We introduce an approach of manufacturing of a field-effect heterotransistor with inhomogenous doping of channel. The inhomogenous distribution of concentration of dopant gives a possibility to change speed of transport of charge carriers and to decrease length of channel.

1. International Journal on Computational Science & Applications (IJCSA) Vol.5,No.6, December 2015
DOI:10.5121/ijcsa.2015.5602 17
ON APPROACH OF OPTIMIZATION OF FORMATION
OF INHOMOGENOUS DISTRIBUTIONS OF DOPANT IN
A FIELD-EFFECT HETEROTRANSISTORS
E.L. Pankratov1
and E.A. Bulaeva1,2
1
Nizhny 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
We introduce an approach of manufacturing of a field-effect heterotransistor with inhomogenous doping of
channel. The inhomogenous distribution of concentration of dopant gives a possibility to change speed of
transport of charge carriers and to decrease length of channel.
KEYWORDS
Field-effect heterotransistor; inhomogenous doping of channel; decreasing of dimensions of transistor
1.INTRODUCTION
Development of solid state electronic devices leads to necessity to decrease dimensions of ele-
ments of integrated circuits. In the present time are have been elaborated several approaches to
decrease dimensions of these elements. One of them is growth thin film devices [1-4]. The second
one is diffusion or ion doping of required areas of homogenous sample or heterostructures and
laser or ion doping of dopant and /or radiation defects [5-7]. Using one of these types of anneal-
ing gives a possibility to generate inhomogenous distribution of temperature. In this situation one
can obtain inhomogeneity of structure, which should be doped. The inhomogeneity leads to de-
creasing of dimensions of the considered devices. Another way to modify properties of materials
is radiation processing [8,9].
In this paper we consider manufacturing a field-effect heterotransistor with inhomogenous doping
of channel. Inhomogenous doping of channel during technological process gives us possibility to
change speed of transport of charge carriers [10] and to prevent the effect of closing the source to
the drain after decreasing of length of channel [11]. To illustrate the introduced approach of opti-
mization of doping of channel we consider heterostructure, which presented in Fig. 1. One of
ends of the channel has been doped by diffusion or ion implantation. After that the dopant and/or
radiation defects have been annealed by using electro-magnetic (microwave) irradiation. In this
situation one can find inhomogenous distribution of concentration of dopant. This type of anneal-
ing gives a possibility to heat near-surficial area to decrease dopant diffusion deep heterostruc-
ture. After finishing of the annealing channel will be overgrown. After the overgrowth it should
be manufactured gate. Main aim of the present paper is optimization of annealing time to dope of
required part of channel.

2. International Journal on Computational Science & Applications (IJCSA) Vol.5,No.6, December 2015
18
Fig. 1. Structure of the field-effect heterotransistor
2.METHOD OF SOLUTION
We solved our aims by solving the following second Fick’s law [10,12-14] to determine spatio-
temporal distribution of concentration of dopant
( ) ( ) ( ) ( )






+





+





=
z
tzyxC
D
zy
tzyxC
D
yx
tzyxC
D
xt
tzyxC
CCC
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂ ,,,,,,,,,,,,
. (1)
Boundary and initial conditions for the equations are
( ) 0
,,,
0
=
∂
∂
=x
x
tzyxC
,
( ) 0
,,,
=
∂
∂
= xLx
x
tzyxC
,
( ) 0
,,,
0
=
∂
∂
=y
y
tzyxC
,
( ) 0
,,,
=
∂
∂
= yLx
y
tzyxC
,
( ) 0
,,,
0
=
∂
∂
=z
z
tzyxC
,
( ) 0
,,,
=
∂
∂
= zLx
z
tzyxC
, C(x,y,z,0)=f (x,y,z). (2)
In the Eqs. (1) and (2) the function C(x,y,z,t) describes the spatio-temporal distribution of concen-
tration of dopant; the parameter DС is the dopant diffusion coefficient. Dopant diffusion coeffi-
cient will be changed with changing of materials of heterostructure, heating and cooling of hete-
rostructure during annealing of dopant or radiation defects (with account Arrhenius law). Depen-
dences of dopant diffusion coefficient on coordinate in heterostructure, temperature of annealing
and concentrations of dopant and radiation defects could be written as [15-17]
( ) ( )
( )
( ) ( )
( ) 







++





+= 2*
2
2*1
,,,,,,
1
,,,
,,,
1,,,
V
tzyxV
V
tzyxV
TzyxP
tzyxC
TzyxDD LC ςςξ γ
γ
. (3)
Here function DL (x,y,z,T) describes dependences of dopant diffusion coefficient on coordinate
and temperature of annealing T. Function P (x,y,z,T) describes the same dependences of the limit
of solubility of dopant. The parameter γ is integer and usually could be varying in the following
interval γ ∈[1,3]. The parameter describes quantity of charged defects, which interacting (in aver-
age) with each atom of dopant. More detailed information about concentrational dependence of
dopant diffusion coefficient is presented in [15]. The function V (x,y,z,t) describes distribution of
concentration of radiation vacancies in space and time. The parameter V*
describes the equili-
brium distribution of concentration of vacancies. It should be noted, that diffusion type of doping
gives a possibility to obtain doped materials without radiation defects. In this situation ζ1=ζ2=0.
We determine spatio-temporal distributions of concentrations of radiation defects by solving the
following system of equations [16,17]

3. International Journal on Computational Science & Applications (IJCSA) Vol.5,No.6, December 2015
19
( ) ( ) ( ) ( ) ( ) ( ) ×−





∂
∂
∂
∂
+





∂
∂
∂
∂
=
∂
∂
Tzyxk
y
tzyxI
TzyxD
yx
tzyxI
TzyxD
xt
tzyxI
IIII ,,,
,,,
,,,
,,,
,,,
,,,
,
( ) ( ) ( ) ( ) ( ) ( )tzyxVtzyxITzyxk
z
tzyxI
TzyxD
z
tzyxI VII ,,,,,,,,,
,,,
,,,,,, ,
2
−





∂
∂
∂
∂
+× (4)
( ) ( ) ( ) ( ) ( ) ( ) ×−





∂
∂
∂
∂
+





∂
∂
∂
∂
=
∂
∂
Tzyxk
y
tzyxV
TzyxD
yx
tzyxV
TzyxD
xt
tzyxV
VVVV ,,,
,,,
,,,
,,,
,,,
,,,
,
( ) ( ) ( ) ( ) ( ) ( )tzyxVtzyxITzyxk
z
tzyxV
TzyxD
z
tzyxV VIV ,,,,,,,,,
,,,
,,,,,, ,
2
−





∂
∂
∂
∂
+× .
Boundary and initial conditions for these equations are
( ) 0
,,,
0
=
∂
∂
=x
x
tzyxρ
,
( ) 0
,,,
=
∂
∂
= xLx
x
tzyxρ
,
( ) 0
,,,
0
=
∂
∂
=y
y
tzyxρ
,
( ) 0
,,,
=
∂
∂
= yLy
y
tzyxρ
,
( ) 0
,,,
0
=
∂
∂
=z
z
tzyxρ
,
( ) 0
,,,
=
∂
∂
= zLz
z
tzyxρ
, ρ (x,y,z,0)=fρ (x,y,z). (5)
Here ρ=I,V. The function I(x,y,z,t) describes the distribution of concentration of radiation intersti-
tials in space and time. The functions Dρ(x,y,z,T) describe dependences of the diffusion coeffi-
cients of point radiation defects on coordinate and temperature. The quadric on concentrations
terms of Eqs. (4) describes generation divacancies and diinterstitials. The function kI,V(x,y,z,T)
describes dependence of the parameter of recombination of point radiation defects on coordinate
and temperature. The function kI,I(x,y,z,T) and kV,V(x,y,z,T) describes dependences of the parame-
ters of generation of simplest complexes of point radiation defects on coordinate and temperature.
Now let us calculate distributions of concentrations of divacancies ΦV(x,y,z,t) and diinterstitials
ΦI(x,y,z,t) in space and time by solving the following system of equations [16,17]
( ) ( ) ( ) ( ) ( ) +




 Φ
+




 Φ
=
Φ
ΦΦ
y
tzyx
TzyxD
yx
tzyx
TzyxD
xt
tzyx I
I
I
I
I
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂ ,,,
,,,
,,,
,,,
,,,
( ) ( ) ( ) ( ) ( ) ( )tzyxITzyxktzyxITzyxk
z
tzyx
TzyxD
z
III
I
I ,,,,,,,,,,,,
,,,
,,, 2
, −+




