Indian Journal of Engineering & Materials Sciences
Vol. 12, August 2005, pp. 356-362
Distribution of pulse type uniform input in aquifers in tropical regions
Naveen Kumara, Vijay Kumar Singha & R R Yadavb
a
Department of Mathematics, Faculty of Science, Banaras Hindu University, Varanasi 221 005, India
b
Department of Mathematics, University of Lucknow, Lucknow 226 002, India
Received 3 December 2004; accepted 1 April 2005
The present work presents mathematical models for the distribution of solute concentration of uniform input of pulse
type along sinusoidally varying unsteady longitudinal velocity through such inhomogeneous aquifers of finite as well as
semi-infinite extents whose main source of replenishment is infiltration during rains. A new time variable is introduced by
using an expression in old time variable. Numerical solutions of such models are obtained using finite difference approximations. At initial stage the dispersion problem is considered steady one in homogeneous aquifer. Its solution is considered as
initial condition for the respective unsteady problems. To achieve minimum time period of computation, the parameters of
the problem are non-dimensionalized in terms of the existing variables in cases of semi-infinite and finite aquifers separately. The effects of inhomogeneity in both cases are shown through illustrations.
IPC Code: F15B
The intensive use of natural resources and the large
production of wastes in the modern society are posing
a great threat to groundwater quality and have resulted in many incidents of groundwater pollution.
With the growing recognition of the importance of
groundwater resources and increasing dependence on
it due to failure of municipal resources, in some developing countries like India, efforts are increasing to
prevent, reduce and eliminate groundwater pollution.
After a pollutant reaches the water table, it penetrates
vertically downward due to much higher density and
from each position of the vertical, it starts spreading
along the groundwater flow in horizontal directions.
As the pollutant concentration decreases due to hydrodynamic dispersion and other attenuating effects
like adsorption, first order decay term, the most important task is to know the position and time, away
from the source at which harmless or very low concentration is existing. It may be accomplished by suitable mathematical modeling and its solution. It may
also predict the time period in which a polluted aquifer may be rehabilitated once the source of pollution
on the ground surface is eliminated. The present work
is an effort in this context.
Solutions to conventional one-dimensional problems have been compiled1,2. Most of these works have
considered the other effects responsible for concentration attenuation. All these works have considered the
porous domains as homogeneous and the flow
through them as time and space independent.
Inhomogeneity of Aquifers
Aquifers are seldom homogeneous. The study of
dispersion problem through inhomogeneous porous
domain was initiated in 1967 (ref. 3) in which the
domain was considered layered one, and the stratification was considered parallel to the longitudinal steady
flow direction. This theory was further pursued4,5. But
this theory has a practical difficulty of involving as
many governing equations, as is the number of parallel layers. This makes difficult to solve these equations either analytically or numerically in case of
large number of such layers. To overcome the difficulty porosity has been proposed as linear continuous
function of distance variable6 and the dispersion problem was solved along steady flow numerically for
some such relationships7. But these relationships limit
the study only to a finite domain. Later two exponential type porosity distance relationships which describe the increasing and decreasing natures of inhomogeneity of a semi-infinite aquifer were given8. In
the present study two continuous functions of space
variable are chosen in each case of semi-infinite and
finite aquifers.
Unsteady Groundwater Velocity
The other aspect nearer to realistic situation may be
that the groundwater velocity may not be steady always, as considered in most of the previous works. In
case of rising and falling water tables the groundwater
velocity will be transient or unsteady9. While consi-
KUMAR et al.: DISTRIBUTION OF PULSE TYPE UNIFORM INPUT IN AQUIFERS IN TROPICAL REGIONS
dering the problem of unsteady flow in porous domain, and longitudinal dispersion in such an unsteady
flow, linear and exponentially decreasing time dependent forms for seepage velocity were derived10,
the latter expression was considered in the study of
flow against dispersion in homogeneous porous medium11. While establishing a direct relationship12 between dispersion coefficient and steady velocity it
was found valid for exponentially and sinusoidally
varying unsteady seepage velocity too. This led to
consider the following sinusoidal time dependent
form of groundwater velocity in the present study:
u = u0 (1 − sin mt ) ,
… (1)
where u0 is the initial velocity and m(T)-1 is the flow
resistance coefficient.
