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Applications and Applied
Mathematics
(AAM):
An International Journal
Vol. 2, No. 1 (June 2007) pp. 1 – 13
ISSN: 1932-9466
PARAMETER ESTIMATION IN NONLINEAR COUPLED
ADVECTION-DIFFUSION EQUATION
Robert R. Ferdinand
Department of Mathematics, East Central University
1100 East 14th Street, Ada, Oklahoma 74820-6899, USA
Email: rferdinand@ecok.edu
Received October 6, 2006; revised February 3, 2007; accepted February 21, 2007
Abstract
In this paper a coupled system of two nonlinear advection-diusion equations is presented.
Such systems of equations have been used in mathematical literature to describe the dynamics of contaminant present in groundwater owing through cracks in a porous rock matrix
and getting absorbed into it. An inverse method procedure that approximates innitedimensional model parameters is described and convergence results for the parameter approximants are proved. This is nally followed by a computational experiment to compare
theoretical and numerical results to verify accuracy of the mathematical analysis presented.
Keywords:
MSC:
advection, coupled, diusion, innite-dimensional, parameter estimation
65M06, 65M32, 65N06, 65N21
1. Introduction
A nonlinear coupled system of advection-diusion PDEs that describes the dynamics of
contaminant concentration in groundwater as it ows through cracks in a rocky, porous
medium is presented below.
8
>
<
u + [a (u)] = [b (u; u )]
>
:
v = [e (v; v )]
t
t
z
z
x
x
z
u + v (t; 0; z )
x
v;
(1)
with initial and boundary conditions
8
>
<
u (0; z ) = (z ) ; u (t; 0) = (t) ; u (t; zmax ) = 0
>
:
v (0; x; z ) = (x; z ) ; v (t; 0; z ) = u (t; z ) ; v (t; xmax ; z ) = 0:
The schematic diagram below illustrates the above phenomenon:
1
(2)
2
Robert R. Ferdinand
Figure 1: Schematic Diagram of Porous Fractured Rock
In the above model, the parameters are described as follows:
z 2 = [0; zmax]: Distance along fracture
x 2 = [0; xmax]: Perpendicular distance into rock matrix
t 2 = [0; Tmax]: Time
u (t; z ): Contaminant concentration in liquid owing through fracture in porous rock
matrix at (t; z )
v (t; x; z): Contaminant concentration in liquid diusing into rock matrix while owing
through fracture at (t; x; z )
a (u): Speed at which contaminant moves forward in liquid owing through fracture
e (v; v ): Rate at which contaminant diuses into rocky matrix
b (u; u ): Rate at which contaminant diuses in liquid owing through fracture
z
: Rate at which contaminant (radioactive) decays in time
: Rate at which contaminant in liquid moving in fracture gets absorbed into rocky
matrix
x
(z ): Initial contaminant concentration in fracture
(t): Contaminant concentration in fracture at x = 0
(x; z ): Initial contaminant concentration in rocky matrix
: Fraction of contaminant diusing into rocky matrix at x = 0
One may note that the generally nonlinear nature of the model parameters a, b and e indicates that the groundwater and rock matrix may not necessarily be homogeneous in their
chemical compositions. This was one of the major reasons for studying this model, since in
Sudicky and Frind (1982), these parameters are linear, indicating total homogeneity in the
composition of the rock and water mediums. For a more detailed and exhaustive description
AAM: Intern. J., Vol. 2, No. 1 (June 2007)
3
of the physical phenomenon described by this model and denitions of the constants , and , the reader may refer to Sudicky and Frind (1982).
A forward Euler nite dierence method is used by the author in Ferdinand (2007) to obtain
a numerical solution of model Eq.(1-2). This method will be described briey, a little later in
this manuscript. The goal of this paper, however, is to estimate innite-dimensional parameters in the model using an inverse method procedure. This innite-dimensional parameter
estimation problem can be formally stated as follows:
Problem:
Given observed model solution data
8
>
>
>
>
<
>
>
>
>
:
=
Z
=
zmax
0
i
Z
i
u (t ; z ) dz
i
zmax
Z
0
xmax
0
v (t ; x; z ) dx dz;
i
i = 0; ; X , nd an innite-dimensional parameter vector q = (a; b; e; ; ; ) so that the
least-squares cost functional J (q ) is minimized over the innite-dimensional parameter space
Q, where
J (q ) =
X
X
=0
i
Z
zmax
0
u (t ; z; q ) dz
i
2
i
+
Z
zmax
0
Z
xmax
0
v (t ; x; z; q ) dx dz
i
2
!
i
(3)
and (u (t; z; q ) ; v (t; x; z; q )) represents the parameter dependent solution of Eqs.(1-2).