 Φ
+ Φ
∂
∂
∂
∂
(6)
( ) ( ) ( ) ( ) ( ) +




 Φ
+




 Φ
=
Φ
ΦΦ
y
tzyx
TzyxD
yx
tzyx
TzyxD
xt
tzyx V
V
V
V
V
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂ ,,,
,,,
,,,
,,,
,,,
( ) ( ) ( ) ( ) ( ) ( )tzyxVTzyxktzyxVTzyxk
z
tzyx
TzyxD
z
VVV
V
V ,,,,,,,,,,,,
,,,
,,, 2
, −+




 Φ
+ Φ
∂
∂
∂
∂
.
Boundary and initial conditions for these equations are
( )
0
,,,
0
=
∂
Φ∂
=x
x
tzyxρ
,
( )
0
,,,
=
∂
Φ∂
= xLx
x
tzyxρ
,
( )
0
,,,
0
=
∂
Φ∂
=y
y
tzyxρ
,
( )
0
,,,
=
∂
Φ∂
= yLy
y
tzyxρ
,
( )
0
,,,
0
=
∂
Φ∂
=z
z
tzyxρ
,
( )
0
,,,
=
∂
Φ∂
= zLz
z
tzyxρ
, ΦI (x,y,z,0)=fΦI (x,y,z), ΦV (x,y,z,0)=fΦV (x,y,z). (7)

4. International Journal on Computational Science & Applications (IJCSA) Vol.5,No.6, December 2015
20
The functions DΦρ(x,y,z,T) describe dependences of the diffusion coefficients of the above com-
plexes of radiation defects on coordinate and temperature. The functions kI(x,y,z,T) and kV(x,y,z,
T) describe the parameters of decay of these complexes on coordinate and temperature.We calcu-
late distributions of concentrations of point radiation defects in space and time by using recently
elaborated approach [18-20]. To use the approach let us transform dependences of diffusion coef-
ficients of point defects in space and time to the following form: Dρ(x,y,z,T) =D0ρ [1+ερ
gρ(x,y,z,T)], where D0ρ are the average values of diffusion coefficients, 0≤ερ<1, |gρ(x,y,z, T)|≤1,ρ
=I,V. Let us also transform dependences of another parameters to the similar form: kI,V(x, y,z,T)=
k0I,V[1+εI,V gI,V(x,y,z,T)], kI,I(x,y,z,T)=k0I,I [1+εI,I gI,I(x,y,z,T)] and kV,V (x,y,z,T) = k0V,V [1+εV,V
gV,V(x,y,z,T)], where k0ρ1,ρ2 are the their average values, 0≤εI,V <1, 0≤εI,I <1, 0≤εV,V<1, |
gI,V(x,y,z,T)|≤1, | gI,I(x,y,z,T)|≤1, |gV,V(x,y,z,T)|≤1. Let us introduce the following dimensionless va-
riables: ( ) ( ) *
,,,,,,
~
ItzyxItzyxI = , χ = x/Lx, η = y /Ly, ( ) ( ) *
,,,,,,
~
VtzyxVtzyxV = ,
2
00 LtDD VI=ϑ , VIVI DDkL 00,0
2
=ω , VI DDkL 00,0
2
ρρρ =Ω , φ= z/Lz. The introduction leads to
transformation of Eqs.(4) and conditions (5) to the following form
( ) ( )[ ] ( ) ( )[ ]{ ×+
∂
∂
+






∂
∂
+
∂
∂
=
∂
∂
Tg
I
Tg
DD
DI
IIII
VI
I
,,,1
,,,
~
,,,1
,,,
~
00
0
φηχε
ηχ
ϑφηχ
φηχε
χϑ
ϑφηχ
( ) ( )[ ] ( ) ( ) ×−






∂
∂
+
∂
∂
+



∂
∂
× ϑφηχ
φ
ϑφηχ
φηχε
φη
ϑφηχ
,,,
~,,,
~
,,,1
,,,
~
00
0
00
0
I
I
Tg
DD
D
DD
DI
II
VI
I
VI
I
( )[ ] ( ) ( )[ ] ( )ϑφηχφηχεϑφηχφηχεω ,,,
~
,,,1,,,
~
,,,1 2
,,,, ITgVTg IIIIIVIVI +Ω−+× (8)
( ) ( )[ ] ( ) ( )[ ]{ ×+
∂
∂
+






∂
∂
+
∂
∂
=
∂
∂
Tg
V
Tg
DD
DV
VVVV
VI
V
,,,1
,,,
~
,,,1
,,,
~
00
0
φηχε
ηχ
ϑφηχ
φηχε
χϑ
ϑφηχ
( ) ( )[ ] ( ) ( ) ×−






∂
∂
+
∂
∂
+



∂
∂
× ϑφηχ
φ
ϑφηχ
φηχε
φη
ϑφηχ
,,,
~,,,
~
,,,1
,,,
~
00
0
00
0
I
V
Tg
DD
D
DD
DV
VV
VI
V
VI
V
( )[ ] ( ) ( )[ ] ( )ϑφηχφηχεϑφηχφηχεω ,,,
~
,,,1,,,
~
,,,1 2
,,,, VTgVTg VVVVVVIVI +Ω−+×
( ) 0
,,,~
0
=
∂
∂
=χ
χ
ϑφηχρ
,
( ) 0
,,,~
1
=
∂
∂
=χ
χ
ϑφηχρ
,
( ) 0
,,,~
0
=
∂
∂
=η
η
ϑφηχρ
,
( ) 0
,,,~
1
=
∂
∂
=η
η
ϑφηχρ
,
( ) 0
,,,~
0
=
∂
∂
=φ
φ
ϑφηχρ
,
( ) 0
,,,~
1
=
∂
∂
=φ
φ
ϑφηχρ
, ( )
( )
*
,,,
,,,~
ρ
ϑφηχ
ϑφηχρ ρf
= . (9)
Let us determine solutions of Eqs.(8) as the following power series [18-20]
( ) ( )∑ ∑ ∑Ω=
∞
=
∞
=
∞
=0 0 0
,,,~,,,~
i j k
ijk
kji
ϑφηχρωεϑφηχρ ρρ . (10)
After substitution of the series (10) into Eqs.(8) and conditions (9) we obtain equations for initial-
order approximations of concentration of point defects ( )ϑφηχ ,,,
~
000I and ( )ϑφηχ ,,,
~
000V , corrections
( )ϑφηχ ,,,
~
ijkI and ( )ϑφηχ ,,,
~
ijkV and conditions for them for all i≥1, j≥1, k≥1. The equations are

5. International Journal on Computational Science & Applications (IJCSA) Vol.5,No.6, December 2015
21
presented in the Appendix. We calculate solutions of the equations by standard Fourier approach
[21,22]. The solutions are presented in the Appendix.
Farther we determine spatio-temporal distributions of concentrations of simplest complexes of
point radiation defects. To determine the distributions we transform approximations of diffusion
coefficients in the following form: DΦρ(x,y,z,T)=D0Φρ[1+εΦρgΦρ(x,y,z,T)], where D0Φρ are the av-
erage values of diffusion coefficients. In this situation the Eqs.(6) could be written as
( )
( )[ ] ( )
( ) ( )++





 Φ
+=
Φ
ΦΦΦ tzyxITzyxk
x
tzyx
Tzyxg
x
D
t
tzyx
II
I
III
I
,,,,,,
,,,
,,,1
,,, 2
,0
∂
∂
ε
∂
∂
∂
∂
( )[ ] ( )
( )[ ] ( )
−





 Φ
++





 Φ
++ ΦΦΦΦΦΦ
z
tzyx
Tzyxg
z
D
y
tzyx
Tzyxg
y
D I
III
I
III
∂
∂
ε
∂
∂
∂
∂
ε
∂
∂ ,,,
,,,1
,,,
,,,1 00
( ) ( )tzyxITzyxkI ,,,,,,−
( )
( )[ ] ( )
( ) ( )++





 Φ
+=
Φ
ΦΦΦ tzyxITzyxk
x
tzyx
Tzyxg
x
D
t
tzyx
II
V
VVV
V
,,,,,,
,,,
,,,1
,,, 2
,0
∂
∂
ε
∂
∂
∂
∂
( )[ ] ( ) ( )[ ] ( ) −





 Φ
++





 Φ
++ ΦΦΦΦΦΦ
z
tzyx
Tzyxg
z
D
y
tzyx
Tzyxg
y
D V
VVV
V
VVV
∂
∂
ε
∂
∂
∂
∂
ε
∂
∂ ,,,
,,,1
,,,
,,,1 00
( ) ( )tzyxITzyxkI ,,,,,,− .
We determine spatio-temporal distributions of concentrations of complexes of radiation defects as
the following power series
( ) ( )∑ Φ=Φ
∞
=
Φ
0
,,,,,,
i
i
i
tzyxtzyx ρρρ ε . (11)
Equations for the functions Φρi(x,y,z,t) and conditions for them could be obtained by substitution
of the series (11) into Eqs.(6) and appropriate boundary and initial conditions. We present the
equations and conditions in the Appendix. Solutions of the equations have been calculated by
standard approaches [21,22] and presented in the Appendix.
Now we calculate distribution of concentration of dopant in space and time by using the same
approach, which was used for calculation the same distributions of another concentrations. To use
the approach we transform spatio-temperature approximation of dopant diffusion coefficient to
the form: DL(x,y,z,T)=D0L[1+εLgL(x,y,z,T)], where D0L is the average value of dopant diffusion
coefficient, 0≤εL<1, |gL(x,y,z,T)|≤1. Now we solve the Eq.(1) as the following power series
( ) ( )∑ ∑=
∞
=
∞
=0 1
,,,,,,
i j
ij
ji
L tzyxCtzyxC ξε .
The equations for the functions Cij(x,y,z,t) and conditions for them have been obtained by substi-
tution of the series into Eq.(1) and conditions (2). We presented the equations and conditions for
them in the Appendix. We solve the equations by standard Fourier approach [21,22]. The solu-
tions have been presented in the Appendix.