This represents the seasonal behaviour of sinusoidal nature of groundwater velocity in tropical regions
like Indian sub-continent. The main source of replenishment of aquifers in such regions is the infiltration
during rainy seasons. For mt = 2, 5, 8, ......, 29, 32, 35,
u is alternatively minimum and maximum. For m =
0.0165 (day)-1, these values of mt provide t(days) =
121.2, 303.0, 484.8, ......, 1757.6, 1939.4, 2121.2, respectively, which are at uniform interval of 181.8
days. So if t = 121.2 days is considered some date in
the month of June, the hottest month of the year, just
before rainy season, when the groundwater level, and
the velocity is minimum, then the next value t = 303.0
days will represent almost same date in the month of
December which is the coldest after the rainy season,
during which the groundwater velocity is maximum.
357
Similarly the next two values of t will correspond to
the months of June and December, respectively, in the
successive years (Table 1).
Initial condition
In most of the works, the initial concentration is
taken zero. It represents the most ideal case where the
groundwater is supposed solute free at the initial
stage. Deviating from this situation, initial concentrations are considered as functions of distance variable,
in many works. But the expressions for those are chosen arbitrarily. To get rid of such arbitraryness, in
each of the two present problems, initial condition is
chosen as solution of the same problem but for steady
concentration distribution along steady flow through
homogeneous aquifer. Similar types of initial conditions have also been considered in some of the problems of unsteady concentration dispersion along
steady flow through homogeneous porous media2.
Boundary conditions
The unsteady concentration distribution along porous media flow is described by a partial differential
equation of parabolic type. To obtain its particular
solution one initial condition and two conditions at
different points of the space domain (boundary conditions) are required. Usually one condition is defined
in terms of input concentration (the pollutant concentration reaching the groundwater domain) at the origin
(the position at which the input meets the groundwater level, in case of point source of pollution). The
second condition is defined at the extreme end of the
porous domain. In the present work the input condi-
Table 1—Values of (i) u = u0(1-sin mt) for u0 = 0.01 hm/day, m = 0.0165 (day)-1 and different values of mt, and
(ii) non-dimensional time variable
mt
u (hm/day)
t (days)
T
Duration
2
5
8
11
14
17
20
23
26
29
32
35
38
0.9070 E-03
0.1959 E-01
0.1064 E-03
0.2000 E-01
0.9393 E-04
0.1961 E-01
0.8705 E-03
0.1846 E-01
0.2374 E-02
0.1664 E-01
0.4486 E-02
0.1428 E-01
0.7036 E-02
121.2
303.0
484.8
667.7
848.5
1030.3
1212.1
1393.9
1575.8
1757.6
1939.4
2121.2
2303.0
0.3539 E-02
0.2596 E-01
0.4154 E-01
0.6063 E-01
0.7962 E-01
0.9530 E-01
0.1176 E+00
0.1301 E+00
0.1554 E+00
0.1652 E+00
0.1929 E+00
0.2006 E+00
0.2300 E+00
June I yr
December Iyr
June II yr.
December II yr
June III yr.
December III yr.
June IV yr.
December IV yr.
June V yr.
December V yr.
June VI yr.
December VI yr.
June VII yr.
358
INDIAN J. ENG. MATER. SCI., AUGUST 2005
tion is of uniform nature and of pulse type. This is
defined as:
x = 0; c = C0; 0 < t ≤ to
… (2a)
= 0; t > t0,
… (2b)
where c(ML-3) is the concentration at any time t and
position x along the longitudinal direction of aquifer,
C0 is reference concentration and t0 is the time at
which the source of pollution on the surface is eliminated. Conditions (2a,b) illustrate that the input concentration remains uniform with time, up to t = t0, and
beyond that it becomes zero. This situation occurs due
to uniform existence of pollution at its source till it is
buried for ever. The second boundary condition is
defined as:
x → ∞, ∂c/∂x = 0 (in case of semi-infinite aquifer)
… (3a)
or x = L, ∂c/∂x = 0 (in case of finite aquifer),
… (3b)
where L is the length of the aquifer.
