Kouli_etal_2008_Groundwater modelling_BOOK.pdf - Pantelis ...

272 Jürgen Geiser
The mass is the integrated:
m i,rest =
⎛
∏i−1
i∑ i∏
λ j ⎜ (
⎝
j=1
∫ 1
v j t
for
j=1
k=1
k≠j
1
λ k − λ j
)
(
a(x − v j t) + (b −
i = 2, . . .,M
i∑
(
k=1
k≠j
i∏ λ jl
)exp(−λ j t) ⎟
λ jl − λ jk ⎠
l=1
l≠k
l≠j
a
λ jk
)(1 − exp(−λ jk (x − v j t)))
⎞
)
dx
(50)
This form could be written as a compressed equation, using the following four expressions.
a.) The polynomes with the constant factors are reduced to:
i∑
(
k=1
k≠j
i∏
l=1
l≠k
l≠j
b.) The polynomes with the linear factor are reduced to:
i∑
(
k=1
k≠j
i∏
l=1
l≠k
l≠j
λ jl
λ jl − λ jk
)1 = 1 (51)
λ jl
λ jl − λ jk
) 1
λ jk
=
i∑
k=1
k≠j
c.) The polynomes with the quadratical factors are reduced to:
i∑
(
k=1
k≠j
i∏ λ jl 1
i∑
)
λ jl − λ jk (λ jk ) 2 = ( 1
l=1
l≠k
l≠j
k=1
k≠j
λ jk
1
λ jk
(52)
i∑
l≥k
l≠j
1
λ jl
) (53)
d.) The symmetry of the terms with the exp expressions are from the form of:
exp(−λ j t)exp(λ jk (1 − v j t)) = exp(−λ k t)exp(−λ kj (1 − v k t)) (54)
By using with the four reductions we could write the equation in an applicable form:
m i,rest (t) =
∏
i∑ i∏
(
i−1
λ j
j=1 j=1 k=1
k≠j
⎛
1
λ k − λ j
) (55)
⎜
exp(−λ j t) ⎝a (1 − v jt) 2
+ b(1 − v j t −
2
i∑
k=1
k≠j
1
λ jk
)
−a(1 − v j t)(
i∑
k=1
k≠j
( i∑
1
) + a
λ jk k=1
k≠j
1
λ jk
(
i∑
l≥k
l≠j
) ⎞ 1 ⎟
) ⎠
λ jl