,,,,,,

,,,,,,,,,2

00

2

00

01

2

11

2

0

11 α

( )( )

( )( )

( )

+

∂

∂

∂

∂+

∂

∂

∂

∂+

∂

∂×

x

tzyxTTzyxg

xz

tzyxTTzyxg

zy

tzyxTTassT

,,,,,,

,,,,,,

,,, 01

0

00

2

00

2

α

( )( )[ ]

( )( )[ ]

( )+

∂

∂++

∂

∂++

∂

∂+

assTTz

tzyxTTzyxg

y

tzyxTTzyxg

x

tzyxT02

01

2

2

01

2

2

01

2 ,,,,,,1

,,,,,,1

,,,α

( )( )

( ) ( )( )

( ) ( )( )

( )×

∂

∂+

∂

∂+

∂

∂+

+++ 2

00

2

1

00

10

2

00

2

1

00

10

2

00

2

1

00

10,,,

,,,

,,,,,,

,,,

,,,,,,

,,,

,,,

z

tzyxT

tzyxT

tzyxT

y

tzyxT

tzyxT

tzyxT

x

tzyxT

tzyxT

tzyxTϕϕϕ

( )( ) ( ) ( )

( )×+

∂

∂+

∂

∂+

∂

∂+×

tzyxT

T

z

tzyxT

y

tzyxT

x

tzyxT

tzyxT

TT dassdass

dass,,,

,,,,,,,,,

,,,00

0

2

10

2

2

10

2

2

10

2

00

0

0 ϕ

ϕ

ϕ

ϕϕ αα

α

( ) ( ) ( ) ( ) ( ) ( )−

∂

∂

∂

∂+

∂

∂+

∂

∂×

z

tzyxTTzyxg

zy

tzyxTTzyxg

x

tzyxTTzyxg

TTT

,,,,,,

,,,,,,

,,,,,, 00

2

10

2

2

10

2

( ) ( ) ( ) ( ) ( ) ( )×

∂

∂

∂

∂+

∂

∂

∂

∂+

∂

∂

∂

∂−

z

tzyxT

z

tzyxT

y

tzyxT

y

tzyxT

x

tzyxT

x

tzyxT ,,,,,,,,,,,,,,,,,,001000100010

( )( )( )

( ) ( ) ( )×

∂

∂+

∂

∂+

∂

∂−×

++

2

00

2

00

2

00

1

00

1

00

0,,,,,,,,,

,,,

,,,

,,, z

tzyxT

y

tzyxT

x

tzyxT

tzyxT

Tzyxg

tzyxT

TTdass

ϕϕ

ϕαϕ

ϕϕα dass T0× .

Conditions for the functions Tij(x,y,z,t) (i≥0, j≥0)

( )0

,,,

0

=∂

∂

=x

ij

x

tzyxT,

( )0

,,,=

∂

∂

= xLx

ij

x

tzyxT,

( )0

,,,

0

=∂

∂

=y

ij

y

tzyxT,

( )0

,,,=

∂

∂

= yLx

ij

y

tzyxT,

( )0

,,,

0

=∂

∂

=z

ij

z

tzyxT,

( )0

,,,=

∂

∂

= zLx

ij

z

tzyxT, T00(x,y,z,0)=fT(x,y,z), Tij(x,y,z,0)=0, i ≥1, j ≥1.

Solutions of the equations for the functions Tij(x,y,z,t) (i≥0, j≥0) with account appropriate boun-

dary and initial conditions could be written as


41

( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫+∫ ∫ ∫=∞

=1 00 0 000

2,,

1,,,

n

L

nnTnnn

zyx

L L L

T

zyx

xx y z

uctezcycxcLLL

udvdwdwvufLLL

tzyxT

( ) ( ) ( ) ( )×+∫ ∫ ∫ ∫+∫ ∫×

zyx

t L L L

asszyx

L L

TnnLLL

dudvdwdwvup

LLLudvdwdwvufwcvc

x y zy z 2,,,1,,

0 0 0 00 0

τν

τ

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ∫ ∫ ∫ ∫−×∞

=1 0 0 0 0

,,,

n

t L L L

ass

nnnnTnTnnn

x y z

dudvdwdwvup

wcvcucetezcycxc τν

ττ ,

where cn(χ) = cos (π nχ/L), ( )

++−=

2220

22 111exp

zyx

assnTLLL

tnte απ ;

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−=∞

=1 0 0 0 02

0

0,,,2,,,

n

t L L L

TnnnnTnTnnn

zyx

ass

i

x y z

TwvugwcvcusetezcycxcnLLL

tzyxT ταπ

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫−+∂

∂×

∞

=

−

1 0 0 02

010 2,,,

n

t L L

nnnTnTnnn

zyx

assix y

vsucetezcycxcnLLL

dudvdwdu

wvuTτ

απτ

τ

( ) ( ) ( ) ( ) ( ) ( ) ( )×∑+∫∂

∂×

∞

=

−

12

0

0

10 2,,,

,,,n

nTnnn

zyx

assL

i

nTtezcycxcn

LLLdudvdwd

v

wvuTwcTwvug

z απτ

τ

( ) ( ) ( ) ( ) ( )( )

∫ ∫ ∫ ∫∂

∂−× −

t L L Li

TnnnnT

x y z

dudvdwdw

wvuTTwvugwsvcuce

0 0 0 0

10 ,,,,,, τ

ττ , i ≥1,

where sn(χ) = sin (π n χ/L);

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )

×∑ ∫ ∫ ∫ ∫∂

∂−=

∞

=1 0 0 0 02

00

2

2001

,,,2,,,

n

t L L L

nnnTnTnnn

zyx

d

ass

x y z

u

wvuTvcusetezcycxcn

LLL

TtzyxT

ττ

πα

ϕ

( )( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−+×∞

=1 0 0 0 020

00

2

,,, n

t L L L

nnnnTnTnnn

zyx

d

assn

x y z

wcvsucetezcycxcnLLL

T

wvuT

dudvdwdwc τ

πα

τ

τ ϕ

ϕ

( )( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫−+∂

∂×

∞

=1 0 0 020

00

2

00

2 2

,,,

,,,

n

t L L

nnnTnTnnn

zyx

d

ass

x y

vcucetezcycxcnLLL

T

wvuT

dudvdwd

v

wvuTτ

πα

τ

ττ ϕ

ϕ

( ) ( )( )

( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫−−∫∂

∂×

∞

=1 0 0

0

0 00

2

00

2

2,,,

,,,

n

t L

nnTnTnnn

zyx

ass

d

L

n

xz

ucetezcycxcLLL

TwvuT

dudvdwd

u

wvuTws τ

αϕ

τ

ττ ϕ

ϕ

( ) ( ) ( )( )

( ) ( ) ( ) ( ) ( ) ×∑ ∫ −−∫ ∫

∂

∂×

∞

=+

1 0

0

0 01

00

2

00 2,,,

,,,

n

t

nTnTnnn

zyx

dass

L L

nnetezcycxc

LLL

T

wvuT

dudvdwd

u

wvuTwcvc

y z

τα

ϕτ

ττ ϕ

ϕ

( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( ) ×∑−∫ ∫ ∫

∂

∂×

∞

=+

1

0

0 0 01

00

2

00 2,,,

,,,

nnTnnn

zyx

ass

d

L L L

nnntezcycxc

LLLT

wvuT

dudvdwd

v

wvuTwcvcuc

x y z αϕ

τ

ττ ϕ

ϕ

( ) ( ) ( ) ( )( )

( )∫ ∫ ∫ ∫

∂

∂−×

+

t L L L

nnnnT

x y z

wvuT

dudvdwd

w

wvuTwcvcuce

0 0 0 01

00

2

00

,,,

,,,

τ

τττ

ϕ;

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )

×∑ ∫ ∫ ∫ ∫∂

∂−=

∞

=1 0 0 0 02

01

2

2002

,,,2,,,

n

t L L L

nnnTnTnnn

zyx

d

ass

x y z

u

wvuTvcusetezcycxcn

LLL

TtzyxT

ττ

πα

ϕ

( )( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫ ∫−+×∞

=1 0 0 0 020

00

2

,,, n

t L L L

nnnnTnTnnn

zyx

d

assn

x y z

wcvsucetezcycxcnLLL

T

wvuT

dudvdwdwc τ

πα

τ

τ ϕ

ϕ

( )( )

( ) ( ) ( ) ( ) ( ) ( )∑ ×∫ ∫−+∂

∂×

∞

=1 0 020

00

2

01

2 2

,,,

,,,

n

t L

nnTnTnnn

zyx

d

ass

x

ucetezcycxcnLLL

T

wvuT

dudvdwd

v

wvuTτ

πα

τ

ττ ϕ

ϕ


42

( ) ( ) ( )( )

( ) ( ) ( ) ( ) ( ) ×∑ ∫ −−∫ ∫∂

∂×

∞

=1 02

0

0 0 00

2

01

2

2,,,

,,,

n

t

nTnTnnn

zyx

ass

L L

nnetezcycxc

LLLwvuT

dudvdwd

w

wvuTwsvc

y z

ταπ

τ

ττϕ

( ) ( ) ( ) ( ) ( )( )