6. International Journal on Computational Science & Applications (IJCSA) Vol.5,No.6, December 2015
22
We analyzed distributions of concentrations of dopant and radiation defects in space and time
analytically by using the second-order approximations on all parameters, which have been used in
appropriate series. The approximations are usually enough good approximations to make qualita-
tive analysis and to obtain quantitative results. We check all results of analytical modeling by
comparison with results of numerical simulation.
3.DISCUSSION
In this section based on recently calculated relations we analyzed redistribution of dopant with
account redistribution of radiation defects. These relations give us possibility to obtain spatial
distributions of concentration of dopant. Typical spatial distributions of concentration of dopant
in directions, which is perpendicular to channel, are presented in Fig. 2. Curve 1 of this paper is a
typical distribution of concentration of dopant in directions of channel. The figure shows, that
presents of interface between layers of heterostructure gives us possibility to obtain more com-
pact and more homogenous distribution of concentration of dopant in direction, which is perpen-
dicular to the interface. However in this situation one shall to optimize annealing time. Reason of
the optimization is following. If annealing time is small, dopant can not achieves the interface. If
annealing time is large, dopant will diffuse into another layers of heterostructure too deep. We
calculate optimal value of annealing time by using recently introduced criterion [18-20]. Frame-
work the criterion we approximate real distribution of concentration of dopant by ideal step-wise
function ψ (x,y,z). Farther the required optimal value of dopant concentration by minimization of
the following mean-squared error
Fig.2. The infused dopant concentration distributions. The considered direction is perpendicular to interface
between epitaxial layer and substrate. Increasing of number of distributions corresponds to increasing of
difference between values of dopant diffusion coefficient in layers of heterostructure. The distributions
have been calculated under condition, when value of dopant diffusion coefficient in epitaxial layer is larger,
than value of dopant diffusion coefficient in substrate

7. International Journal on Computational Science & Applications (IJCSA) Vol.5,No.6, December 2015
23
x
0.0
0.5
1.0
1.5
2.0
C(x,Θ)
2
3
4
1
0 L/4 L/2 3L/4 L
Epitaxial layer Substrate
Fig.3. Fig.2. The implanted dopant concentration distributions. The considered direction is perpendicular to
interface between epitaxial layer and substrate. Curves 1 and 3 corresponds to annealing time Θ = 0.0048
(Lx
2
+Ly
2
+Lz
2
)/D0. Curves 2 and 4 corresponds to annealing time Θ = 0.0057(Lx
2
+Ly
2
+Lz
2
)/D0. Curves 1 and
2 have been calculated for homogenous sample. Curves 3 and 4 have been calculated for heterostructure
under condition, when value of dopant diffusion coefficient in epitaxial layer is larger, than value of dopant
diffusion coefficient in substrate
( ) ( )[ ]∫ ∫ ∫ −Θ=
x y zL L L
zyx
xdydzdzyxzyxC
LLL
U
0 0 0
,,,,,
1
ψ . (12)
Dependences of optimal value of annealing time are presented on Figs. 3. Optimal values of an-
nealing time of implanted dopant should be smaller in comparison with the same annealing time
of infused dopant. Reason of the difference is necessity to anneal radiation defects before anneal-
ing of dopant.
0.0 0.1 0.2 0.3 0.4 0.5
a/L, ξ, ε, γ
0.0
0.1
0.2
0.3
0.4
0.5
ΘD0L
-2
3
2
4
1
Fig.3a. Optimal annealing time of infused dopant as dependences of several parameters. Curve 1 is the de-
pendence of the considered annealing time on dimensionless thickness of epitaxial layer a/L and ξ=γ=0 for
equal to each other values of dopant diffusion coefficient in all parts of heterostructure. Curve 2 is the de-
pendence of the considered annealing time on the parameter ε for a/L=1/2 and ξ=γ=0. Curve 3 is the de-
pendence of the considered annealing time on the parameter ξ for a/L=1/2 and ε=γ=0. Curve 4 is the de-
pendence of the considered annealing time on parameter γ for a/L=1/2 and ε=ξ=0

8. International Journal on Computational Science & Applications (IJCSA) Vol.5,No.6, December 2015
24
0.0 0.1 0.2 0.3 0.4 0.5
a/L, ξ, ε, γ
0.00
0.04
0.08
0.12
ΘD0L
-2
3
2
4
1
Fig.3a. Optimal annealing time of implanted dopant as dependences of several parameters. Curve 1 is the
dependence of the considered annealing time on dimensionless thickness of epitaxial layer a/L and ξ=γ=0
for equal to each other values of dopant diffusion coefficient in all parts of heterostructure. Curve 2 is the
dependence of the considered annealing time on the parameter ε for a/L=1/2 and ξ=γ=0. Curve 3 is the
dependence of the considered annealing time on the parameter ξ for a/L=1/2 and ε=γ=0. Curve 4 is the
dependence of the considered annealing time on parameter γ for a/L=1/2 and ε=ξ=0
4.CONCLUSIONS
In this paper we introduce an approach to manufacture of a field-effect heterotransistor with in-
homogenous doping of channel. Some recommendations to optimize technological process to
manufacture more compact distribution of concentration of dopant have been formulated.
ACKNOWLEDGEMENTS
This work is supported by the agreement of August 27,2013 № 02.В.49.21.0003 between
The Ministry of education and science of the Russian Federation and Lobachevsky State Univer-
sity of Nizhni Novgorod, educational fellowship for scientific research of Government of Russia,
educational fellowship for scientific research of Government of Nizhny Novgorod Region and of
Nizhny Novgorod State University of Architecture and Civil Engineering.
REFERENCES
[1] G. Volovich. Modern chips UM3Ch class D manufactured by firm MPS. Modern Electronics. Issue 2.
P. 10-17 (2006).
[2] A. Kerentsev, V. Lanin. Constructive-technological features of MOSFET-transistors. Power Electron-
ics. Issue 1. P. 34-38 (2008).
[3] O.A. Ageev, A.E. Belyaev, N.S. Boltovets, V.N. Ivanov, R.V. Konakova, Ya.Ya. Kudryk, P.M. Lyt-
vyn, V.V. Milenin, A.V. Sachenko. Influence of displacement of the electron-hole equilibrium on the
process of transition metals diffusion in GaAs. Semiconductors. Vol. 43 (7). P. 897-903 (2009).
[4] N.I. Volokobinskaya, I.N. Komarov, T.V. Matioukhina, V.I. Rechetniko, A.A. Rush, I.V. Falina, A.S.
Yastrebov. Investigation of technological processes of manufacturing of the bipolar power high-
voltage transistors with a grid of inclusions in the collector region. Semiconductors. Vol. 35 (8). P.
1013-1017 (2001).
[5] K.K. Ong, K.L. Pey, P.S. Lee, A.T.S. Wee, X.C. Wang, Y.F. Chong. Dopant distribution in the re-
crystallization transient at the maximum melt depth induced by laser annealing. Appl. Phys. Lett. 89
(17), 172111-172114 (2006).