Concentration Distribution Along Unsteady Flow
Through Inhomogeneous Aquifer
The concentration distribution behaviour along unsteady groundwater flow through inhomogenous aquifer, under the effects of adsorption, first order decay
and zero-order production in one space dimension,
may be described by the following partial differential
equation13:
R
∂c 1 ∂ ⎡
∂c
⎤
=
n (D
− uc) ⎥ − μ * (t )c + γ * (t ) ,
⎢
∂t n ∂ x ⎣
∂x
⎦
… (4)
where D and u are dispersion coefficient (L2T-1) and
groundwater velocity (LT-1), respectively, n(x) is the
porosity of the aquifer, μ*(T-1) is the first order decay
term, γ*(ML-3 T-1) is zero order production term and R
= 1 + K [{1 − n( x)}/n( x)] , K being a constant, is the
retardation factor due to adsorption. Let porosity, dispersion and velocity be of following degenerate
forms:
n = n0 F(x); D = D(t) F(x) and u = u(t) F(x),
… (5)
where n0 is porosity of homogeneous aquifer. Using
Eq. (5), Eq. (4) may be written as
⎡
∂ 2c
∂c ⎤
dF ( x)
∂c
= F ( x) ⎢ D (t ) 2 − u (t ) ⎥ + 2
R
∂t
∂x
∂x ⎦
dx
⎣
∂c
⎡
⎤
*
*
⎢ D (t ) ∂x − u (t )c ⎥ − μ (t ) c + γ (t )
⎣
⎦
… (6)
Let u(t) = u0V(t), u0 be the initial velocity of
groundwater, and V(t) be a non-dimensional expression of time. Owing to the direct relationship12 between dispersion coefficient and velocity, D = αu,
may be considered, where α (L) is a constant which
depends upon the pore-geometry of the aquifer. Thus
D = αu0V(t) = D0V(t), D0 = αu0 being the initial dispersion coefficient. As other attenuation effects will
also depend upon the groundwater velocity directly,
μ* (t) = μ0 V(t) and γ*(t) = γ 0 V(t) may also be considered, where μ0 and γ0 are the initial values. Using
these relationships and introducing a new time variable by a transformation14:
T* =
∫
t
0
V (t )dt ,
Eq. (6) may be written as
⎡ ∂2c
∂c
∂c ⎤
R * = F ( x) ⎢ D0 2 − u0 ⎥
∂T
∂
∂x ⎦
x
⎣
dF ( x) ⎡ ∂c
⎤
+2
− u0 c ⎥ − μ 0 c + γ 0
D0
⎢
dx ⎣ ∂x
⎦
… (7)
… (8)
Now the dispersion problem is dealt with for the
following two types of aquifers:
Semi-infinite aquifer
In case of semi-infinite aquifer the initial condition
to solve Eq. (8) is taken as the solution of the following boundary value problem (as discussed earlier):
D0
d 2c
dc
− u0
− μ0 c + γ 0 = 0 ,
2
dx
dx
… (9)
x = 0; c = Ci and x → ∞; (dc/dx) = 0 ,
which may be obtained as
⎛
⎡ u
⎤
γ
γ ⎞
c( x,0) = 0 + ⎜ Ci − 0 ⎟ exp ⎢ 0 (1 − δ 0 ) x ⎥ ,
μ0 ⎝
μ0 ⎠
⎣ 2 D0
⎦
δ 0 = (1 + 4μ0 D0 /u0 2 ) .