( ) ×∑−∫ ∫ ∫∂

∂

∂

∂×

∞

=+

12

0

0 0 01

00

0100 2,,,

,,,,,,

nn

zyx

ass

d

L L L

nnndxc

LLLT

wvuT

dudvdwd

u

wvuT

u

wvuTwcvcucT

x y z απ

τ

τττ ϕ

ϕ

ϕ

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )

−∫ ∫ ∫ ∫∂

∂

∂

∂−×

+

t L L L

nnnnTnTnn

x y z

wvuT

dudvdwd

v

wvuT

v

wvuTwcvcucetezcyc

0 0 0 01

00

0100

,,,

,,,,,,

τ

ττττ

ϕ

( ) ( ) ( ) ( ) ( ) ( )( )

∑ ×∫ ∫ ∫ ∫∂

∂

∂

∂−−

∞

=+

10 0 0 01

00

0100

2

0

,,,

,,,,,,2

n

t L L L

nnnnT

zyx

assdx y z

wvuT

dudvdwd

w

wvuT

w

wvuTwcvcuce

LLL

T

τ

ττττ

απ

ϕ

ϕ

( ) ( ) ( ) ( )tezcycxc nTnnn× ;

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )

∑ ∫ ∫ ∫ ∫

×∂

∂−=

∞

=1 0 0 0 02

00

2

0

11

,,,2,,,

n

t L L L

nnnnTnTnnn

zyx

assx y z

w

wvuTwcvcucetezcycxc

LLLtzyxT

ττ

α

( ) ( ) ( ) ( ) ( ) ( )( )

+

∂

∂

∂

∂+

∂

∂+× τ

τ

τττdudvdwd

wvuT

wvuT

w

wvuTwg

wv

wvuTTwvugTwvug

TTT,,,

,,,,,,,,,,,,,,,

00

0100

2

00

2

( )( )[ ] ( )

( )( )

+

∂

∂

∂

∂+

∂

∂++

∂

∂× τ

τττdudvdwd

w

wvuTTwvug

wv

wvuTTwvug

u

wvuTTT

,,,,,,

,,,,,,1

,,,01

2

01

2

2

01

2

( ) ( ) ( ) ( ) ( ) ( ) ( )( )( )

( )∑ ∫ ∫ ∫ ∫

+

∂

∂−+

∞

=+

1 0 0 0 02

00

2

1

00

100,,,

,,,

,,,2

n

t L L L

nnnTnTnnn

zyx

dassx y z

u

wvuT

wvuT

wvuTvcucetezcycxc

LLL

T τ

τ

ττ

αϕ

ϕ

( )( )

( ) ( )( )

( )( ) ×+

∂

∂+

∂

∂×

++

yx

ass

nLL

dudvdwdwcw

wvuT

wvuT

wvuT

v

wvuT

wvuT

wvuT0

2

00

2

1

00

10

2

00

2

1

00

10,,,

,,,

,,,,,,

,,,

,,, ατ

τ

τ

ττ

τ

τϕϕ

( ) ( ) ( ) ( )( ) ( ) ( )

∑ ×∫ ∫ ∫ ∫

∂

∂+

∂

∂+

∂

∂−×

∞

=1 0 0 0 02

10

2

2

10

2

2

10

2 ,,,,,,,,,2

n

t L L L

nnnnT

z

dx y z

w

wvuT

v

wvuT

u

wvuTwcvcuce

L

T ττττ

ϕ

( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫−+×

∞

=1 0 0 0

0

00

22,,, n

t L L

nnnTnTnnn

zyx

dass

nTnnn

x y

vcucetezcycxcLLL

Ttezcycxc

wvuT

dudvdwdτ

α

τ

τ ϕ

ϕ

( ) ( )( )

( )( )

( )∫

×+

∂

∂

∂

∂+

∂

∂×

zL

TTTnTwvug

w

wvuTTwvug

wu

wvuTTwvugwc

0

00

2

00

2

,,,,,,

,,,,,,

,,,ττ

( )( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫−−

∂

∂×

∞

=1 0 0 0

0

00

2

00

2

2,,,

,,,

n

t L L

nnnTnTnnn

zyx

dassx y

vcucetezcycxcLLL

T

wvuT

dudvdwd

v

wvuTτ

αϕ

τ

ττ ϕ

ϕ

( ) ( ) ( ) ( )( )∫

∂

∂+

∂

∂+

∂

∂×

+

zL

nwvuT

dudvdwd

w

wvuT

v

wvuT

u

wvuTwc

01

00

2

00

2

00

2

00

,,,

,,,,,,,,,

τ

ττττϕ

.

Integro-differential form of equations of systems (4) could be written as

( ) ( ) ( ) ( ) ( ) ( )

∫ ∫ ×

∂∂

∂−

∂∂

∂−

∂

∂−+=

t zzyx

xxwx

wyxu

yx

wyxu

x

wyxuttzyxutzyxu

0 0

22

2

2,,,,,,,,,

5,,,,,,τττ

τφ

( )( )[ ]

( ) ( ) ( ) ( )( )

−∫ ∫+

∂∂

∂−

∂∂

∂−

∂

∂−

+×

∞

0 0

22

2

2

1

,,,,,,,,,5

616

zzyx d

w

wdwE

wx

wyxu

yx

wyxu

x

wyxut

w

dwdwEτ

σ

τττ

σ

τ

( ) ( ) ( ) ( ) ( )∫ ∫

+

∂∂

∂+∫ ∫

∂∂

∂+

∂∂

∂+

∂

∂−

∞ t zx

zzyx

yx

wyxudwd

wx

wyxu

yx

wyxu

x

wyxuwKt

0 0

2

0 0

22

2

2,,,

2

1,,,,,,,,, ττ

τττ


43

( ) ( )( )

( ) ( )( ) ( )

×∫ ∫

∂

∂+

∂∂

∂−+−

+

∂

∂+

t zyxy

w

wyxu

wx

wyxutdt

w

wdwE

y

wyxu

0 02

22

2

2 ,,,,,,

2

1

1

,,, τττττ

σ

τ

( )( )

( ) ( ) ( )( )

( )∫ ∫

+

∂∂

∂−∫ ∫

+

∂

∂+

∂∂

∂−

+×

∞∞

0 0

2

0 02

22 ,,,

1

,,,,,,

21

zx

zyx

wx

wyxud

w

wdwE

y

wyxu

yx

wyxutd

w

wdwE ττ

σ

τττ

σ

( ) ( )( )

( ) ( ) ( )( )

( ) ×∫ ∫+∫ ∫∂

∂−−

+

∂

∂+

∞

0 00 02

2,,,

21

,,, zt zy

wKtdwdx

wyxTwKwt

td

w

wdwE

w

wyxuτ

τβττ

σ

τ

( ) ( ) ( ) ( ) ( ) ( )−Φ−∫ ∫∂

∂−+

∂

∂× tzyxdwd

wyxuwtdwd

x

wyxTw

x

t zx ,,,

,,,,,,1

0 02

2

ττ

τρττ

τβ

( )( )

∫ ∫∂

∂−

∞

0 02

2 ,,,zx dwd

wyxuwt τ

τ

τρ ,

( ) ( ) ( )( ) ( ) ( )

( )

×−∫ ∫+

∂∂

∂+

∂

∂−+=

21

,,,,,,

2

1,,,,,,

0 0

2

2

2

td

w

wdwE

yx

wyxu

x

wyxuttzyxutzyxu

t zxy

yyτ

σ

τττφ

( ) ( ) ( )( )

( ) ( )∫ ∫

−

∂∂

∂−

∂

∂+∫ ∫

+

∂∂

∂+

∂

∂×

∞ t zxy

zxy

yx

wyxu

y

wyxud

w

wdwE

yx

wyxu

x

wyxu

0 0

2

2

2

0 0

2

2

2,,,,,,

56

1

1

,,,,,, τττ

σ

ττ

( ) ( )( )

( )( ) ( ) ( )

×∫ ∫

∂∂

∂−

∂∂

∂−

∂

∂−−

+

∂∂

∂−

∞

0 0

22

2

22,,,,,,,,,

51

,,, zzxyz

wy

wyxu

yx

wyxu

y

wyxudt

w

wdwE

wy

wyxu τττττ

σ

τ

( )( )

( ) ( ) ( ) ( ) ( ) ( ) ( )+∫ ∫

∂

∂+∫ ∫

∂

∂−−

+×

∞

0 00 0

,,,,,,

61

zt z

dwdy

wyxTwKwtdwd

y

wyxTwwKt

td

w

wdwEτ

τβτ

τβττ

σ

( ) ( )( ) ( ) ( )

( ) ×∫ ∫−∫ ∫

∂∂

∂+

∂

∂+

∂∂

∂−+

∞

0 00 0

2

2

22 ,,,,,,,,, zt zyyy

wKtdwdwy

wyxu

y

wyxu

yx

wyxuwKt τ

ττττ

( ) ( ) ( )( )

( )∫

+

∂

∂−+

∂∂

∂+

∂

∂+

∂∂

∂×

tyyyy

z

zyxutdwd

wy

wyxu

y

wyxu

yx

wyxu

0

2

2

22 ,,,,,,,,,,,, τττ

τττ

( ) ( )( )[ ]

( )( )[ ]