To solve the above problem, one proceeds in this paper as follows. In the next section, an inverse method procedure is described which numerically approximates the innite-dimensional
parameter set q . Convergence results for these parameter approximants are proved while a
numerical example illustrating the accuracy of these theoretical results is presented in section
3. Some concluding remarks are reported in section 4 while section 5 contains the reference
information.
2. Inverse Method
To begin this section, the space D is dened as
D = C [0; 1) C ([0; 1) (
1; 1)) C ([0; 1) ( 1; 1)) C (
) C ( ) C ( ) ;
with Q being a compact subset of D. Further, the following conditions are also satised by
elements of set q 2 Q.
(B1) a Lipschitz in u
(B2) b Lipschitz in u; u
(B3) e Lipschitz in v; v
z
x
4
Robert R. Ferdinand
(B4) 2 C 1 (
)
(B5) 2 C 1 ( )
(B6) 2 C 1 ( )
Techniques similar to those used in Ackleh (1999) and Ferdinand (2004) are used to solve
this inverse problem. This procedure is well established and involves two steps which follow.
Step I: Numerically approximate parameter dependent model solution (u (t; z; q ) ; v (t; x; z; q ))
of Eqs.(1-2) using the following nite dierence method from Ferdinand (2007):
Let
T
x
z
t = max ;
x = max
and
z = max ;
Y
N
R
be uniform mesh sizes used for discretizing the t, x and z axes, respectively. This leads to
t = k t;
x = j x
k
z = l z;
and
j
l
for k = 0; ; Y , j = 0; ; N and l = 0; ; R being the axes mesh points. Use notation
such as v to represent v (t ; x ; z ) and so on to arrive
at the following
numerical scheme
which computes the parameter dependent solution u (q ) ; v (q ) :
k
k
j;l
j
l
k
l
k
j;l
8
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
<
h u +1 (q ) = (1 t) u (q ) + 1 a u (q ) a u +1 (q )
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
:
v +1 (q ) = (1
k
k
k
k
l
l
l
l
u (q ) u (q )
+b u (q ) ; +1
z
k
l
k
k
l
!
u (q ) u 1 (q )
b u 1 (q ) ;
z
k
l
k
k
l
l
l
+ 2 v1 (q ) u (q ) for l = 1; ; R 1
k
k
;l
l
"
v (q ) v (q )
t) v (q ) + 2 e v (q ) ; +1
x
k
j;l
k
k
k
j;l
j;l
j
v (q ) v 1 (q )
e v 1 (q ) ;
x
k
j
k
k
j;l
j
k
;l
!
(4)
j;l
!#
;l
;l
for j = 1; ; N
where
!#
1 and l = 0; ; R;
t
t
and
2 = ;
z
x
while initial and boundary conditions give
8
0
>
u0 (q ) = ; u (q ) = 0
< u (q ) = ;
1 =
l
>
:
l
k
k
k
R
v 0 (q ) = ; v0 (q ) = u (q ) ; v
j;l
j;l
k
;l
k
k
l
N;l
(q ) = 0:
(5)
AAM: Intern. J., Vol. 2, No. 1 (June 2007)
5
o
n
Use (B1)-(B6) (see Ferdinand (2007)) to get that the solution approximants u (q ) ; v (q )
obtained from Eqs.(4-5) above, converge to a unique parameter dependent solution, (u (q ) ; v (q ))
of Eqs.(1-2) as t; x; z ! 0. In fact, one can extend these approximants to a family of
functions as follows:
k
l
n
(U (t; z; q ) ; V (t; x; z; q )) =
t;
when
z
t;
x;
z
t 2 [t 1 ; t ) ;
k
for
j
t;
x;
Z
X
X
z
i
zmax
0
=0
Z
+
k
zmax
Z
0
xmax
0
z
j;l
z 2 [z 1 ; z ) ;
j
l
l
l = 1; ; R:
U (t ; z; q ) dz
t;
o
k
l
k = 1; ; Y;
j = 1; N;
This leads to the following computational form for J :
J (q ) =
n
u (q ) ; v (q ) ;
x 2 [x 1 ; x ) ;
k
o
i
2
i
V (t ; x; z; q ) dx dz
t;
x;
k
j;l
z
i
(6)
2
!
i
used in theorem 1 below.
u1 ! 0 and v
Using the convergence result u
0 in Ferdinand (2007), one arrives at
Theorem 1.
k
l
v 1 ! 0 as t; x; z !
k
j;l
J (q ) ! J (q ) uniformly 8q 2 Q:
t;
x;
(7)
z
Follows from convergence results for the solution approximants
sented in detail in Ferdinand (2007).