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[6] H.T. Wang, L.S. Tan, E. F. Chor. Pulsed laser annealing of Be-implanted GaN. J. Appl. Phys. 98 (9),
094901-094905 (2006).
[7] Yu.V. Bykov, A.G. Yeremeev, N.A. Zharova, I.V. Plotnikov, K.I. Rybakov, M.N. Drozdov, Yu.N.
Drozdov, V.D. Skupov. Diffusion processes in semiconductor structures during microwave annealing.
Radiophysics and Quantum Electronics. Vol. 43 (3). P. 836-843 (2003).
[8] V.V. Kozlivsky. Modification of semiconductors by proton beams (Nauka, Sant-Peterburg, 2003, in
Russian).
[9] V.L. Vinetskiy, G.A. Kholodar', Radiative physics of semiconductors. ("Naukova Dumka", Kiev,
1979, in Russian).
[10] I.P. Stepanenko. Basis of Microelectronics (Soviet Radio, Moscow, 1980).
[11] M.V. Dunga, L. Chung-Hsun, X. Xuemei, D.D. Lu, A.M. Niknejad, H. Chenming. Modeling Ad-
vanced FET Technology in a Compact Model. IEEE Transactions on Electron Devices. Vol. 53 (9). P.
157-162 (2006).
[12] V.G. Gusev, Yu.M. Gusev. Electronics (Moscow: Vysshaya shkola, 1991, in Russian).
[13] N.A. Avaev, Yu.E. Naumov, V.T. Frolkin. Basis of microelectronics (Radio and communication,
Moscow, 1991).
[14] V.I. Lachin, N.S. Savelov. Electronics (Phoenix, Rostov-na-Donu, 2001).
[15] Z.Yu. Gotra. Technology of microelectronic devices (Radio and communication, Moscow, 1991).
[16] P.M. Fahey, P.B. Griffin, J.D. Plummer. Diffusion and point defects in silicon. Rev. Mod. Phys.
1989. V. 61. № 2. P. 289-388.
[17] V.L. Vinetskiy, G.A. Kholodar', Radiative physics of semiconductors. ("Naukova Dumka", Kiev,
1979, in Russian).
[18] E.L. Pankratov. Influence of mechanical stress in semiconductor heterostructure on density of p-n-
junctions. Applied Nanoscience. Vol. 2 (1). P. 71-89 (2012).
[19 ]E.L. Pankratov, E.A. Bulaeva. Doping of materials during manufacture p-n-junctions and bipolar
transistors. Analytical approaches to model technological approaches and ways of optimization of dis-
tributions of dopants. Reviews in Theoretical Science. Vol. 1 (1). P. 58-82 (2013).
[20] E.L. Pankratov, E.A. Bulaeva. Increasing of sharpness of diffusion-junction heterorectifier by using
radiation processing. Int. J. Nanoscience. Vol. 11 (5). P. 1250028-1--1250028-8 (2012).
[21] A.N. Tikhonov, A.A. Samarskii. The mathematical physics equations (Moscow, Nauka 1972) (in
Russian).
[22] H.S. Carslaw, J.C. Jaeger. Conduction of heat in solids (Oxford University Press, 1964).
Authors:
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 110 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 74 published papers in area of her researches.

10. International Journal on Computational Science & Applications (IJCSA) Vol.5,No.6, December 2015
26
APPENDIX
Equations for the functions ( )ϑφηχ ,,,
~
ijkI and ( )ϑφηχ ,,,
~
ijkV , i ≥0, j ≥0, k ≥0 and conditions for
them
( ) ( ) ( ) ( )
2
000
2
0
0
2
000
2
0
0
2
000
2
0
0000 ,,,
~
,,,
~
,,,
~
,,,
~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂ I
D
DI
D
DI
D
DI
V
I
V
I
V
I
( ) ( ) ( ) ( )
2
000
2
0
0
2
000
2
0
0
2
000
2
0
0000 ,,,
~
,,,
~
,,,
~
,,,
~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂ V
D
DV
D
DV
D
DV
I
V
I
V
I
V
;
( ) ( ) ( ) ( )
( )


×
∂
∂
+








∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
Tg
III
D
DI
I
iii
V
Ii
,,,
,,,
~
,,,
~
,,,
~
,
~
2
00
2
2
00
2
2
00
2
0
000
φηχ
χφ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑχ
( )
( )
( )
( )


×
∂
∂
+








∂
∂
∂
∂
+




∂
∂
× −−
Tg
I
Tg
D
D
D
DI
I
i
I
V
I
V
Ii
,,,
,,,
~
,,,
,,,
~
100
0
0
0
0100
φηχ
φη
ϑφηχ
φηχ
ηχ
ϑφηχ
( )
V
Ii
D
DI
0
0100 ,,,
~




∂
∂
× −
φ
ϑφηχ
, i≥1,
( ) ( ) ( ) ( )
( )


×
∂
∂
+








∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
Tg
VVV
D
DV
V
iii
I
Vi
,,,
,,,
~
,,,
~
,,,
~
,
~
2
00
2
2
00
2
2
00
2
0
000
φηχ
χφ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑχ
( )
( )
( )
( )


×
∂
∂
+








∂
∂
∂
∂
+




∂
∂
× −−
Tg
V
Tg
D
D
D
DV
V
i
V
I
V
I
Vi
,,,
,,,
~
,,,
,,,
~
100
0
0
0
0100
φηχ
φη
ϑφηχ
φηχ
ηχ
ϑφηχ
( )
I
Vi
D
DV
0
0100 ,,,
~




∂
∂
× −
φ
ϑφηχ
, i≥1,
( ) ( ) ( ) ( ) −





∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
010
2
2
010
2
2
010
2
0
0010 ,,,
~
,,,
~
,,,
~
,,,
~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ III
D
DI
V
I
( )[ ] ( ) ( )ϑφηχϑφηχφηχε ,,,
~
,,,
~
,,,1 000000,, VITg VIVI+−
( ) ( ) ( ) ( ) −





∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
010
2
2
010
2
2
010
2
0
0010 ,,,
~
,,,
~
,,,
~
,,,
~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ VVV
D
DV
I
V
( )[ ] ( ) ( )ϑφηχϑφηχφηχε ,,,
~
,,,
~
,,,1 000000,, VITg VIVI+− ;
( ) ( ) ( ) ( ) −





∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
020
2
2
020
2
2
020
2
0
0020 ,,,
~
,,,
~
,,,
~
,,,
~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ III
D
DI
V
I
( )[ ] ( ) ( ) ( ) ( )[ ]ϑφηχϑφηχϑφηχϑφηχφηχε ,,,
~
,,,
~
,,,
~
,,,
~
,,,1 010000000010,, VIVITg VIVI ++−

11. International Journal on Computational Science & Applications (IJCSA) Vol.5,No.6, December 2015
27
( ) ( ) ( ) ( ) −





∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
020
2
2
020
2
2
020
2
0
0020 ,,,
~
,,,
~
,,,
~
,,,
~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ VVV
D
DV
V
I
( )[ ] ( ) ( ) ( ) ( )[ ]ϑφηχϑφηχϑφηχϑφηχφηχε ,,,
~
,,,
~
,,,
~
,,,
~
,,,1 010000000010,, VIVITg VIVI ++− ;
( ) ( ) ( ) ( ) −





∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
001
2
2
001
2
2
001
2
0
0001 ,,,
~
,,,
~
,,,
~
,,,
~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ III
D
DI
V
I
( )[ ] ( )ϑφηχφηχε ,,,
~
,,,1 2
000,, ITg IIII+−
( ) ( ) ( ) ( ) −





∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
001
2
2
001
2
2
001
2
0
0001 ,,,
~
,,,
~
,,,
~
,,,
~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ VVV
D
DV
I
V
( )[ ] ( )ϑφηχφηχε ,,,
~
,,,1 2
000,, VTg IIII+− ;
( ) ( ) ( ) ( )
×+








∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
V
I
V
I
D
DIII
D
DI
0
0
2
110
2
2
110
2
2
110
2
0
0110 ,,,
~
,,,
~
,,,
~
,,,
~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ
( )
( )
( )
( )
( )[




×
∂
∂
+








∂
∂
∂
∂
+








∂
∂
∂
∂
× Tg
I
Tg
I
Tg III ,,,
,,,
~
,,,
,,,
~
,,, 010010
φηχ
φη
ϑφηχ
φηχ
ηχ
ϑφηχ
φηχ
χ
( )
( ) ( ) ( ) ( )[ ]×+−








∂
∂
× ϑφηχϑφηχϑφηχϑφηχ
φ
ϑφηχ
,,,
~
,,,
~
,,,
~
,,,
~,,,
~
100000000100
010
VIVI
I
( )[ ]Tg IIII ,,,1 ,, φηχε+×
( ) ( ) ( ) ( )
×+








∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
I
V
I
V
D
DVVV
D
DV
0
0
2
110
2
2
110
2
2
110
2
0
0110 ,,,
~
,,,
~
,,,
~
,,,
~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ
( )
( )
( )
( )
( )[




×
∂
∂
+








∂
∂
∂
∂
+








∂
∂
∂
∂
× Tg
V
Tg
V
Tg IVV ,,,
,,,
~
,,,
,,,
~
,,, 010010
φηχ
φη
ϑφηχ
φηχ
ηχ
ϑφηχ
φηχ
χ
( )
( ) ( ) ( ) ( )[ ]×+−








∂
∂
× ϑφηχϑφηχϑφηχϑφηχ
φ
ϑφηχ
,,,
~
,,,
~
,,,
~
,,,
~,,,
~
100000000100
010
IVIV
V
( )[ ]Tg VVVV ,,,1 ,, φηχε+× ;
( ) ( ) ( ) ( ) −





∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
002
2
2
002
2
2
002
2
0
0002 ,,,
~
,,,
~
,,,
~
,,,
~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ III
D
DI
V
I
( )[ ] ( ) ( )ϑφηχϑφηχφηχε ,,,
~
,,,
~
,,,1 000001,, IITg IIII+−

12. International Journal on Computational Science & Applications (IJCSA) Vol.5,No.6, December 2015
28
( ) ( ) ( ) ( ) −





∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
002
2
2
002
2
2
002
2
0
0002 ,,,
~
,,,
~
,,,
~
,,,
~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ VVV
D
DV
I
V
( )[ ] ( ) ( )ϑφηχϑφηχφηχε ,,,
~
,,,
~
,,,1 000001,, VVЕg VVVV+− ;
( ) ( ) ( ) ( )
+








∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
101
2
2
101
2
2
101
2
0
0101 ,,,
~
,,,
~
,,,
~
,,,
~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ III
D
DI
V
I
( ) ( ) ( ) ( )




+





∂
∂
∂
∂
+





∂
∂
∂
∂
+
η
ϑφηχ
φηχ
ηχ
ϑφηχ
φηχ
χ
,,,
~
,,,
,,,
~
,,, 001001
0
0 I
Tg
I
Tg
D
D
II
V
I
( ) ( ) ( )[ ] ( ) ( )ϑφηχϑφηχφηχε
φ
ϑφηχ
φηχ
φ
,,,
~
,,,
~
,,,1
,,,
~
,,, 000100
001
VITg
I
Tg III +−










∂
∂
∂
∂
+
( ) ( ) ( ) ( ) +





∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
101
2
2
101
2
2
101
2
0
0101 ,,,
~
,,,
~
,,,
~
,,,
~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ VVV
D
DV
I
V
( ) ( ) ( ) ( )




+





∂
∂
∂
∂
+





∂
∂
∂
∂
+
η
ϑφηχ
φηχ
ηχ
ϑφηχ
φηχ
χ
,,,
~
,,,
,,,
~
,,, 001001
0
0 V
Tg
V
Tg
D
D
VV
I
V
( ) ( ) ( )[ ] ( ) ( )ϑφηχϑφηχφηχε
φ
ϑφηχ
φηχ
φ
,,,
~
,,,
~
,,,1
,,,
~
,,, 100000
001
VITg
V
Tg VVV +−










∂
∂
∂
∂
+ ;
( ) ( ) ( ) ( )
( ) ×−








∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
ϑφηχ
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ
,,,
~,,,
~
,,,
~
,,,
~
,,,
~
0102
011
2
2
011
2
2
011
2
0
0011
I
III
D
DI
V
I
( )[ ] ( ) ( )[ ] ( ) ( )ϑφηχϑφηχφηχεϑφηχφηχε ,,,
~
,,,
~
,,,1,,,
~
,,,1 000001,,000,, VITgITg VIVIIIII +−+×
( ) ( ) ( ) ( )
( )×−








∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
ϑφηχ
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ
,,,
~,,,
~
,,,
~
,,,
~
,,,
~
0102
011
2
2
011
2
2
011
2
0
0011
V
VVV
D
DV
I
V
( )[ ] ( ) ( )[ ] ( ) ( )ϑφηχϑφηχφηχεϑφηχφηχε ,,,
~
,,,
~
,,,1,,,
~
,,,1 001000,,000,, VItgVTg VIVIVVVV +−+× ;
( )
0
,,,~
0
=
∂
∂
=x
ijk
χ
ϑφηχρ
,
( )
0
,,,~
1
=
∂
∂
=x
ijk
χ
ϑφηχρ
,
( )
0
,,,~
0
=
∂
∂
=η
η
ϑφηχρijk
,
( )
0
,,,~
1
=
∂
∂
=η
η
ϑφηχρijk
,
( )
0
,,,~
0
=
∂
∂
=φ
φ
ϑφηχρijk
,
( )
0
,,,~
1
=
∂
∂
=φ
φ
ϑφηχρijk
(i≥0, j≥0, k≥0);
( ) ( ) *
000 ,,0,,,~ ρφηχφηχρ ρf= , ( ) 00,,,~ =φηχρijk (i≥1, j≥1, k≥1).
Solutions of the above equations could be written as

13. International Journal on Computational Science & Applications (IJCSA) Vol.5,No.6, December 2015
29
( ) ( ) ( ) ( ) ( )∑+=
∞
=1
000
21
,,,~
n
nn ecccF
LL
ϑφηχϑφηχρ ρρ ,
where ( ) ( ) ( ) ( )∫ ∫ ∫=
1
0
1
0
1
0
*
,,coscoscos
1
udvdwdwvufwnvnunF nn ρρ πππ
ρ
, ( ) ( )IVnI DDne 00
22
exp ϑπϑ −= ,
cn(χ)=cos(πnχ), ( ) ( )VInV DDne 00
22
exp ϑπϑ −= ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
×∑ ∫ ∫ ∫ ∫
∂
∂
−−=
∞
=
−
1 0
1
0
1
0
1
0
100
0
0
00
,,,
~
2,,,
~
n
i
nnnInIn
V
I
i
u
wvuI
vcuseecccn
D
D
I
ϑ τ
τϑφηχπϑφηχ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫−−×
∞
=1 0
1
0
1
00
0
2,,,
n
nnnInIn
V
I
In vsuceecccn
D
D
dudvdwdTwvugwc
ϑ
τϑφηχπτ
( ) ( )
( )
( ) ( ) ( ) ( ) ( ) ×∑ ∫ −−∫
∂
∂
×
∞
=
−
1 00
0
1
0
100
2
,,,
~
,,,
n
nInIn
V
Ii
In eecccn
D
D
dudvdwd
v
wvuI
Twvugwc
ϑ
τϑφηχπτ
τ
( ) ( ) ( ) ( )
( )
∫ ∫ ∫
∂
∂
× −
1
0
1
0
1
0
100 ,,,
~
,,, τ
τ
dudvdwd
w
wvuI
Twvugwsvcuc i
Innn , i ≥1,
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ×∫ ∫ ∫ ∫−−=
∞
=1 0
1
0
1
0
1
00
0
00 ,,,2,,,
~
n
VnnnInVn
I
V
i Twvugvcuseecccn
D
D
V
ϑ
τϑφηχπϑφηχ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ×∫ ∫ ∫−−
∂
∂
×
∞
=
−
1 0
1
0
1
00
0100 ,
~
n
nnnInVn
I
Vi
n vsuceecccn
D
D
dudvdwd
u
uV
wc
ϑ
τϑφηχτ
τ
( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ×−∫
∂
∂
×
∞
=
−
1
0
0
1
0
100
2
,
~
,,,2
n
nVn
I
Vi
Vn ecccn
D
D
dudvdwd
v
uV
Twvugwc ϑφηχπτ
τ
π
( ) ( ) ( ) ( ) ( ) ( )
∫ ∫ ∫ ∫
∂
∂
−× −
ϑ
τ
τ
τ
0
1
0
1
0
1
0
100 ,
~
,,, dudvdwd
w
uV
Twvugwsvcuce i
VnnnnI , i≥1,
where sn(χ)=sin(π nχ);
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ∫ ∫ ∫ ∫ ×−−=
∞
=1 0
1
0
1
0
1
0
010 2,,,~
n
nnnnnnnn wcvcuceeccc
ϑ
ρρ τϑφηχϑφηχρ
( )[ ] ( ) ( ) τττε dudvdwdwvuVwvuITwvug VIVI ,,,
~
,,,
~
,,,1 000000,,+× ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ×∑ ∫ ∫ ∫ ∫ +−−=
∞
=1 0
1
0
1
0
1
0
,
0
0
020 12,,,~
n
VInnnnnnnn
V
I
wcvcuceeccc
D
D ϑ
ρρ ετϑφηχϑφηχρ
( )] ( ) ( ) ( ) ( )[ ] τττττ dudvdwdwvuVwvuIwvuVwvuITwvug VI ,,,
~
,,,
~
,,,
~
,,,
~
,,, 010000000010, +× ;