( ) ( )+∫

∂

∂+

∂

∂

+−

+

∂

∂+

∞

0

,,,,,,

1212

,,,τ

ττ

σστ

τd

y

zyxu

z

zyxu

z

zEt

z

zEd

y

zyxuzyz

( )( )

( ) ( )( )

( )

Φ−∫ ∫∂

∂−−∫ ∫

∂

∂+

∞

tzyxdwdwyxu

wtdwdwyxu

wy

t zy

zy

,,,,,,,,,

10 0

2

2

0 02

2

ττ

τρττ

τ

τρ ,

( ) ( ) ( )( ) ( ) ( )

( )

×−∫ ∫+

∂∂

∂+

∂

∂−+=

21

,,,,,,

2

1,,,,,,

0 0

2

2

2t

dw

wdwE

wx

wyxu

x

wyxuttzyxutzyxu

t zxz

zzτ

σ

τττφ

( ) ( ) ( )( )

( ) ( )×∫ ∫

∂∂

∂+

∂

∂+∫ ∫

+

∂∂

∂+

∂

∂×

∞ t zyz

zxz

wy

wyxu

y

wyxud

w

wdwE

wx

wyxu

x

wyxu

0 0

2

2

2

0 0

2

2

2 ,,,,,,

1

,,,,,, τττ

σ

ττ

( )( )

( ) ( ) ( ) ( )( )

( ) ( ) ×∫ ∫−−∫ ∫+

∂∂

∂+

∂

∂−

−

+×

∞ t zzyz wtd

w

wdwE

wy

wyxu

y

wyxutd

t

w

wdwE

0 00 0

2

2

2

1

,,,,,,

221βττ

σ

τττ

τ

σ

( ) ( ) ( ) ( ) ( ) ( ) ( )∫

−

∂

∂−+∫ ∫

∂

∂+

∂

∂×

∞ tz

z

z

zyxutdwd

w

wyxTwwKtdwd

w

wyxTwK

00 0

,,,5

,,,,,, τττ

τβτ

τ

( ) ( ) ( )( )[ ]

( )( )

( ) ( )∫

−

∂

∂−

∂

∂

+−

+

∂

∂−

∂

∂−

∞

0

,,,,,,5

116

,,,,,,

x

zyxu

z

zyxu

z

zE

z

zEd

y

zyxu

x

zyxuxzyx

ττ

σστ

ττ


44

( )( ) ( ) ( ) ( ) ( )

−∫

∂

∂+

∂

∂+

∂

∂−+

∂

∂−

tzyxy

dz

zyxu

y

zyxu

x

zyxutzK

td

y

zyxu

0

,,,,,,,,,

6

,,,τ

τττττ

τ

( ) ( ) ( ) ( ) ( )

Φ−∫

∂

∂+

∂

∂+

∂

∂−

∞

tzyxdz

zyxu

y

zyxu

x

zyxuzKt

z

zyx ,,,,,,,,,,,,

10

ττττ

,

where ( ) ( ) ( )[ ]∫=Φz

i

iswdtwyxuwtzyx

s0

,,,,,, ρ , s =x,y,z, φ =L/Θ2E0, E0 is the average value of Young

modulus.

The second-order approximations of components of displacement vector could be written as

( ) ( ) ( ) ( ) ( )

∫ ∫

−

∂∂

∂−

∂

∂−++=

t zyx

xuxxyx

wyxu

x

wyxuttzyxutzyxu

0 0

1

2

2

1

2

1122

,,,,,,5

6

1,,,,,,

τττφα

( ) ( )( )

( ) ( ) ( )∫ ∫ ×

∂∂

∂−

∂∂

∂−

∂

∂−

+

∂∂

∂−

∞

0 0

1

21

2

2

1

2

1

2 ,,,,,,,,,5

1

,,, zzyxz

wx

wyxu

yx

wyxu

x

wyxud

w

wdwE

wx

wyxu ττττ

σ

τ

( )( )

( ) ( ) ( ) ( ) ( )−∫ ∫

∂∂

∂+

∂∂

∂+

∂

∂−+

+×

t zzyx dwd

wx

wyxu

yx

wyxu

x

wyxuwKt

td

w

wdwE

0 0

1

21

2

2

1

2,,,,,,,,,

61τ

τττττ

σ

( ) ( ) ( ) ( ) ( )∫ ∫

+

∂∂

∂+∫ ∫

∂∂

∂+

∂∂

∂+

∂

∂−

∞ t zx

zzyx

yx

wyxudwd

wx

wyxu

yx

wyxu

x

wyxuwKt

0 0

1

2

0 0

1

21

2

2

1

2,,,,,,,,,,,, τ

ττττ

( ) ( )( )

( ) ( ) ( ) ( )( )

×∫ ∫+

∂

∂+

∂∂

∂+

−

+

∂

∂+

t zyxy

w

wdwE

y

wyxu

yx

wyxud

t

w

wdwE

y

wyxu

0 02

1

2

1

2

2

1

2

1

,,,,,,

2

1

21

,,,

σ

τττ

τ

σ

τ

( ) ( ) ( ) ( ) ( )( )

( )( )

×∫ ∫+

−∫ ∫+

∂

∂+

∂∂

∂−+−×

∞

0 00 02

1

2

1

2

121

,,,,,,

2

1 zt zyx

w

wEtd

w

wdwE

w

wyxu

wx

wyxutdt

στ

σ

τττττ

( ) ( ) ( ) ( )×∫ ∫

∂

∂+

∂∂

∂−

∂

∂+

∂∂

∂×

∞

0 02

1

2

1

2

2

1

2

1

2 ,,,,,,

2

,,,,,, zyxyx

w

wyxu

wx

wyxutdwd

y

wyxu

yx

wyxu τττ

ττ

( )( )

( ) ( ) ( ) ( ) ( ) ( ) (∫ −∫∂

∂−∫ ∫

∂

∂+

+×

∞ t zz

twdx

wyxTwwKdwd

x

wyxTwKwtd

w

wdwE

0 00 0

,,,,,,

1

τχτ

τχτ

σ

) ( ) ( ) ( ) ( ) ( )−∫ ∫

∂

∂+∫ ∫

∂

∂−−−

∞

0 02

1

2

0 02

1

2 ,,,,,, zx

t zx dwd

wyxuwtdwd

wyxuwtd τ

τ

τρτ

τ

τρτττ

( ) ( )tzyxtzyxxxux

,,,,,,102

Φ−Φ− α ,

( ) ( )( ) ( ) ( )

( )

−∫ ∫+

∂∂

∂+

∂

∂−++=

t zxy

yuyyd

w

wdwE

yx

wyxu

x

wyxuttzyxutzyxu

0 0

1

2

2

1

2

1221

,,,,,,

2,,,,,, τ

σ

τττα

( ) ( ) ( )( )

( ) ( )∫ ∫

−

∂∂

∂−

∂

∂+∫ ∫

+

∂∂

∂+

∂

∂−

∞ t zxy

zxy

yx

wyxu

y

wyxud

w

wdwE

yx

wyxu

x

wyxut

0 0

1

2

2

1

2

0 0

1

2

2

1

2,,,,,,

51

,,,,,,

2

τττ

σ

ττ

( ) ( )( )

( ) ( ) ( )×∫ ∫

∂∂

∂−

∂∂

∂−

∂

∂−

−

+

∂∂

∂−

∞

0 0

1

2

1

2

2

1

2

1

2 ,,,,,,,,,5

61

,,, zzxyz

wy

wyxu

yx

wyxu

y

wyxud

t

w

wdwE

wy

wyxu ττττ

τ

σ

τ

( )( )

( ) ( ) ( ) ( ) ( ) ( ) ( )−∫ ∫

∂

∂+∫ ∫

∂

∂−−

+×

∞

0 00 0

,,,,,,

61

zt z

dwdy

wyxTwwKtdwd

y

wyxTwKwt

td

w

wdwEτ

τχτ

τχττ

σ


45

( ) ( )( ) ( ) ( )

( )∫ ∫ ×−∫ ∫

∂∂

∂+

∂

∂+

∂∂

∂−−

∞

0 00 0

1

2

2

1

2

1

2 ,,,,,,,,, zt zyyy

wKtdwdwy

wyxu

y

wyxu

yx

wyxuwKt τ

ττττ

( ) ( ) ( )( )

( )∫

+

∂

∂−+

∂∂

∂+

∂

∂+

∂∂

∂×

tyyyy

z

zyxutdwd

wy

wyxu

y

wyxu

yx

wyxu

0

11

2

2

1

2

1

2 ,,,

2

1,,,,,,,,, τττ

τττ

( ) ( )( )

( )( )[ ]

( ) ( )×−∫

∂

∂+

∂

∂

+−

+

∂

∂+

∞

20

111 ,,,,,,

121

,,,uy

zyz dy

zyxu

z

zyxu

z

zEt

z

zEd

y

zyxuατ

ττ

σστ

τ

( ) ( )110

,,,,,, φtzyxtzyxyy

Φ−Φ× ,

( ) ( )( ) ( ) ( )

( )

×∫ ∫+

∂∂

∂+

∂

∂++=

t zxz

zzzw

wdwE

wx

wyxu

x

wyxutzyxutzyxu

0 0

1

2

2

1

2

11221

,,,,,,

2

1,,,,,,

σ

ττφα

( ) ( ) ( ) ( )( )