… (10)
While obtaining it another reference concentration
Ci is introduced in the first boundary condition, so
that the variation in initial concentration may be studied by changing its value. Now following nondimensional variables are introduced:
C = c/C0 ; X = xu0 /D0 ; T = T *u0 2 /D0 ; μ = μ0 D0 /u0 2 ;
γ = γ 0 D0 /(C0 u0 2 )
… (11)
KUMAR et al.: DISTRIBUTION OF PULSE TYPE UNIFORM INPUT IN AQUIFERS IN TROPICAL REGIONS
Y = 0; C = 1; 0 < T ≤ T0
As a result, Eq. (8) may be written as
R( X )
+2
⎡ ∂ 2C ∂ C ⎤
∂C
= F(X )⎢
−
2
∂T
∂ X ⎥⎦
⎣∂ X
dF ( X ) ⎡ ∂C
⎤
− C ⎥ − μC + γ ,
⎢
dX ⎣ ∂X
⎦
= 0; T > T0 ,
Y = 1; ∂C/∂Y = 0 ; T ≥ 0
… (12)
γ ⎛ Ci γ ⎞
⎡1
⎤
+⎜
− ⎟ exp ⎢ (1 − δ ) X ⎥ ,
μ ⎝ C0 μ ⎠
⎣2
⎦
δ = δ 0 /u0 = 1 + 4μ .
F (Y ) = 0.8 +
… (13)
The boundary conditions (2a), (2b) and (3a) assume
the forms:
X = 0; C = 1; 0 < T ≤ T0
= 0; T > T0
X → ∞; ∂C/∂X = 0 .
… (14a)
…(14b)
… (15)
The two expressions of F(X) in Eq. (12) are chosen
as
0.5 exp(− X )
…(16a)
1.5 + exp(− X )
0.05exp(− X )
and F ( X ) = 0.8 +
… (16b)
1.25 − exp( − X )
Expression (16a) shows the variation of increasing
nature, along the longitudinal direction from the value
0.8 at X = 0 to the value 1.0 as X → ∞, while the second expression (16b) shows the reverse nature. Such
relationships explain the variation of permeability in
flow fields of large scales of many kilometers, for
example in sedimentary basins15.
To implement the finite difference scheme on
above problem, the semi-infinite domain X ∈(0,∞) is
converted into a finite domain Y ∈(0,1) by the following transformation:
F(X ) =1−
Y = 1 − exp(− X )
… (17)
As a result, Eq. (12)-(16) are changed as follows:
R(Y )
0.5 (1 − Y )
2.5 − Y
… (20a)
… (20b)
… (21)
… (22a)
and
and the initial condition (10) becomes
C ( X ,0) =
F (Y ) = 1 −
359
⎡
∂C
∂ 2C
∂C ⎤
= F (Y ) ⎢(1 − Y )2 2 − 2(1 − Y )
∂T
∂Y
∂Y ⎥⎦
⎣
+2(1 − Y )
dF (Y ) ⎡
∂C
⎤
− C ⎥ − μC + γ
(1 − Y )
dY ⎢⎣
∂Y
⎦
… (18)
C (Y ,0) =
γ ⎛ Ci γ ⎞
+⎜
− ⎟ (1 − Y ) −0.5(1−δ ) ,
μ ⎝ C0 μ ⎠
… (19)
0.05 (1 − Y )
0.25 +Y
… (22b)
where R(Y ) = 1 + K ⎡⎣1 − n0 F (Y ) ⎤⎦ / ⎡⎣n0 F (Y )⎤⎦ . Y = 1 corresponds to x → ∞ , however, it is not possible to compute concentration values at infinity. The values can
be evaluated up to some finite position only. This position may be assumed as much far away from the
input at which the assumption that the concentration
remains steady even after large time is fulfilled so that
this may be equivalent to the similar condition at infinity. Let this position be x = x0 , then condition (21)
may be used in the form:
Y = Y0 ; ∂C/∂Y = 0 for T ≥ 0
… (23)
Finite aquifer
In case of finite aquifer, the following set of nondimensional variables are introduced:
C = c/C0 ; X = x/L;U = u0 L/D0 ;
T = T * D0 /L2 ; μ = μ0 L2 /D0 ; γ = γ 0 L2 /(C0 D0 )
… (24)
The partial differential equation (Eq. 8) along with
boundary conditions (2) and (3b) may be written as
⎡ ∂ 2C
∂C
∂C ⎤
= F(X ) ⎢ 2 −U
⎥
∂T
∂X ⎦
⎣ ∂X
dF ( X ) ⎡ ∂C
⎤
+2
− UC ⎥ − μ C + γ