Proof.
n
o
n
o
u ; v
k
k
l
j;l
pre-
n
o
Approximate the innite-dimensional parameter space Q by a sequence Q
n
o
of nite-dimensional compact subsets of Q in the topology of D. An example of Q is
given in section 3. This further leads to the nite-dimensional computational form
Step II:
M
M
J q
t;
x;
z
M
=
Z
X
X
0
=0
i
+
q
M
2Q
M
zmax
Z
zmax
U t ; z; q
t;
Z
0
xmax
0
z
M
i
dz
V t ; x; z; q
t;
x;
z
i
M
2
i
dx dz
2
!
i
(8)
;
being a nite-dimensional parameter vector, and one arrives at Theorem 2.
Theorem 2.
For xed values of t; x and z
J q
t;
x;
z
M
! J t;
(q )
x;
z
6
as q
Robert R. Ferdinand
M
! q in Q.
Proof.
Let
n
u
k;M
o
l
n
; v
k;M
o
j;l
and
n
o
and
o
k
k
l
j;l
U t; z; q
t;
n
u ; v
M
z
represent the families of functions
; V t; x; z; q
t;
x;
M
z
(U (t; z; q ) ; V (t; x; z; q )) ;
t;
respectively. Then let
z
t;
x;
z
u ; v
= u
u ;v
v
for k = 0; ; Y , j = 0; ; N and l = 0; ; R. Boundary and initial conditions from Eq.(5)
give
8
>
0 = ;
u0 = ;
u = 0
< u
(9)
>
:
0
v = ;
v0 = u ;
v = 0:
;M
l
k;M
k;M
l
j;l
;M
k
k;M
k
l
l
j;l
j;l
k;M
M
k;M
l
l
j;l
k;M
R
k;M
M
j;l
j;l
k;M
k
;l
k;M
k;M
l
N;l
From the rst of the two coupled equations in Eq.(4) get the operator forms
u +1
= A1 u
(10)
u +1 = A1 u
and subtract Eq.(11) from Eq.(10) to get
u
+1 j1 tj u (11)
k
;M
l
and
k
;M
k;M
l
k
k
l
l
k;M
l
l
+1 a
u +1
M
k;M
M
l
M
M
+1 b
u
k;M
l
+1 b
;
u 1;
k;M
u +1
k;M
;l
u
l
u
z
k;M
l
u
u
k;M
k;M
!
l
a u +1 + a u
k
k
l
l
u
u
b u ; +1
z
k
k
k
l
l
l
u 1
k;M
z
l
k;M
l
k;M
l
+ 2 v1
a
l
!
b u 1;
k
l
u
k
l
!
u 1
z
k
l
(12)
!
v1 + u ;
k
;l
k
l
which gives
u +1
k
l
;M j1 tj u
k;M l
+ 1 I + 1 II + 1 III + 2 IV:
Bounds need to be established for I , II , III and IV in Eq.(13). First let !; ! ; !1
positive constants. Then start with I , add and subtract terms to get
(13)
!9 be
AAM: Intern. J., Vol. 2, No. 1 (June 2007)
7
a u +1
I
k;M
M
+a
k;M
k;M
l
u
M
a u +1 + a u +1
l
k;M
a u
l
a u +1
k
l
l
k;M
+a u
l
k;M
a u
k
l
l
(14)
:
(B2) yields
I a
M
u +1
k;M
a u +1
l
k;M
l
+ a
M
u
k;M
l
a u
k;M
l
+ !1 u +1 + !2 u
k;M
k;M l
l
:
(15)
Add and subtract terms in II again to get
b
II
u
M
k;M
l
+b u
;
u +1
;
u +1
k;M
l
k;M
l
k;M
z
b u
k;M
l
l
!
u
k;M
l
!
u
k;M
k;M
;
l
k
!
u
k;M
z
k
k
l
l
l
!
u
u
b u ; +1
z
l
z
u +1
l
:
Similarly (B3) gives
M
M
II b
u
k;M
l
;
u +1
k;M
l
z
u
k;M
!
b u
k;M
l
l
;
u +1
k;M
u
k;M
l
z
l
!
+ !3 u
!