( ) ( )∫ ∫

+

∂

∂−+∫ ∫

+

∂∂

∂+

∂

∂−−×

∞ t zz

zxz

y

wyxutd

w

wdwE

wx

wyxu

x

wyxutdt

0 02

1

2

0 0

1

2

2

1

2 ,,,

2

1

1

,,,,,,

2

τττ

σ

ττττ

( ) ( )( )

( ) ( ) ( ) ( ) ( )∫ ∫

+

∂

∂−∫ ∫

∂

∂−−

+

∂∂

∂+

∞

0 02

1

2

0 0

1

2,,,,,,

1

,,, zz

t zy

y

wyxudwd

w

wyxTwwKtd

w

wdwE

wy

wyxu ττ

τχττ

σ

τ

( ) ( )( )

( ) ( ) ( ) ( )( )[ ]

( )×∫ −+

+∫ ∫∂

∂+

+

∂∂

∂+

∞ tzy

tz

zEdwd

w

wyxTwwKt

td

w

wdwE

wy

wyxu

00 0

1

2

16

,,,

21

,,,τ

στ

τχτ

σ

τ

( ) ( ) ( ) ( ) ( )∫

−

∂

∂−

∂

∂−

∂

∂−

∂

∂−

∂

∂×

∞

0

11111,,,,,,

56

,,,,,,,,,5

x

zyxu

z

zyxutd

y

zyxu

x

zyxu

z

zyxuxzyxz

τττ

τττ

( ) ( )( )[ ]

( ) ( ) ( ) ( )×∫

∂

∂+

∂

∂+

∂

∂−+

+

∂

∂−

tzyxy

dz

zyxu

y

zyxu

x

zyxut

z

zEtd

y

zyxu

0

1111 ,,,,,,,,,

16

,,,τ

ττττ

στ

τ

( ) ( )( ) ( ) ( )

( ) −Φ−∫

∂

∂+

∂

∂+

∂

∂−×

∞

tzyxdz

zyxu

y

zyxu

x

zyxuzKtzK

zz

zyx ,,,,,,,,,,,,

020

111 αττττ

( )tzyxz

,,,1

Φ− .

Calculation of the average values αuβ2 leads to the following results

( ) ( ) ( ) ( ) ( )( )

(

∫ ∫ ∫ ∫ −+

∂∂

∂−

∂∂

∂−

∂

∂−Θ=

Θ

0 0 0 0

1

21

2

2

1

22

21

,,,,,,,,,5

12

1 x y zL L L

z

zyx

zuxL

z

zdzE

zx

tzyxu

yx

tzyxu

x

tzyxutL

σα

) ( ) ( ) ( ) ( ) ( )( )

×∫ ∫ ∫ ∫+

−

∂∂

∂−

∂∂

∂−

∂

∂Θ−−

∞

0 0 0 0

1

21

2

2

1

22

1

,,,,,,,,,5

12

x y zL L Lzzyx

z

zL

zx

tzyxu

yx

tzyxu

x

tzyxuzEtdxdydz

σ

( ) ( ) ( ) ( ) ( )×∫ ∫ ∫ ∫

∂∂

∂+

∂∂

∂+

∂

∂−−Θ+×

Θ

0 0 0 0

1

21

2

2

1

22 ,,,,,,,,,

2

1 x y zL L Lzyx

zzx

tzyxu

yx

tzyxu

x

tzyxuzLttdxdydzd

( ) ( ) ( ) ( ) ( )×∫ ∫ ∫ ∫

∂∂

∂+

∂∂

∂+

∂

∂−

Θ−×

∞

0 0 0 0

1

21

2

2

1

22 ,,,,,,,,,

2

x y zL L Lzyx

zzx

tzyxu

yx

tzyxu

x

tzyxuzLtdxdydzdzK

( ) ( ) ( ) ( ) ( )( )

×∫ ∫ ∫ ∫+

−

∂

∂+

∂∂

∂−Θ+×

Θ

0 0 0 02

1

2

1

22

1

,,,,,,

2

1 x y zL L Lzyx ydzd

z

zL

y

tzyxu

yx

tzyxuzEttdxdydzdzK

σ


46

( ) ( ) ( ) ( ) ( )( )

×Θ

−∫ ∫ ∫ ∫+

∂

∂+

∂∂

∂−−Θ+×

Θ

41

,,,,,,

2

1 2

0 0 0 02

1

2

1

22

x y zL L Lyx

ztdxdyd

z

zdzE

z

tzyxu

zx

tzyxuzLttdxd

σ

( ) ( )( )

( ) ( )( )

( )∫ ∫ ∫ ∫ ×+

−−Θ+∫ ∫ ∫ ∫

∂

∂+

∂∂

∂

+

−×

Θ∞

0 0 0 0

2

0 0 0 02

1

2

1

2

14

1,,,,,,

1

x y zx y z L L Lz

L L Lyxz

z

zLttdxdydzd

y

tzyxu

yx

tzyxu

z

zLzE

σσ

( ) ( )( ) ( ) ( ) ( )

×∫ ∫ ∫ ∫∂

∂−+

∂

∂+

∂∂

∂×

∞

0 0 0 02

1

2

2

1

2

1

2 ,,,

2

,,,,,, x y zL L Lx

z

yx

t

tzyxuzzLtdxdydzdzE

z

tzyxu

zx

tzyxu ρ

( ) ( ) ( ) ( ) ( ) ×∫ ∫ ∫ ∫ −Θ

+∫ ∫ ∫ ∫ −−Θ−Θ×∞Θ

0 0 0 0

2

0 0 0 01

2

2,,,

x y zx y z L L L

z

L L L

xzzLtdxdydzdtzyxuzzLttdxdydzd ρ

( )( ) ( ) ( ) ( )

∫ −Θ

ΘΧ−∞ΧΘ

+∂

∂×

zL

zxx

x zdzzLLtdxdydzdt

tzyxu

020

2

2

1

2

42

1

2

,,,ρ ,

( ) ( )( )

( )( ) ( )

×Θ

−∫ ∫ ∫ ∫

∂∂

∂+

∂

∂

+

−−Θ=

Θ

2

,,,,,,

12

12

0 0 0 0

1

2

2

1

2

2

2

x y zL L Lxyz

zuytdxdydzd

yx

tzyxu

x

tzyxu

z

zLzEL

στα

( )( )

( )( ) ( )

( )( )

×∫ ∫ ∫ ∫+

−+∫ ∫ ∫ ∫

∂∂

∂+

∂

∂

+

−×

Θ∞

0 0 0 00 0 0 0

1

2

2

1

2

112

1,,,,,,

1

x y zx y z L L Lz

L L Lxyz

z

zLzEtdxdydzd

yx

tzyxu

x

tzyxu

z

zLzE

σσ

( ) ( ) ( )( ) ( ) ×∫ ∫ ∫ ∫

Θ−−Θ

∂∂

∂−

∂∂

∂−

∂

∂×

∞

0 0 0 0

221

2

1

2

2

1

2

12

,,,,,,,,,5

x y zL L Lzxy

zEtdxdydzdzy

tzyxu

yx

tzyxu

y

tzyxuτ

( )( ) ( ) ( ) ( ) ( )

×∫ ∫ ∫ ∫−Θ+

∂∂

∂−

∂∂

∂−

∂

∂

+

−×

Θ

0 0 0 0

21

2

1

2

2

1

2

2

,,,,,,,,,5

1

x y zL L Lzxyz

zKttdxdydzd

zy

tzyxu

yx

tzyxu

y

tzyxu

z

zL

σ

( )( ) ( ) ( )

( ) ( ) ×∫ ∫ ∫ ∫ −−

∂∂

∂+

∂

∂+

∂∂

∂−×

∞

0 0 0 0

1

2

2

1

2

1

2 ,,,,,,,,, x y zL L L

z

yyy

zzLzKtdxdydzd

zy

tzyxu

y

tzyxu

yx

tzyxuzL

( ) ( ) ( ) ( )∫ ∫ ∫ ∫

+

∂

∂+

∂∂

∂+

∂

∂+

∂∂

∂Θ×

Θ

0 0 0 0

11

2

2

1

2

1

22 ,,,

2

1,,,,,,,,,

2

x y zL L Lyyyy

z

tzyxutdxdydzd

zy

tzyxu

y

tzyxu

yx

tzyxu

( ) ( )( )

( )( ) ( ) ( )

( )×∫ ∫ ∫ ∫

+

∂

∂+

∂

∂Θ−−Θ

+

∂

∂+

∞

0 0 0 0

112

21

1

,,,,,,

21

,,, x y zL L Lzyz

z

zdzE

y

tzyxu

z

tzyxutdxdyd

z

zdzE

y

tzyxu

στ

σ

( ) ( )( ) ( )

×∫ ∫ ∫ ∫∂

∂Θ+∫ ∫ ∫ ∫

∂

∂−Θ−×

∞Θ

0 0 0 02

1

22

0 0 0 02

1

2

2,,,

2

,,,

2

1 x y zx y z L L Ly

L L Ly

t

tzyxutdxdydzd

t

tzyxuzttdxdyd ρ

( ) ( ) ( ) ( )( ) ( )