dX ⎢⎣ ∂X
⎦
R( X )
X = 0; C = 1; 0 > T ≤ T0
= 0; T > T0
X = 1; ∂C / ∂X ; T ≥ 0,
… (25)
… (26a)
… (26b)
… (27)
where R( X ) = 1 − K [{1 − n0 F ( X )}/n0 F ( X )] . The initial condition will be solution of above equations under the condition of steady concentration distribution
and groundwater flow through homogeneous finite
aquifer [ F ( X ) = 1] , i.e., the solution of the following
ordinary boundary value problem:
INDIAN J. ENG. MATER. SCI., AUGUST 2005
360
d 2C
dC
−U
− μ C+γ =0 ,
2
dX
dX
X = 0; C = 1 and X = 1; dC/dX = 0 , i.e.,
… (28)
γ ⎛C γ ⎞
C ( X ,0) = + ⎜ i − ⎟ ,
μ ⎝ C0 μ ⎠
1
δ −U
1
exp (U − δ ) X +
exp {(U + δ ) X − 2δ }
δ +U
2
2
δ −U
1+
exp( −δ )
δ +U
… (29)
where δ = U (1 + 4 μ/U ) . Two expressions for
F ( X ) in Eq. (25) be chosen as follows:
0.5 X
… (30a)
F(X ) =1−
1.5 + X
0.05 X
and F ( X ) = 0.8 +
… (30b)
1.25 − X
The first expression shows decreasing tendency
while the second one has reverse trend.
Numerical Solutions and Discussion
The above two unsteady dispersion problems in
semi-finite and finite inhomogeneous aquifers are
solved numerically, using finite difference two level
explicit schemes. In each case two numerical solutions, one for increasing nature and other for decreasing nature of inhomogeneity of aquifer are compared
with those for homogeneous aquifers. The numerical
solutions for homogeneous aquifers are obtained for
F ( X ) = 1 in Eqs (12) and (25), and these are compared with the respective analytical solutions, given in
the Appendix. It has been found that both analytical
and numerical solutions for a homogeneous aquifer
are in good agreement, tallying up to fourth decimal
places at each position and time. This ensures the
convergence of the numerical scheme chosen. To
achieve the stability of the solutions the sizes of intervals along non-dimensional space and time variable
axes X and T are chosen as ΔX = 0.05 and ΔT =
2
0.0001, respectively. The ratio, ΔT /(Δ X ) is 0.04 and
thus the stability condition (the ratio should be less
than 0.5) is very much satisfied. The larger values of
ΔT and ΔX satisfying the stability condition may be
chosen which will reduce the number of intervals
along the two axes and may not achieve the same accuracy. The expressions for T may be obtained by
using transformation (7) and the expression of
V (t ) = u/u0 from Eq. (1) as follows:
2
T=
u0 2 ⎡
1
⎤
t−
(1 − cos mt ) ⎥
⎢
D0 ⎣
m
⎦
and T =
D0 ⎡
1
⎤
t − (1 − cos mt) ⎥ ,
2
⎢
m
L ⎣
⎦
… (31)
… (32)
in cases of semi-finite and finite aquifers, respectively. For m = 0.0165(days)-1, the values of T are
given in Table 1, for different values of mt. The other
parameters have been assigned values as: C0 = 1.0, Ci
= 0.2, K = 0.1, n0 = 0.35, u0 = 0.01 hm/day, D0 = 1.0
hm2/day, μ0 = 0.0002, γ0 = 0.0001 and L = 100 hm. In
case of semi-infinite aquifer, concentration values are
computed up to x0 = 100 hm. Thus in both cases of
aquifer T and X will have same values for above set of
data.