14. International Journal on Computational Science & Applications (IJCSA) Vol.5,No.6, December 2015
30
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=
∞
=1 0
1
0
1
0
1
0
001 2,,,~
n
nnnnnnnn wcvcuceeccc
ϑ
ρρ τϑφηχϑφηχρ
( )[ ] ( ) ττρε ρρρρ dudvdwdwvuTwvug ,,,~,,,1 2
000,,+× ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ∫ ∫ ×∫ ∫−−=
∞
=1 0
1
0
1
0
1
0
002 2,,,~
n
nnnnnnnn wcvcuceeccc
ϑ
ρρ τϑφηχϑφηχρ
( )[ ] ( ) ( ) ττρτρε ρρρρ dudvdwdwvuwvuTwvug ,,,~,,,~,,,1 000001,,+× ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=
∞
=1 0
1
0
1
0
1
00
0
110 2,,,
~
n
nnnnInInnn
V
I
ucvcuseecccn
D
D
I
ϑ
τϑφηχπϑφηχ
( ) ( ) ( ) ( ) ( ) ( ) ×∑−
∂
∂
×
∞
=
−
1
0
0100
2
,,,
~
,,,
n
nInnn
V
Ii
I ecccn
D
D
dudvdwd
u
wvuI
Twvug ϑφηχπτ
τ
( ) ( ) ( ) ( ) ( ) ( ) ×−∫ ∫ ∫ ∫
∂
∂
−× −
V
Ii
InnnnI
D
D
dudvdwd
v
wvuI
Twvugucvsuce
0
0
0
1
0
1
0
1
0
100
2
,,,
~
,,, πτ
τ
τ
ϑ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫
∂
∂
−×
∞
=
−
1 0
1
0
1
0
1
0
100 ,,,
~
,,,
n
i
InnnnInI dudvdwd
w
wvuI
Twvugusvcuceen
ϑ
τ
τ
τϑ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[∑ ∫ ∫ ∫ ∫ ×+−−×
∞
=1 0
1
0
1
0
1
0
,12
n
VInnnnInnnInnnn vcvcuceccecccc
ϑ
ετφηϑχφηχ
( )] ( ) ( ) ( ) ( )[ ] τττττ dudvdwdwvuVwvuIwvuVwvuITwvug VI ,,,
~
,,,
~
,,,
~
,,,
~
,,, 100000000100, +×
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=
∞
=1 0
1
0
1
0
1
00
0
110 2,,,
~
n
nnnnVnVnnn
I
V
ucvcuseecccn
D
D
V
ϑ
τϑφηχπϑφηχ
( ) ( ) ( ) ( ) ( ) ( )×∑−
∂
∂
×
∞
=
−
1
0
0100
2
,,,
~
,,,
n
nVnnn
I
Vi
V ecccn
D
D
dudvdwd
u
wvuV
Twvug ϑφηχπτ
τ
( ) ( ) ( ) ( ) ( ) ( ) ×−∫ ∫ ∫ ∫
∂
∂
−× −
I
Vi
VnnnnV
D
D
dudvdwd
v
wvuV
Twvugucvsuce
0
0
0
1
0
1
0
1
0
100
2
,,,
~
,,, πτ
τ
τ
ϑ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫
∂
∂
−×
∞
=
−
1 0
1
0
1
0
1
0
100 ,,,
~
,,,
n
i
VnnnnVnV dudvdwd
w
wvuV
Twvugusvcuceen
ϑ
τ
τ
τϑ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]∑ ∫ ∫ ∫ ∫ ×+−−×
∞
=1 0
1
0
1
0
1
0
,, ,,,12
n
VIVInnnVnnnInnnn Twvugvcuceccecccc
ϑ
ετφηϑχφηχ
( ) ( ) ( ) ( ) ( )[ ] τττττ dudvdwdwvuVwvuIwvuVwvuIwcn ,,,
~
,,,
~
,,,
~
,,,
~
100000000100 +× ;

15. International Journal on Computational Science & Applications (IJCSA) Vol.5,No.6, December 2015
31
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=
∞
=1 0
1
0
1
0
1
00
0
101 ,,,2,,,
~
n
InnnInInnn
V
I
Twvugvcuseecccn
D
D
I
ϑ
τϑφηχπϑφηχ
( )
( )
( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫−−
∂
∂
×
∞
=1 0
1
00
0001
2
,,,
~
n
nnInInnn
V
I
n uceecccn
D
D
dudvdwd
u
wvuI
wc
ϑ
τϑφηχπτ
τ
( ) ( ) ( )
( )
( ) ( ) ( ) ( ) ×∑−∫ ∫
∂
∂
×
∞
=1
0
0
1
0
1
0
001
2
,,,
~
,,,
n
nnnnI
V
I
Inn cccen
D
D
dudvdwd
v
wvuI
Twvugwcvs φηχϑπτ
τ
( ) ( ) ( ) ( ) ( )
( )
( ) ( ) ( ) ×∑−∫ ∫ ∫ ∫
∂
∂
−×
∞
=10
1
0
1
0
1
0
001
2
,,,
~
,,,
n
nnnInnnnI cccdudvdwd
w
wvuI
Twvugwsvcuce φηχτ
τ
τ
ϑ
( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( )∫ ∫ ∫ ∫ +−×
ϑ
τττετϑ
0
1
0
1
0
1
0
000100,, ,,,
~
,,,
~
,,,1 dudvdwdwvuVwvuITwvugwcvcucee VIVInnnnInI
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=
∞
=1 0
1
0
1
0
1
00
0
101 ,,,2,,,
~
n
VnnnVnVnnn
I
V
Twvugvcuseecccn
D
D
V
ϑ
τϑφηχπϑφηχ
( )
( )
( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫−−
∂
∂
×
∞
=1 0
1
00
0001
2
,,,
~
n
nnVnInnn
I
V
n uceecccn
D
D
dudvdwd
u
wvuV
wc
ϑ
τϑφηχπτ
τ
( ) ( ) ( )
( )
( ) ( ) ( ) ( ) ×∑−∫ ∫
∂
∂
×
∞
=1
0
0
1
0
1
0
001
2
,,,
~
,,,
n
nnnnV
I
V
Vnn cccen
D
D
dudvdwd
v
wvuV
Twvugwcvs φηχϑπτ
τ
( ) ( ) ( ) ( ) ( )
( )
( ) ( ) ( ) ×∑−∫ ∫ ∫ ∫
∂
∂
−×
∞
=10
1
0
1
0
1
0
001
2
,,,
~
,,,
n
nnnVnnnnV cccdudvdwd
w
wvuV
Twvugwsvcuce φηχτ
τ
τ
ϑ
( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( )∫ ∫ ∫ ∫ +−×
ϑ
τττετϑ
0
1
0
1
0
1
0
000100,, ,,,
~
,,,
~
,,,1 dudvdwdwvuVwvuITwvugwcvcucee VIVInnnnVnV
;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ){∑ ∫ ∫ ∫ ∫ ×−−=
∞
=1 0
1
0
1
0
1
0
000011 ,,,
~
2,,,
~
n
nnnnInInnn wvuIwcvcuceecccI
ϑ
ττϑφηχϑφηχ
( )[ ] ( ) ( )[ ] ( ) ( )} τττετε dudvdwdwvuVwvuITwvugwvuITwvug VIVIIIII ,,,
~
,,,
~
,,,1,,,
~
,,,1 000001,,010,, +++×
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ){∑ ∫ ∫ ∫ ∫ ×−−=
∞
=1 0
1
0
1
0
1
0
000011 ,,,
~
2,,,
~
n
nnnnVnVnnn wvuIwcvcuceecccV
ϑ
ττϑφηχϑφηχ
( )[ ] ( ) ( )[ ] ( ) ( )} τττετε dudvdwdwvuVwvuITwvugwvuITwvug VIVIVVVV ,,,
~
,,,
~
,,,1,,,
~
,,,1 000001,,010,, +++× .
Equations for the functions Φρi(x,y,z,t), boundary and initial conditions for them could be written
as
( ) ( ) ( ) ( ) +




 Φ
+
Φ
+
Φ
=
Φ
Φ 2
0
2
2
0
2
2
0
2
0
0 ,,,,,,,,,,,,
z
tzyx
y
tzyx
x
tzyx
D
t
tzyx III
I
I
∂
∂
∂
∂
∂
∂
∂
∂

16. International Journal on Computational Science & Applications (IJCSA) Vol.5,No.6, December 2015
32
( ) ( ) ( ) ( )tzyxITzyxktzyxITzyxk III ,,,,,,,,,,,, 2
, −+
( ) ( ) ( ) ( ) +




 Φ
+
Φ
+
Φ
=
Φ
Φ 2
0
2
2
0
2
2
0
2
0
0 ,,,,,,,,,,,,
z
tzyx
y
tzyx
x
tzyx
D
t
tzyx VVV
V
V
∂
∂
∂
∂
∂
∂
∂
∂
( ) ( ) ( ) ( )tzyxVTzyxktzyxVTzyxk VVV ,,,,,,,,,,,, 2
, −+ ;
( ) ( ) ( ) ( )
+







 Φ
+
Φ
+
Φ
=
Φ
Φ 2
2
2
2
2
2
0
,,,,,,,,,,,,
z
tzyx
y
tzyx
x
tzyx
D
t
tzyx iIiIiI
I
iI
∂
∂
∂
∂
∂
∂
∂
∂
( )
( )
( )
( )