×∞Χ

Θ+ΘΧ

−∫ ∫ ∫ ∫ −−×Θ

22,,,

022

0 0 0 01

yyL L L

yz

x y z

tdxdydzdtzyxuzzLtdxdydzdz ρρ

( ) ( )1

0

4

−

∫ −Θ×zL

zzdzzLL ρ ,

( ) ( )( )

( )( ) ( )

×Θ

−∫ ∫ ∫ ∫

∂∂

∂+

∂

∂

+

−−Θ=

Θ

2

,,,,,,

12

12

0 0 0 0

1

2

2

1

22

2

x y zL L Lxzz

zztdxdydzd

zx

tzyxu

x

tzyxu

z

zLzEtL

σα

( ) ( )( )

( ) ( ) ( ) ( ) ×∫ ∫ ∫ ∫−Θ+∫ ∫ ∫ ∫

∂∂

∂+

∂

∂

+

−×

Θ∞

0 0 0 0

2

0 0 0 0

1

2

2

1

2

2

1,,,,,,

1

x y zx y z L L LL L Lxzz zEttdxdydzd

zx

tzyxu

x

tzyxu

z

zLzE

σ

( ) ( ) ( ) ( )∫ ∫ ∫ ∫ ×

∂∂

∂+

∂

∂−

∂∂

∂+

∂

∂×

∞

0 0 0 0

1

2

2

1

21

2

2

1

2 ,,,,,,,,,,,, x y zL L Lyzyz

zy

tzyxu

y

tzyxutdxdydzd

zy

tzyxu

y

tzyxu


47

( ) ( )( )[ ]

( ) ( ) ( )∫ ∫ ∫ ∫ ×

∂

∂−

∂

∂−

∂

∂+

+

−Θ×

Θ

0 0 0 0

1112,,,,,,,,,

512

1

12

x y zL L Lyxzz

y

tzyxu

x

tzyxu

z

tzyxutdxdydzd

z

zLzE

σ

( )( )

( )( ) ( ) ( )

∫ ∫ ∫ ∫ ×

∂

∂−

∂

∂−

∂

∂Θ−−Θ

+×

∞

0 0 0 0

111

22

,,,,,,,,,5

121

x y zL L Lyxz

y

tzyxu

x

tzyxu

z

tzyxutdtxdyd

z

zdzE

σ

( )( )

( )( ) ( ) ( ) ( )

×∫ ∫ ∫ ∫

∂

∂+

∂

∂+

∂

∂−Θ+

+×

Θ

0 0 0 0

1112 ,,,,,,,,,

21

x y zL L Lzyx

z

tzyxu

y

tzyxu

x

tzyxuzKttdxdyd

z

zdzE

σ

( )( )

( ) ( ) ( )×∫ ∫ ∫ ∫

∂

∂+

∂

∂+

∂

∂Θ−

ΘΧ−×

∞

0 0 0 0

111

22 ,,,,,,,,,

22

x y zL L Lzyxz

z

tzyxu

y

tzyxu

x

tzyxuzKtdxdydzd

( ) ( ) ( )( )

( ) ×

∫ −∞Χ

Θ+∫ ∫ ∫ ∫ −−×Θ zx y z L

z

zL L L

zzzLtdxdydzdztzyxuzLtdxdydzd

0

02

0 0 0 01

42

,,, ρ

( ) 1−Θ× Lzdzρ .

System of equations for the functions Cijk(x,y,z,t) (i ≥0, j ≥0, k ≥0) could be written as

( ) ( ) ( ) ( )

∂

∂+

∂

∂+

∂

∂=

∂

∂2

000

2

2

000

2

2

000

2

0

000,,,,,,,,,,,,

z

tzyxC

y

tzyxC

x

tzyxCD

t

tzyxCL

;

( )( )

( )( )

( )

+

∂

∂

∂

∂+

∂

∂

∂

∂=

∂

∂ −−

y

tzyxCTzg

yx

tzyxCTzg

xD

t

tzyxCi

L

i

LL

i,,,

,,,,

,,,,

100100

0

00

( )( ) ( ) ( ) ( )

∂

∂+

∂

∂+

∂

∂+

∂

∂

∂

∂+ −

2

00

2

2

00

2

2

00

2

100,,,,,,,,,,,,

,z

tzyxC

y

tzyxC

x

tzyxC

z

tzyxCTzg

z

iiii

L, i ≥1;

( ) ( ) ( ) ( )+

∂

∂+

∂

∂+

∂

∂=

∂

∂2

010

2

2

010

2

2

010

2

0

010,,,,,,,,,,,,

z

tzyxC

y

tzyxC

x

tzyxCD

t

tzyxCL

( )( )

( ) ( )( )

( )

+

∂

∂

∂

∂+

∂

∂

∂

∂+

y

tzyxC

TzyxP

tzyxC

yx

tzyxC

TzyxP

tzyxC

xD

L

,,,

,,,

,,,,,,

,,,

,,,000000000000

0 γ

γ

γ

γ

( )( )

( )

∂

∂

∂

∂+

z

tzyxC

TzyxP

tzyxC

z

,,,

,,,

,,, 000000

γ

γ

;

( ) ( ) ( ) ( )×+

∂

∂+

∂

∂+

∂

∂=

∂

∂LL

Dz

tzyxC

y

tzyxC

x

tzyxCD

t

tzyxC02

020

2

2

020

2

2

020

2

0

020,,,,,,,,,,,,

( )( )

( )( )

( )( )

×

∂

∂

∂

∂+

∂

∂

∂

∂×

−

y

tzyxCtzyxC

yy

tzyxC

TzyxP

tzyxCtzyxC

x

,,,,,,

,,,

,,,

,,,,,, 000

010

000

1

000

010 γ

γ

( )( )

( )( )

( )( ) ( )

( )

×∂

∂+

∂

∂

∂

∂+

×

−−

TzyxP

tzyxC

xD

z

tzyxC

TzyxP

tzyxCtzyxC

zTzyxP

tzyxCL

,,,

,,,,,,

,,,

,,,,,,

,,,

,,,000

0

000

1

000

010

1

000

γ

γ

γ

γ

γ

γ

( ) ( )( )

( ) ( )( )

( )

∂

∂

∂

∂+

∂

∂

∂

∂+

∂

∂×

z

tzyxC

TzyxP

tzyxC

zy

tzyxC

TzyxP

tzyxC

yx

tzyxC ,,,

,,,

,,,,,,

,,,

,,,,,,010000010000010

γ

γ

γ

γ

;

( ) ( ) ( ) ( )+

∂

∂+

∂

∂+

∂

∂=

∂

∂2

001

2

2

001

2

2

001

2

0

001,,,,,,,,,,,,

z

tzyxC

y

tzyxC

x

tzyxCD

t

tzyxCL


48

( )[ ] ( )( )

( )( )

+

∇

∫

++

∂

∂Ω+

Tk

tzyxWdtWyxC

TzyxP

tzyxCTzyxg

xD S

L

SLSLSSL

z ,,,,,,

,,,

,,,1,,,1

0000

000

0

µξε

γ

γ

( )[ ] ( )( )

( )( )

∇

∫

++

∂

∂Ω+

Tk

tzyxWdtWyxC

TzyxP

tzyxCTzyxg

yD S

L

SLSLSSL

z ,,,,,,

,,,

,,,1,,,1

0000

000

0

µξε

γ

γ

;

( ) ( ) ( ) ( )+

∂

∂+

∂

∂+

∂

∂=

∂

∂2

002

2

2

002

2

2

002

2

0

002,,,,,,,,,,,,

z

tzyxC

y

tzyxC

x

tzyxCD

t

tzyxCL

( )[ ] ( )( )

( )( )

+

∇

∫

++

∂

∂Ω+

Tk

tzyxWdtWyxC

TzyxP

tzyxCTzyxg

xD S

L

SLSLSSL

z ,,,,,,

,,,

,,,1,,,1

0001

000

0

µξε

γ

γ

( )[ ] ( )( )

( )( )

+

∇

∫

++

∂

∂Ω+

Tk

tzyxWdtWyxC

TzyxP

tzyxCTzyxg

yD S

L

SLSLSSL

z ,,,,,,

,,,

,,,1,,,1

0001

000

0

µξε

γ

γ

( )[ ] ( )( )

( )( )

×∫

++

∂

∂Ω+

−zL

SLSLSSLWdtWyxC

TzyxP

tzyxCtzyxCTzyxg

xD

0000

1

000

0010 ,,,,,,

,,,,,,1,,,1

γ

γγξε

( ) ( ) ( )( )

( ) ( ) ×

∫∇

+

∂

∂+∇

×−

zLS

S

S WdtWyxCTk

tzyx

TzyxP

tzyxCtzyxC

yTk

tzyx

0000

1

000

001,,,

,,,

,,,

,,,,,,1

,,, µξ

µγ

γγ

( )[ ]SLLSLS

DTzyxg0

,,,1 Ω+× ε ;

( ) ( ) ( ) ( ) ( )

×

∂

∂

∂

∂+

∂

∂+

∂

∂+

∂

∂=

∂

∂

x

tzyxC

xz

tzyxC

y

tzyxC

x

tzyxCD

t

tzyxCL

,,,,,,,,,,,,,,,010

2

110

2

2

110

2

2

110

2

0

110

( ) ( )( )