Different curves in Figs 1a and 1b show the solute
concentration values along the flow through semiinfinite aquifers of uniform porosity (full line curves),
of decreasing porosity (dotted line curves) and of increasing porosity (semi-dotted curves) at days less
than t0 = 1333 days, the time when the source of the
pollution is eliminated for ever, and those at days
greater than it, respectively. In the former figure the
non-dimensional input concentration is 1.0 while that
in the latter one is zero. The respective attenuation
patterns in Fig. 1a are in decreasing order in the above
mentioned three semi-infinite aquifers and are more
evident at larger days. Fig. 1b shows the peak concentration lowers down and drifts away from the input
Fig. 1—Concentration distributions in semi-infinite aquifers of
uniform porosity (____), of decreasing porosity (……..) and of
increasing porosity ( -.-.-.-.- ).
KUMAR et al.: DISTRIBUTION OF PULSE TYPE UNIFORM INPUT IN AQUIFERS IN TROPICAL REGIONS
361
drogeology (Prentice Hall), 1998.
14 Crank J, The mathematics of diffusion, (Oxford Univ Press
UK), 1975.
15 Fowler A C, Mathematical models in the applied sciences,
section 13.6, (Cambridge, (UK) 1997.
Appendix
Analytical solutions may be obtained in case of homogeneous
aquifers, for which the expressions in Eq. (5) will reduce to the
respective forms:
n = n0; D = D(t) and u = u(t)
… (A1)
Thus in terms of non-dimensional variables introduced in case
of semi-infinite aquifer, the partial differential equation (12) will
reduce to
R
∂C ∂ 2C ∂C
=
−
− μ C+ γ
∂T ∂X 2 ∂X
… (A2)
The initial and boundary conditions will be the same as given
by Eqs (13) - (15). These equations are written in new dependent
variable using the following transformation:
K ( X , T ) = C ( X , T )exp ⎣⎡0.5 X − ( μ + 0.25)T /R ⎦⎤
… (A3)
Fig. 2—Concentration distributions in semi-infinite aquifer
(_______) and finite aquifer ( ………).
Further Laplace transformation technique may be applied to
get the final solution as follows:
source as the time passes away. The curves for finite
aquifers of uniform, decreasing and increasing porosities are not drawn because of the similar patterns.
Instead the concentration values in both types of homogeneous aquifers are drawn at the same times before and after t0 in Figs 2a and 2b, respectively. These
two figures show the attenuation in concentration being faster in semi-infinite aquifer than finite aquifer.
Also the concentration values at all days converge to
initial concentration in semi-infinite aquifer while
these increase with time in finite aquifer as the extreme boundary is approached.
C ( X ,T ) =
References
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241.