+




 Φ
+




 Φ
+
−
Φ
−
ΦΦ
y
tzyx
Tzyxg
yx
tzyx
Tzyxg
x
D
iI
I
iI
II
∂
∂
∂
∂
∂
∂
∂
∂ ,,,
,,,
,,,
,,,
11
0
( )
( )









 Φ
+
−
Φ
z
tzyx
Tzyxg
z
iI
I
∂
∂
∂
∂ ,,,
,,,
1
, i≥1,
( ) ( ) ( ) ( )
+







 Φ
+
Φ
+
Φ
=
Φ
Φ 2
2
2
2
2
2
0
,,,,,,,,,,,,
z
tzyx
y
tzyx
x
tzyx
D
t
tzyx iViViV
V
iV
∂
∂
∂
∂
∂
∂
∂
∂
( )
( )
( )
( )




+




 Φ
+




 Φ
+
−
Φ
−
ΦΦ
y
tzyx
Tzyxg
yx
tzyx
Tzyxg
x
D
iV
V
iV
VV
∂
∂
∂
∂
∂
∂
∂
∂ ,,,
,,,
,,,
,,,
11
0
( )
( )









 Φ
+
−
Φ
z
tzyx
Tzyxg
z
iV
V
∂
∂
∂
∂ ,,,
,,,
1
, i≥1;
( )
0
,,,
0
=
∂
Φ∂
=x
i
x
tzyxρ
,
( )
0
,,,
=
∂
Φ∂
= xLx
i
x
tzyxρ
,
( )
0
,,,
0
=
∂
Φ∂
=y
i
y
tzyxρ
,
( )
0
,,,
=
∂
Φ∂
= yLy
i
y
tzyxρ
,
( )
0
,,,
0
=
∂
Φ∂
=z
i
z
tzyxρ
,
( )
0
,,,
=
∂
Φ∂
= zLz
i
z
tzyxρ
, i≥0; Φρ0(x,y,z,0)=fΦρ (x,y,z), Φρi(x,y,z,0)=0, i≥1.
Solutions of the above equations could be written as
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑+∑+=Φ
∞
=
∞
=
ΦΦ
11
0
221
,,,
n
nnn
n
nnnnn
zyxzyx
zcycxcn
L
tezcycxcF
LLLLLL
tzyx ρρρ
( ) ( ) ( ) ( ) ( ) ( ) ( )[∫ ∫ ∫ ∫ −−× ΦΦ
t L L L
IInnnnn
x y z
wvuITwvukwcvcucete
0 0 0 0
2
, ,,,,,, ττρρ
( ) ( )] ττ dudvdwdwvuITwvukI ,,,,,,− ,

17. International Journal on Computational Science & Applications (IJCSA) Vol.5,No.6, December 2015
33
where ( ) ( ) ( ) ( )∫ ∫ ∫= ΦΦ
x y zL L L
nnnn udvdwdwvufwcvcucF
0 0 0
,,ρρ
, ( )
















++−= ΦΦ 2220
22 111
exp
zyx
n
LLL
tDnte ρρ
π ,
cn(x)=cos(π n x/Lx);
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=Φ
∞
=
ΦΦΦ
1 0 0 0 0
2
,,,
2
,,,
n
t L L L
nnnnnnn
zyx
i
x y z
Twvugvcusetezcycxcn
LLL
tzyx ρρρ
τ
π
ρ
( )
( )
( ) ( ) ( ) ( ) ( ) ×∑ ∫ −−
Φ
×
∞
=
ΦΦ
−
1 0
2
1 2,,,
n
t
nnnnn
zyx
iI
n etezcycxcn
LLL
dudvdwd
u
wvu
wc τ
π
τ
∂
τ∂
ρρ
ρ
( ) ( ) ( ) ( ) ( )
( )
×∑−∫ ∫ ∫ ∫
Φ
−×
∞
=
−
ΦΦ
1
2
0 0 0 0
1 2,,,
,,,
n
zyx
t L L L
iI
nnnn n
LLL
dudvdwd
v
wvu
Twvugwcvsuce
x y z π
τ
∂
τ∂
τ ρ
ρρ
( ) ( ) ( ) ( ) ( )
( )
( ) ×∫ ∫ ∫ ∫
Φ
−× Φ
−
ΦΦ
t L L L
iI
nnnnn
x y z
dudvdwdTwvug
w
wvu
wsvcucete
0 0 0 0
1
,,,
,,,
τ
∂
τ∂
τ ρ
ρ
ρρ
( ) ( ) ( )zcycxc nnn× , i≥1,
where sn(x)=sin(π n x/Lx).
Equations for the functions Cij(x,y,z,t), boundary and initial conditions for them could be written as
( ) ( ) ( ) ( )
2
00
2
02
00
2
02
00
2
0
00 ,,,,,,,,,,,,
z
tzyxC
D
y
tzyxC
D
x
tzyxC
D
t
tzyxC
LLL
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
;
( ) ( ) ( ) ( )
+





∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
0
2
2
0
2
2
0
2
0
0 ,,,,,,,,,,,,
z
tzyxC
y
tzyxC
x
tzyxC
D
t
tzyxC iii
L
i
( )
( )
( )
( )
+





∂
∂
∂
∂
+





∂
∂
∂
∂
+ −−
y
tzyxC
Tzyxg
y
D
x
tzyxC
Tzyxg
x
D i
LL
i
LL
,,,
,,,
,,,
,,, 10
0
10
0
( ) ( )






∂
∂
∂
∂
+ −
z
tzyxC
Tzyxg
z
D i
LL
,,,
,,, 10
0 , i≥1;
( ) ( ) ( ) ( )+
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
01
2
02
01
2
02
01
2
0
01 ,,,,,,,,,,,,
z
tzyxC
D
y
tzyxC
D
x
tzyxC
D
t
tzyxC
LLL
( )
( )
( ) ( )
( )
( ) +





∂
∂
∂
∂
+





∂
∂
∂
∂
+
y
tzyxC
TzyxP
tzyxC
y
D
x
tzyxC
TzyxP
tzyxC
x
D LL
,,,
,,,
,,,,,,
,,,
,,, 0000
0
0000
0 γ
γ
γ
γ
( )
( )
( )






∂
∂
∂
∂
+
z
tzyxC
TzyxP
tzyxC
z
D L
,,,
,,,
,,, 0000
0 γ
γ
;

18. International Journal on Computational Science & Applications (IJCSA) Vol.5,No.6, December 2015
34
( ) ( ) ( ) ( )+
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
02
2
02
02
2
02
02
2
0
02 ,,,,,,,,,,,,
z
tzyxC
D
y
tzyxC
D
x
tzyxC
D
t
tzyxC
LLL
( )
( )
( )
( )
( )
( )
( )






×
∂
∂
+





∂
∂
∂
∂
+
−−
TzyxP
tzyxC
tzyxC
yx
tzyxC
TzyxP
tzyxC
tzyxC
x
D L
,,,
,,,
,,,
,,,
,,,
,,,
,,,
1
00
01
00
1
00
010 γ
γ
γ
γ
( )
( )
( )
( )
( )
+










∂
∂
∂
∂
+


∂
∂
×
−
z
tzyxC
TzyxP
tzyxC
tzyxC
zy
tzyxC ,,,
,,,
,,,
,,,
,,, 00
1
00
01
00
γ
γ
( )
( )
( )
( )
( ) ( )
( )






×
∂
∂
+










∂
∂
∂
∂
+


∂
∂
×
−
TzyxP
tzyxC
x
D
z
tzyxC
TzyxP
tzyxC
tzyxC
zy
tzyxC
L
,,,
,,,,,,
,,,
,,,
,,,
,,, 00
0
00
1
00
01
00
γ
γ
γ
γ
( ) ( )
( )
( ) ( )
( )
( )










∂
∂
∂
∂
+





∂
∂
∂
∂
+


∂
∂
×
z
tzyxC
TzyxP
tzyxC
zy
tzyxC
TzyxP
tzyxC
yx
tzyxC ,,,
,,,
,,,,,,
,,,
,,,,,, 0100010001
γ
γ
γ
γ
;
( ) ( ) ( ) ( )+
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
11
2
02
11
2
02
11
2
0
11 ,,,,,,,,,,,,
z
tzyxC
D
y
tzyxC
D
x
tzyxC
D
t
tzyxC
LLL
( ) ( )
( )
( ) ( ) ( )
( )





×
∂
∂
+





∂
∂
∂
∂
+
−−
TzyxP
tzyxC
tzyxC
yx
tzyxC
TzyxP
tzyxC
tzyxC
x ,,,
,,,
,,,
,,,
,,,
,,,
,,,
1
00
10
00
1
00
10 γ
γ
γ
γ
( ) ( ) ( )
( )
( ) +