( )( )

+

∂

∂

∂

∂+

∂

∂

∂

∂+

×

LLLLD

z

tzyxCTzyxg

zy

tzyxCTzyxg

yTzyxg

0

010010,,,

,,,,,,

,,,,,,

( )( )

( ) ( )( )

( ) ( )( )

×

∂

∂+

∂

∂

∂

∂+

∂

∂

∂

∂+

TzyxP

tzyxC

zy

tzyxC

TzyxP

tzyxC

yx

tzyxC

TzyxP

tzyxC

x ,,,

,,,,,,

,,,

,,,,,,

,,,

,,,000100000100000

γ

γ

γ

γ

γ

γ

( ) ( ) ( )( )

( ) ( )( )

×

∂

∂+

∂

∂

∂

∂+

∂

∂×

−−

TzyxP

tzyxC

yx

tzyxC

TzyxP

tzyxCtzyxC

xD

z

tzyxCL

,,,

,,,,,,

,,,

,,,,,,

,,,1

000000

1

000

1000

100

γ

γ

γ

γ

( )( )

( )( )

( )( )

LD

z

tzyxC

TzyxP

tzyxCtzyxC

zy

tzyxCtzyxC 0

000

1

000

100

000

100

,,,

,,,

,,,,,,

,,,,,,

∂

∂

∂

∂+

∂

∂×

−

γ

γ

;

( ) ( ) ( ) ( ) ( )

×

∂

∂

∂

∂+

∂

∂+

∂

∂+

∂

∂=

∂

∂

y

tzyxC

xz

tzyxC

y

tzyxC

x

tzyxCD

t

tzyxCL

,,,,,,,,,,,,,,,000

2

100

2

2

100

2

2

100

2

0

101

( ) ( ) ( ) ( ) ( )+

∂

∂

∂

∂+

∂

∂

∂

∂+

×

LLLLD

z

tzyxCTzyxg

zy

tzyxCTzyxg

yTzyxg

0

000000,,,

,,,,,,

,,,,,,

( )[ ] ( ) ( )( )

( )

×∫

++

∂

∂Ω+

−zL

SLSLSSLWdtWyxC

TzP

tzyxCtzyxCTzyxg

xD

0100

1

000

1000,,,

,

,,,,,,1,,,1

γ

γ

ξε

( )( )[ ] ( )

( )( )

( )( )

( )[ ] ( )

×∫+∂

∂+

∇

∫×

×

++

∂

∂Ω+

∇

×−

zz L

LSLS

SL

SLSLSSL

S

WdtWyxCTzyxgxTk

tzyxWdtWyxC

TzP

tzyxCtzyxCTzyxg

yD

Tk

tzyx

0100

0100

1

000

1000

,,,,,,1,,,

,,,

,

,,,,,,1,,,1

,,,

εµ

ξεµ

γ

γ


49

( ) ( ) ( )[ ] ( )

×∇

+∂

∂+Ω

∫∇

×Tk

tzyxTzyxg

yDWdtWyxC

Tk

tzyxS

LSLSSL

LS

z ,,,,,,1,,,

,,,0

0100

µε

µ

( ) ( )SL

LL

DWdtWyxCWdtWyxCzz

00

1000

100,,,,,, Ω

∫∫× ;

( ) ( ) ( ) ( ) ( )( )

×

∂

∂+

∂

∂+

∂

∂+

∂

∂=

∂

∂

TzyxP

tzyxC

xz

tzyxC

y

tzyxC

x

tzyxCD

t

tzyxCL

,,,

,,,,,,,,,,,,,,,000

2

011

2

2

011

2

2

011

2

0

011

γ

γ

( ) ( )( )

( ) ( )( )

( )+

∂

∂

∂

∂+

∂

∂

∂

∂+

∂

∂×

LD

z

tzyxC

TzyxP

tzyxC

zy

tzyxC

TzyxP

tzyxC

yx

tzyxC0

001000001000001 ,,,

,,,

,,,,,,

,,,

,,,,,,γ

γ

γ

γ

( ) ( )( )

( ) ( ) ( )

×

∂

∂

∂

∂+

∂

∂

∂

∂+

−

y

tzyxCtzyxC

yx

tzyxC

TzyxP

tzyxCtzyxC

xD

L

,,,,,,

,,,

,,,

,,,,,, 000

001

000

1

000

0010 γ

γ

( )( )

( )( )

( )( ) ( )

×∇

∂

∂+

∂

∂

∂

∂+

×

−−

Tk

tzyx

xz

tzyxC

TzyxP

tzyxCtzyxC

zTzyxP

tzyxCS

,,,,,,

,,,

,,,,,,

,,,

,,, 000

1

000

001

1

000 µγ

γ

γ

γ

( )[ ] ( )( )

( ) ( )×

∇

∂

∂+Ω

∫

++×

Tk

tzyx

xDWdtWyxC

TzyxP

tzyxCTzyxg S

SL

L

SLSLS

z ,,,,,,

,,,

,,,1,,,1 0

0010

000µ

ξεγ

γ

( )[ ] ( ) ( )( )

( ) ×Ω+Ω

∫

++×

−

SLSL

L

SLSLSDDWdtWyxC

TzyxP

tzyxCtzyxCTzyxg

z

000

000

1

000

010,,,

,,,

,,,,,,1,,,1

γ

γ

ξε

( )[ ] ( )( )

( )( )

( )

∫∇

++

∂

∂×

−zL

S

SLSLSWdtWyxC

Tk

tzyx

TzyxP

tzyxCtzyxCTzyxg

y 0000

1

000

010,,,

,,,

,,,

,,,,,,1,,,1

µξε

γ

γ

.

Boundary and initial conditions for functions Cijk(x,t) (i ≥0, j ≥0, k ≥0) could be written as

( )0

,,,

0

==x

ijk

x

tzyxC

∂

∂;

( )0

,,,=

= xLx

ijk

x

tzyxC

∂

∂;

( )0

,,,

0

==y

ijk

y

tzyxC

∂

∂;

( )0

,,,=

= yLy

ijk

y

tzyxC

∂

∂;

( )0

,,,

0

==z

ijk

z

tzyxC

∂

∂;

( )0

,,,=

= zLz

ijk

z

tzyxC

∂

∂; C000(x,y,z,0)=fC (x,y,z); Cijk(x,y,z,0)=0.

Solutions of equations for functions Cijk(x,t) (i ≥0, j ≥0, k ≥0) could be written as

( ) ( ) ( ) ( ) ( )∑+=∞

=1

0

000

2,,,

nnCnnnnC

zyxzyx

C tezcycxcFLLLLLL

FtzyxC ,

where ( )

++−=

2220

22 111exp

zyx

LnCLLL

tDnte π , ( ) ( ) ( ) ( )∫ ∫ ∫=x y zL L L

CnnnnCudvdwdwvufwcvcucF

0 0 0

,, ;

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=∞

=1 0 0 0 02

0

00 ,,,2

,,,n

t L L L

LnnnCnCnnnnC

zyx

L

i

x y z

TwvugvcusetezcycxcFnLLL

DtzyxC τ

π

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫−−∂

∂×

∞

=

−

1 0 02

0100 2,,,

n

t L

nnCnCnnnnC

zyx

Li

n

x

ucetezcycxcFnLLL

Ddudvdwd

u

wvuCwc τ

πτ

τ

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑−∫ ∫∂

∂×

∞

=

−

12

0

0 0

1002,,,

,,,n

nCnnn

zyx

L

L Li

Lnntezcycxc

LLL

Ddudvdwd

v

wvuCTwvugwcvs

y z πτ

τ


50

( ) ( ) ( ) ( ) ( )( )

∫ ∫ ∫ ∫∂

∂−× −

t L L Li

LnnnnCnC

x y z

dudvdwdw

wvuCTwvugwsvcuceFn

0 0 0 0

100 ,,,,,, τ

ττ , i ≥1;

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )

×∑ ∫ ∫ ∫ ∫−−=∞

=1 0 0 0 0

000

2

0

010,,,

,,,2,,,

n

t L L L

nnnCnCnnnnC

zyx

Lx y z

TwvuP

wvuCvcusetezcycxcFn

LLL

DtzyxC

γ

γ ττ

π

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫−−∂

∂×

∞

=1 0 02

00002,,,

n

t L

nnCnCnnnnC

zyx

L

n

x

ucetezcycxcFnLLL

Ddudvdwd

u

wvuCwc τ

πτ

τ

( ) ( ) ( ) ( )( )

( ) ( ) ( ) ×∑−∫ ∫ ∫∂

∂×

∞

=12

0

0 0 0

0000002,,,

,,,

,,,

nnn

zyx

L

L L L

nnnycxcn

LLL

Ddudvdwd

v

wvuC

TwvuP

wvuCwcwcvs

y z z πτ

ττγ

γ

( ) ( ) ( ) ( ) ( ) ( )( )

( )( )

∫ ∫ ∫ ∫∂

∂−×

t L L L

nnnnCnCnnC

x y z

dudvdwdw

wvuC

TwvuP

wvuCwsvcucetezcF

0 0 0 0

000000 ,,,

,,,

,,,τ

τττ

γ

γ

;