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13 Dominico P A & Schwartz F W, Physical and chemical hy-
1⎛ γ ⎞
1 ⎛ Ci γ ⎞
− ⎟ C2 ( X , T )
⎜1 − ⎟ C1 ( X , T ) − ⎜⎜
μ⎠
2⎝
2 ⎝ C0 μ ⎟⎠
+C3 ( X , T ); 0 < T ≤ T0
= C ( X , T ) − 0.5C1 ( X , T − T0 ) ; T > T0,
… (A4)
where
C1 ( X , T ) = exp− X ( Rβ1 − 0.5)erfc E1 + exp X ( Rβ1 + 0.5)erfc E2 ,
C2 ( X , T ) = exp( β 2 − β1 )T
{exp− X (
R β 2 − 0.5) erfc E3 + exp X ( R β 2 + 0.5)erfc E4
}
γ ⎛ Ci γ ⎞
+⎜
− ⎟ exp ( β 2 − β1 )T − X (−0.5 + β 2 R ;
μ ⎜⎝ C0 μ ⎟⎠
β1 = ( μ + 0.25) / R; β 2 = δ 2 / 4 R; δ = 1 + 4μ ;
{
C3 ( X , T ) =
(
= 0.5 ( X
}
)
βT)
E1 , E2 = 0.5 X R / T m β1T ;
E3 , E4
R /T m
2
Similarly in case of finite homogeneous aquifer, Eq. (25) will
reduce to
R
∂C ∂ 2C
∂C
=
−U
− μC + γ
∂T ∂X 2
∂X
… (A5)
Now above equation along with boundary conditions (26)-(27)
and initial condition (28) are transformed in K ( X , T ) by the
following transformation
K ( X , T ) = C ( X , T ) exp ⎡⎣0.5UX − ( μ + 0.25U 2 )T /R ⎤⎦
… (A6)
The Laplace transformation method is applied on the resulting
equations to get the solution as follows:
INDIAN J. ENG. MATER. SCI., AUGUST 2005
362
⎛C
⎛ γ ⎞
γ ⎞
C ( X , T ) = ⎜1 − ⎟ C1 ( X , T ) − ⎜⎜ i − ⎟⎟ C2 ( X , T ) + C3 ( X , T );
μ⎠
⎝
⎝ C0 μ ⎠
0 < T ≤ T0
= C ( X , T ) − C1 ( X , T − T0 ) ; T > T0,
C1 ( X , T ) = F1 ( X , T ) + F2 (2 − X , T ) − F2 (2 + X , T ) ,
C2 ( X , T ) = G1 ( X , T ) + G2 (2 − X , T ) − G2 (2 + X , T ) ,
C3 ( X , T ) =
γ ⎡ Ci γ ⎤ ⎡ δ − U
⎤
+⎢
− ⎥ 1+
exp − δ ⎥
μ ⎣ C0 μ ⎦ ⎢⎣ δ + U
⎦
−1
⎧⎪ δ 2T ⎫⎪
exp ( 0.5UX − θ1T ) exp ⎨
⎬×
⎩⎪ 4 R ⎭⎪
δ −U
⎡
⎤
⎢exp − 0.5δ X + δ + U exp δ (0.5 X − 1) ⎥ ,
⎣
⎦
F1 ( X , T ) = 0.5
⎛
b ⎞
⎟ exp X (0.5U − θ1 )erfc E1 +
F2 ( X , T ) = ⎜ 0.5 −
⎜
b + θ1 ⎟⎠
⎝
⎛
⎞
b
⎜ 0.5 +
⎟ exp X (0.5U + θ1 )erfc E2
⎜
⎟
b
θ
−
+
1 ⎠
⎝
b
−
exp(0.5UX − Tθ1)exp(Tb2 + Xb)erfc E3 ,
θ1 − b2
G2 ( X , T ) = exp( β 2 − θ1 )T
,
⎡⎛
⎤
⎞
b
⎢⎜ 0.5 −
⎟ exp X (0.5U − β 2 )erfc E4 + ⎥
b + β 2 ⎟⎠
⎢⎜⎝
⎥
⎢
⎥
⎢⎛
⎥
⎞
b
⎟ exp X (0.5U + β 2 )erfc E5 ⎥
⎢⎜ 0.5 +
⎜
⎟
−b + β 2 ⎠
⎢⎣⎝
⎥⎦
b
−
exp(0.5UX − Tθ1 )exp(Tb 2 + Xb)erfc E3 ,
β 2 − b2
(
= 0.5 ( X
)
β T);E
⎡ exp X (0.5U − θ1 ) erfc E1 + exp X (0.5U + θ1 )erfc E2 ⎤ ,
⎣
⎦
E1 , E2 = 0.5 X R / T m θ1T ;
G1 ( X , T ) = 0.5exp( β 2 − θ1 )T
E4 , E5
⎡ exp X (0.5U − β 2 )erfc E4 + exp X (0.5U + β 2 )erfc E5 ⎤
⎣
⎦
2
b = 0.5U / R ; θ1 = ( μ + 0.25U ) /R ; β 2 = δ 2 / 4R .
R /T m
2
3
= 0.5( X R / T + bT ) ;