∂
∂
∂
∂
+


∂
∂
×
−
LD
z
tzyxC
TzyxP
tzyxC
tzyxC
zy
tzyxC
0
00
1
00
10
00 ,,,
,,,
,,,
,,,
,,,
γ
γ
( )
( )
( ) ( )
( )
( )



+





∂
∂
∂
∂
+





∂
∂
∂
∂
+
y
tzyxC
TzyxP
tzyxC
yx
tzyxC
TzyxP
tzyxC
x
D L
,,,
,,,
,,,,,,
,,,
,,, 10001000
0 γ
γ
γ
γ
( )
( )
( ) ( ) ( )



+





∂
∂
∂
∂
+









∂
∂
∂
∂
+
x
tzyxC
Tzyxg
x
D
z
tzyxC
TzyxP
tzyxC
z
LL
,,,
,,,
,,,
,,,
,,, 01
0
1000
γ
γ
( ) ( ) ( ) ( )









∂
∂
∂
∂
+





∂
∂
∂
∂
+
z
tzyxC
Tzyxg
zy
tzyxC
Tzyxg
y
LL
,,,
,,,
,,,
,,, 0101
;
( )
0
,,,
0
=
=x
ij
x
tzyxC
∂
∂
,
( )
0
,,,
=
= xLx
ij
x
tzyxC
∂
∂
,
( )
0
,,,
0
=
=y
ij
y
tzyxC
∂
∂
,
( )
0
,,,
=
= yLy
ij
y
tzyxC
∂
∂
,
( )
0
,,,
0
=
=z
ij
z
tzyxC
∂
∂
,
( )
0
,,,
=
= zLz
ij
z
tzyxC
∂
∂
, i ≥0, j ≥0;
C00(x,y,z,0)=fC (x,y,z), Cij(x,y,z,0)=0, i ≥1, j ≥1.

19. International Journal on Computational Science & Applications (IJCSA) Vol.5,No.6, December 2015
35
Solutions of the above equations with account boundary and initial have been calculated by
Fourier approach. The result of calculation could be written as
( ) ( ) ( ) ( ) ( )∑+=
∞
=1
00
21
,,,
n
nCnnnnC
zyxzyx
tezcycxcF
LLLLLL
tzyxC ,
where ( )
















++−= 2220
22 111
exp
zyx
CnC
LLL
tDnte π , ( ) ( ) ( ) ( )∫ ∫ ∫=
x y zL L L
nCnnnC udvdwdwcwvufvcucF
0 0 0
,, ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=
∞
=1 0 0 0 0
20 ,,,
2
,,,
n
t L L L
LnnnCnCnnnnC
zyx
i
x y z
TwvugvcusetezcycxcFn
LLL
tzyxC τ
π
( )
( )
( ) ( ) ( ) ( ) ( ) ×∑ ∫ −−
∂
∂
×
∞
=
−
1 0
2
10 2,,,
n
t
nCnCnnnnC
zyx
i
n etezcycxcFn
LLL
dudvdwd
u
wvuC
wc τ
π
τ
τ
( ) ( ) ( ) ( )
( )
( )∑ ×−∫ ∫ ∫
∂
∂
×
∞
=
−
1
2
0 0 0
10 2,,,
,,,
n
nCnC
zyx
L L L
i
Lnnn teFn
LLL
dudvdwd
v
wvuC
Twvugvcvsuc
x y z π
τ
τ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
∫ ∫ ∫ ∫
∂
∂
−× −
t L L L
i
LnnnnCnnn
x y z
dudvdwd
w
wvuC
Twvugvsvcucezcycxc
0 0 0 0
10 ,,,
,,, τ
τ
τ , i ≥1;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫ ∫−−=
∞
=1 0 0 0 0
201
2
,,,
n
t L L L
nnnnCnCnnnnC
zyx
x y z
wcvcusetezcycxcFn
LLL
tzyxC τ
π
( )
( )
( ) ( ) ( ) ( ) ( )×∑−
∂
∂
×
∞
=1
2
0000 2,,,
,,,
,,,
n
nCnnnnC
zyx
tezcycxcFn
LLL
dudvdwd
u
wvuC
TwvuP
wvuC π
τ
ττ
γ
γ
( ) ( ) ( ) ( )
( )
( )
( )
( )×∑−∫ ∫ ∫ ∫
∂
∂
−×
∞
=1
2
0 0 0 0
0000 2,,,
,,,
,,,
n
nC
zyx
t L L L
nnnnC ten
LLL
dudvdwd
v
wvuC
TwvuP
wvuC
wcvsuce
x y z π
τ
ττ
τ γ
γ
( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
( )
( )
∫ ∫ ∫ ∫
∂
∂
−×
t L L L
nnnnCnnnnC
x y z
dudvdwd
w
wvuC
TwvuP
wvuC
wsvcucezcycxcF
0 0 0 0
0000 ,,,
,,,
,,,
τ
ττ
τ γ
γ
;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=
∞
=1 0 0 0 0
202
2
,,,
n
t L L L
nnnnCnCnnnnC
zyx
x y z
wcvcusetezcycxcFn
LLL
tzyxC τ
π
( ) ( )
( )
( ) ( ) ( )×∑−
∂
∂
×
∞
=
−
1
2
00
1
00
01
2,,,
,,,
,,,
,,,
n
nnnC
zyx
ycxcF
LLL
dudvdwd
u
wvuC
TwvuP
wvuC
wvuC
π
τ
ττ
τ γ
γ
( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
( ) ×∫ ∫ ∫ ∫
∂
∂
−×
−t L L L
nnnCnCn
x y z
v
wvuC
TwvuP
wvuC
wvuCvsucetezcn
0 0 0 0
00
1
00
01
,,,
,,,
,,,
,,,
ττ
ττ γ
γ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫−−×
∞
=1 0 0 0
2
2
n
t L L
nnnCnCnnnnC
zyx
n
x y
vcucetezcycxcFn
LLL
dudvdwdwc τ
π
τ

20. International Journal on Computational Science & Applications (IJCSA) Vol.5,No.6, December 2015
36
( ) ( ) ( )
( )
( ) ( )×∑−∫
∂
∂
×
∞
=
−
1
2
0
00
1
00
01
2,,,
,,,
,,,
,,,
n
n
zyx
L
n xcn
LLL
dudvdwd
w
wvuC
TwvuP
wvuC
wvuCws
z π
τ
ττ
τ γ
γ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∫ ∫ ∫ ∫
∂
∂
−×
t L L L
nnnnCnCnnnC
x y z
u
wvuC
wvuCwcvcusetezcycF
0 0 0 0
00
01
,,,
,,,
τ
ττ
( )
( )
( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫−−×
∞
=
−
1 0 0
2
1
00 2
,,,
,,,
n
t L
nnCnCnnnnC
zyx
x
ucetezcycxcFn
LLL
dudvdwd
TwvuP
wvuC
τ
π
τ
τ
γ
γ
( ) ( ) ( ) ( )
( )
( ) ×∑−∫ ∫
∂
∂
×
∞
=
−
1
2
0 0
00
1
00
01
2,,,
,,,
,,,
,,,
n
zyx
L L
nn n
LLL
dudvdwd
v
wvuC
TwvuP
wvuC
wvuCwcvs
y z π
τ
ττ
τ γ
γ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
×∫ ∫ ∫ ∫−×
−t L L L
nnnnCnCnnnnC
x y z
TwvuP
wvuC
wvuCwsvcucetezcycxcF
0 0 0 0
1
00
01
,,,
,,,
,,, γ
γ
τ
ττ
( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫−−
∂
∂
×
∞
=1 0 0
2
00 2,,,
n
t L
nnCnCnnnnC
zyx
x
usetezcycxcF
LLL
dudvdwd
w
wvuC
τ
π
τ
τ
( ) ( ) ( )
( )
( ) ( ) ( )∑ ×−∫ ∫
∂
∂
×
∞
=1
2
0 0
0100 2,,,
,,,
,,,
n
nCn
zyx
L L
nn texc
LLL
dudvdwd
u
wvuC
TwvuP
wvuC
wcvcn
y z π
τ
ττ
γ
γ
( ) ( ) ( ) ( ) ( ) ( )
( )
( ) ×∫ ∫ ∫ ∫
∂
∂
−×
t L L L
nnnnCnnC
x y z
dudvdwd
v
wvuC
TwvuP
wvuC
wcvsuceycF
0 0 0 0
0100 ,,,
,,,
,,,
τ
ττ
τ γ
γ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫ ∫−−×
∞
=1 0 0 0 0
2
2
n
t L L L
nnnnCnCnnnnC
zyx
n
x y z
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2
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LLL
zcycxcF τ
π

21. International Journal on Computational Science & Applications (IJCSA) Vol.5,No.6, December 2015
37
( ) ( )
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