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=∞

=1 0 0 0 00102

0

020 ,,,2

,,,n

t L L L

nnnCnCnnnnC

zyx

Lx y z

wvuCvcusetezcycxcFnLLL

DtzyxC ττ

π

( ) ( )( )

( ) ( ) ( ) ( ) ( ) ×∑−∂

∂×

∞

=

−

12

0000

1

0002,,,

,,,

,,,

nnCnnnnC

zyx

L

ntezcycxcFn

LLL

Ddudvdwd

u

wvuC

TwvuP

wvuCwc

πτ

ττγ

γ

( ) ( ) ( ) ( ) ( ) ( )( )

( )−∫ ∫ ∫ ∫

∂

∂−×

−t L L L

nnnnC

x y z

dudvdwdv

wvuC

TwvuP

wvuCwvuCwcvcuse

0 0 0 0

000

1

000

010

,,,

,,,

,,,,,, τ

ττττ

γ

γ

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )

×∑ ∫ ∫ ∫ ∫−−∞

=

−

1 0 0 0 0

1

000

010,,,

,,,,,,

n

t L L L

nnnnCnCnnnnC

x y z

TwvuP

wvuCwvuCwcvcusetezcycxcFn

γ

γ τττ

( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ −−∂

∂×

∞

=1 02

0000

2

0 2,,,2

n

t

nCnCnnnnC

zyx

L

zyx

L etezcycxcFnLLL

Ddudvdwd

w

wvuC

LLL

Dτ

πτ

τπ

( ) ( ) ( ) ( )( )

( ) ( ) ×∑−∫ ∫ ∫∂

∂×

∞

=12

0

0 0 0

0100002,,,

,,,

,,,

nnnC

zyx

LL L L

nnnxcFn

LLL

Ddudvdwd

u

wvuC

TwvuP

wvuCwcvcus

x y z πτ

ττγ

γ

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )

( )−∫ ∫ ∫ ∫

∂

∂−×

t L L L

nnnnCnCnn

x y z

dudvdwdv

wvuC

TwvuP

wvuCwcvsucetezcyc

0 0 0 0

010000,,,

,,,

,,,τ

τττ

γ

γ

( ) ( ) ( ) ( ) ( )( )

( )×∑ ∫ ∫ ∫ ∫

∂

∂−−

∞

=1 0 0 0 0

010000

2

0,,,

,,,

,,,2

n

t L L L

nnnnCnC

zyx

Lx y z

dudvdwdw

wvuC

TwvuP

wvuCwsvcuceFn

LLL

Dτ

τττ

πγ

γ

( ) ( ) ( ) ( )tezcycxcnCnnn

× ;

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫ ∫

∇−

Ω−=

∞

=1 0 0 0 02

0

001

,,,2,,,

n

t L L LS

nnCnCnnnnC

zyx

Lx y z

Tk

wvuusetezcycxcFn

LLL

DtzyxC

τµτ

π

( )[ ] ( )( )

( ) ( ) ( ) −∫

++× ττ

τξε

γ

γ

dudvdvcwdwcWdWvuCTwvuP

wvuCTwvug

nn

L

SLSLS

z

0000

000 ,,,,,,

,,,1,,,1

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫ ∫−Ω

−∞

=1 0 0 0 0 00002

0 ,,,2

n

t L L L L

nnnnCnCnnnnC

zyx

Lx y z z

WdWvuCwcvsucetezcycxcFnLLL

Dττ

π

( )[ ] ( )( )

( )τ

τµτξε

γ

γ

dudvdwdTk

wvu

TwP

wvuCTwvug S

SLSLS

,,,

,

,,,1,,,1 000

∇

++× ;

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫ ∫

∇−

Ω−=

∞

=1 0 0 0 02

0

002

,,,2,,,

n

t L L LS

nnnCnCnnnnC

zyx

Lx y z

Tk

wvuvcusetezcycxcFn

LLL

DtzyxC

τµτ

π

( )[ ] ( )( )

( ) ( ) ×Ω

−∫

++×

zyx

L

n

L

SLSLSLLL

DdudvdwdwcWdWvuC

TwvuP

wvuCTwvug

z

2

0

0001

000 ,,,,,,

,,,1,,,1

πττ

τξε

γ

γ


51

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫ ∫∇

−×∞

=1 0 0 0 0 0001

,,,,,,

n

t L L L LS

nnnnCnCnnnnC

x y z z

WdWvuCTk

wvuwcvsucetezcycxcFn τ

τµτ

( )[ ] ( )( )

( ) ( ) ( ) ×∑Ω

−

++×

∞

=12

00002

,,,

,,,1,,,12

nnnnnC

zyx

L

SLSLSzcycxcF

LLL

Ddudvdwd

TwvuP

wvuCTwvug

πτ

τξε

γ

γ

( ) ( ) ( ) ( ) ( ) ( ) ( )( )

( ) ×∫ ∫ ∫ ∫ ∫

+−×

−t L L L L

SnnnnCnC

x y z z

WdWvuCTwvuP

wvuCtzyxCwcvcuseten

0 0 0 0 0000

1

000

001,,,

,,,

,,,,,,1 τ

τξτ

γ

γγ

( )[ ] ( ) ( ) ( ) ( ) ( ) ×∑Ω

−∇

+×∞

=12

02,,,

,,,1n

nCnnnnC

zyx

LS

LSLStezcycxcF

LLL

Ddudvdwd

Tk

wvuTwvug

πτ

τµε

( ) ( ) ( ) ( ) ( ) ( )( )

( )∫ ∫ ∫ ∫ ×∫

+

∇×

−t L L L L

S

S

nnn

x y z z

WdWvuCTwP

wvuCtzyxC

Tk

wvuwcvcusn

0 0 0 0 0000

1

000

001,,,

,

,,,,,,1

,,,τ

τξ

τµγ

γγ

( )[ ] ( ) ττε deudvdwdTwvugnCLSLS

−+× ,,,1 ;

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=∞

=1 0 0 0 02

0

110 ,,,2

,,,n

t L L L

LnnnCnCnnnnC

zyx

Lx y z


DtzyxC τ

π

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫−−∂

∂×

∞

=1 0 02

00102,,,

n

t L

nnCnCnnnnC

zyx

L

n

x

ucetezcycxcFnLLL

Ddudvdwd

u

wvuCwc τ

πτ

τ

( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ×−∫ ∫∂

∂×

∞

=12

0

0 0

0102,,,

,,,n

nnnnC

zyx

L

L L

LnnzcycxcFn

LLL

Ddudvdwd

v

wvuCTwvugwcvs

y z πτ

τ

( ) ( ) ( ) ( ) ( ) ( ) ( )×−∫ ∫ ∫ ∫

∂

∂−×

zyx

Lt L L L

LnnnnCnCLLL

Ddudvdwd

w

wvuCTwvugwsvcucete

x y z

2

0

0 0 0 0

0102,,,

,,,π

ττ

τ

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−×∞

=1 0 0 0 0100 ,,,,,,

n

t L L L

LnnnnCnCnnnnC

x y z

wvuCTwvugwcvcusetezcycxcFn ττ

( )( )

( ) ( ) ( ) ( ) ( ) ×∑−∂

∂×

∞

=

−

12

0000

1

0002,,,

,,,

,,,

nnCnnnnC

zyx

L tezcycxcFnLLL

Ddudvdwd

u

wvuC

TwvuP

wvuC πτ

ττγ

γ

( ) ( ) ( ) ( ) ( ) ( )( )

( )×∫ ∫ ∫ ∫

∂

∂−×

−t L L L

LnnnC

x y z

v

wvuC

TwvuP

wvuCwvuCTwvugvsuce

0 0 0 0

000

1

000

100

,,,

,,,

,,,,,,,,,

ττττ

γ

γ

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫−−×∞

=1 0 0 02

02

n

t L L

nnnCnCnnnnC

zyx

L

n

x y

vcucetezcycxcFnLLL

Ddudvdwdwc τ

πτ

( ) ( ) ( ) ( )( )

( )−∫

∂

∂×

−zL

Lndudvdwd

w

wvuC

TwvuP

wvuCwvuCTwvugws

0

000

1

000

100

,,,

,,,

,,,,,,,,, τ

τττ

γ

γ

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−∞

=1 0 0 0 02

0 ,,,2

n

t L L L

LnnnnCnCnnnnC

zyx

Lx y z

TwvugwcvcusetezcycxcFnLLL

Dτ

π

( )( )

( ) ( ) ( ) ( ) ( )×∑−∂

∂×

∞

=12

01000002,,,

,,,

,,,

nnCnnn

zyx

L tezcycxcnLLL

Ddudvdwd

u

wvuC

TwvuP

wvuC πτ

ττγ

γ

( ) ( ) ( ) ( ) ( )( )

( )×∫ ∫ ∫ ∫

∂

∂−×

t L L L

LnnnCnC

x y z

vdwdu

wvuC

TwvuP

wvuCTwvugwcvseF

0 0 0 0

100000,,,

,,,

,,,,,,

τττ

γ

γ

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−×∞

=1 0 0 0 02

02

n

t L L L

nnnnCnCnnnnC

zyx

L

n

x y z

wsvcucetezcycxcFnLLL

Dduduc τ

πτ

( )( )

( )( )

τττ

γ

γ

dudvdwdu

wvuC

TwP

wvuCTwvug

L∂

∂×

,,,

,

,,,,,, 100000 ;


52

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=∞

=1 0 0 0 02

0

101 ,,,2

,,,n

t L L L

LnnnCnCnnnnC

zyx

Lx y z


DtzyxC τ

π

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫−−∂

∂×

∞

=1 0 02

0000 2,,,

n

t L

nnCnCnnnnC

zyx

L

n

x

ucetezcycxcFLLL

Ddudvdwd

u

wvuCwc τ

πτ

τ

( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑−∫ ∫ ∫∂

∂×

∞

=12

0

0 0 0

0002,,,

,,,n

nn

zyx

L

L L L

Lnnnycxcn

LLL

Ddudvdwd

v

wvuCTwvugwcwcvsn

y z z πτ

τ

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )−∫ ∫ ∫ ∫

∂

∂−×

t L L L

LnnnnCnCnnC

x y z

dudvdwdw

wvuCTwvugwsvcucetezcF

0 0 0 0

000,,,

,,, ττ

τ

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]×∑ ∫ ∫ ∫ ∫ +−Ω−∞

=1 0 0 0 02

0 ,,,12

n

t L L L

LSLSnnnCnCnnnnC

zyx

Lx y z


Dετ

π

( ) ( )( )

( ) ( )−

∇∫

+

∇× τ

τµτ

τξ

τµγ

γ

dudvdwdTk

wvuWdWvuC

TwvuP

wvuC

Tk

wvuS

L

S

Sz ,,,

,,,,,,

,,,1

,,,

0100

000

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]∑ ∫ ∫ ∫ ∫ ×+−Ω−∞

=1 0 0 0 02

0 ,,,12

n

t L L L

LSLSnnnCnCnnnnC

zyx

Lx y z


Dετ

π

( ) ( ) ( )( )

( ) ( ) −∫∇

+×

−

τττµτ

τξγ

γ

dudvdwdWdWvuCTk

wvu

TwvuP

wvuCwvuCwc

zLS

Sn0

000

1

000

100 ,,,,,,

,,,

,,,,,,1

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ×∑ ∫ ∫ ∫ ∫ +−Ω−∞

=1 0 0 0 02

0 ,,,12

n

t L L L

LSLSnnnCnCnnnnC

zyx

Lx y z

TwvugvsucetezcycxcFnLLL

Dετ

π

( )( )

( )( )

( )( ) ττ

ττξ

τµγ

γ

dudvdwdWdWvuCTwvuP

wvuCwvuC

Tk

wvuwc

zL

S

S

n ∫

+

∇×

−

0000

1

000

100 ,,,,,,

,,,,,,1

,,, ;

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )

×∑ ∫ ∫ ∫ ∫−−=∞

=1 0 0 0 0

000

2

0

011,,,

,,,2,,,

n

t L L L

nnnCnCnnnnC

zyx

Lx y z

TwvuP

wvuCvcusetezcycxcFn

LLL

DtzyxC

γ

γ ττ

π

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫−−∂

∂×

∞

=1 0 02

00012,,,

n

t L

nnCnCnnnnC

zyx

L

n

x

ucetezcycxcFnLLL

Ddudvdwd

u

wvuCwc τ

πτ

τ

( ) ( ) ( ) ( )( )

( ) ( ) ( ) ×∑−∫ ∫ ∫∂

∂×

∞

=12

0

0 0 0

0010002,,,

,,,

,,,

nnnnC

zyx

L

L L L

nnnycxcFn

LLL

Ddudvdwd

v

wvuC

TwvuP

wvuCwcwcvs

y z z πτ

ττγ

γ

( ) ( ) ( ) ( ) ( ) ( ) ( )( )

( )×−∫ ∫ ∫ ∫

∂

∂−×

zyx

Lt L L L

nnnnCnCnLLL

Ddudvdwd

w

wvuC

TwvuP

wvuCwsvcucetezc

x y z

2

0

0 0 0 0

0010002,,,

,,,

,,, πτ

τττ

γ

γ

( ) ( ) ( ) ( ) ( ) ( )( )

( )×∑ ∫ ∫ ∫ ∫

∂

∂−×

∞

=

−

1 0 0 0 0

000

1

000

001

,,,

,,,

,,,,,,

n

t L L L

nnnnCnC

x y z

dudvdwdu

wvuC

TwvuP

wvuCwvuCwcvcuseF τ

ττττ

γ

γ

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−×∞

=1 0 0 0 00012

0 ,,,2

n

t L L L

nnnCnCnnn

zyx

L

nCnnn

x y z

wvuCvsucetezcycxcLLL

Dtezcycxcn ττ

π

( ) ( )( )

( ) ( ) ( ) ( ) ( ) ×∑−∂

∂×

∞

=

−

12

0000

1

0002,,,

,,,

,,,

nnCnnn

zyx

L

nnCtezcycxcn

LLL

Ddudvdwd

v

wvuC

TwvuP

wvuCwcFn

πτ

ττγ

γ

( ) ( ) ( ) ( ) ( ) ( )( )

( )−∫ ∫ ∫ ∫

∂

∂−×

−t L L L

nnnnCnC

x y z

dudvdwdw

wvuC

TwvuP

wvuCwvuCwsvcuceF

0 0 0 0

000

1

000

001

,,,

,,,

,,,,,, τ

ττττ

γ

γ

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ×∑ ∫ ∫ ∫ ∫ +−Ω−∞

=1 0 0 0 02

0 ,,,12

n

t L L L

LSLSnnnnCnCnnnnC

zyx

Lx y z

TwvugwcvcusetezcycxcFnLLL

Dετ

π


53

( )( )

( ) ( ) ×∑Ω−∫∇

+×

∞

=12

0

0010

0002

,,,,,,

,,,

,,,1

nnC

zyx

LL

S

SF

LLL

DdudvdwdWdWvuC

Tk

wvu

TwvuP

wvuC z πττ

τµτξ

γ

γ

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( )( )

×∫ ∫ ∫ ∫

++−×

t L L L

SLSLSnnnCnCnnn

x y z

TwvuP

wvuCTwvugvsucetezcycxcn

0 0 0 0

000

,,,

,,,1,,,1

γ

γ τξετ

( ) ( ) ( ) ( ) ( ) ( )∑ ×Ω−∫∇

×∞

=12

0

0010

2,,,

,,,

nnnnnC

zyx

LL

S

nzcycxcF

LLL

DdudvdwdWdWvuC

Tk

wvuwc

z πττ

τµ

( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( )( )

×∫ ∫ ∫ ∫

++−×

−t L L L

SLSLSnnnnCnC

x y z

TwvuP

wvuCwvuCTwgwcvcuseten

0 0 0 0

1

000

010,,,

,,,,,,1,1

γ

γ ττξετ

( ) ( ) ( ) ( ) ( ) ( ) ×∑Ω−∫∇

×∞

=12

0

0000

2,,,

,,,

nnCnnn

zyx

LL

S tezcycxcnLLL

DdudvdwdWdWvuC

Tk

wvu z πττ

τµ

( ) ( ) ( ) ( ) ( )[ ] ( ) ×∫ ∫ ∫ ∫ ∫+−×t L L L L

LSLSnnnnCnC

x y z z

WdWvuCTwvugwcvsuceF0 0 0 0 0

000 ,,,,,,1 τετ

( )( )

( )( )

ττµτ

τξγ

γ

dudvdwdTk

wvu

TwP

wvuCwvuC S

S

,,,

,

,,,,,,1

1

000

010

∇

+×

−

.

Author

Pankratov Evgeny Leonidovich was born at 1977. From 1985 to 1995 he was educated in a secondary

school in Nizhny Novgorod. From 1995 to 2004 he was educated in Nizhny Novgorod State University:

from 1995 to 1999 it was bachelor course in Radiophysics, from 1999 to 2001 it was master course in Ra-

diophysics with specialization in Statistical Radiophysics, from 2001 to 2004 it was PhD course in Radio-

physics. From 2004 to 2008 E.L. Pankratov was a leading technologist in Institute for Physics of Micro-

structures. From 2008 to 2012 E.L. Pankratov was a senior lecture/Associate Professor of Nizhny Novgo-

rod State University of Architecture and Civil Engineering. Now E.L. Pankratov is in his Full Doctor

course in Radiophysical Department of Nizhny Novgorod State University. He has 96 published papers in

area of his researches.

Bulaeva Elena Alexeevna was born at 1991. From 1997 to 2007 she was educated in secondary school of

village Kochunovo of Nizhny Novgorod region. From 2007 to 2009 she was educated in boarding school

“Center for gifted children”. From 2009 she is a student of Nizhny Novgorod State University of Architec-

ture and Civil Engineering (spatiality “Assessment and management of real estate”). At the same time she

is a student of courses “Translator in the field of professional communication” and “Design (interior art)” in

the University. E.A. Bulaeva was a contributor of grant of President of Russia (grant MK-548.2010.2).

She has 29 published papers in area of her researches.

MODELING OF MANUFACTURING OF A FIELDEFFECT TRANSISTOR TO DETERMINE CONDITIONS TO DECREASE LENGTH OF...

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Transcript of MODELING OF MANUFACTURING OF A FIELDEFFECT TRANSISTOR TO DETERMINE CONDITIONS TO DECREASE LENGTH OF...