+ !4 u 1 + u
l
+ u +1 k;M k;M
(16)
l
and
III b
u 1;
k;M
u
k;M
l
k;M
z
l
u 1
!
b u 1;
k;M
l
u
k;M
l
k;M
z
l
u 1
l
k;M
l
k;M l
;
(17)
while
IV
v1
;l
;M
l
l
+b
M
u
k;M
l
+b
M
;
u +1
k;M
l
u
k;M
l
k;M l
M
k;M
z
+ a
u
u
k;M
b u
;
!
b u 1;
k;M
u +1
l
k;M
l
u
k;M ;l
:
!
u 1
!
k;M
z
l
k;M
l
(18)
u
k;M
l
k;M
:
l
a u
!
l
k;M l
+ u +1 + u 1 + v1
l
k;M
u 1
z
k;M
+ u
l
l
k;M
;l
l
l
+!5 u
k;M
l
u 1;
k;M
l
a u +1
k;M k
l
k;M
M
u = v1
k;M
k
;l
Now Eqs.(15-18) make Eq.(13) yield
u
+1 a u +1
k
v1 + u
k;M
k;M
z
l
(19)
8
Robert R. Ferdinand
Next, proceed with the second of the two equations in Eq.(4) to arrive at
v +1
= A2 v
j;l
(20)
v +1 = A2 v :
(21)
k
;M
j;l
and
Subtract Eq.(21) from Eq.(20) to get
v
+1 j1 tj v
k
k;M
k
k
j;l
j;l
k;M ;M
j;l
j;l
0
+2 e
v
@
M
k;M
j;l
;
v +1
k;M
j
0
+2 e
v
v 1;
@
M
x
k;M
j;l
k;M
j
;l
v
k;M
j;l
v
v
e v ; +1
x
k
A
v 1
x
j
k
;l
j;l
1
k;M
j
;l
1
;l
v
e v 1;
A
;l
(22)
!
;l
v 1
x
k
j;l
k
j
k
!
j;l
k
j
;
which gives
v +1
k
j1 tj v
;M j;l
k;M j;l
+ 2 V + 2 V I;
and bounds for V and V I are hence required. Start with V , use (B4) and follow techniques
similar to those used in obtaining bounds for II and III to get
V
0
e
M
v
@
k;M
j;l
;
v +1
k;M
j
;l
x
v
k;M
j;l
1
0
A
e @v
;
k;M
j;l
v +1
k;M
j
;l
x
v
1
A
k;M
j;l
+ !6 v
k;M j;l
+ v +1 1 k;M
j
;
(23)
and
0
VI e
v 1;
M
@
k;M
j
;l
v
k;M
j;l
1
v 1
x
k;M
j
;l
0
e @v 1 ;
k;M
A
j
v
k;M
j;l
1
;l
A
v 1
x
k;M
j
;l
+ !7 v 1 + v
k;M
j
;l
k;M j;l
: (24)
Thus Eq.(22) yields
v +1
k
j;l
;M 0
e
v
@
M
k;M
j;l
;
v +1
k;M
j
+e
0
M
v 1;
@
+!8 v
k;M
j
x
k;M
j;l
;l
k;M j;l
v
;l
v
k;M
j;l
1
0
A
e @v
v 1
x
k;M
j
;l
1
A
;l
+ v +1 + v 1
k;M
j
;l
k;M
j
k;M
j;l
;
v +1
k;M
j
0
e @v 1 ;
k;M
j
:
;l
;l
x
v
k;M
j;l
v
k;M
j;l
1
A
v 1
x
k;M
j
;l
1
A
(25)
AAM: Intern. J., Vol. 2, No. 1 (June 2007)
Now pass the limit q
9
! q in Q.
M
First, boundary and initial conditions in Eq.(9) yield
u0
;M l
; u0
; v0
; u
k;M k;M R
;M j;l
; v
k;M N;l
! 0:
(26)
Next, Eq.(19) gives
u +1
k
l
!5 u
;M k;M l
+ u +1 + u 1 + v1
k;M
k;M
l
k;M l
;l
:
(27)
Further, Eq.(25) leads to
v +1
k
;M j;l
!8 v
j;l
;l
+ v +1 + v 1
k;M k;M
j
;l
k;M
j
;
(28)
and nally
v0 +1
k
;l
Thus Eqs.(26-29) give
u
+1 + v +1
k
;M
k
l
!5 u
;M ;M j;l
l
+ u +1 + u 1 + v1
k;M k;M
k;M
l
l
!9 u
l
k;M
k;M
l
:
l
(29)
;l
k;M
j
k;M ;l
;l
+ v +1 + v 1
k;M j;l
;l
+ u +1 + u 1 + v1
k;M + v
k;M k;M
j
(30)
:
Now dene
k;M
= =0 max
; =0 j
;
;N
l
;
u
;R
k;M l
+ v
k;M j;l
:
This results in Eq.(30) giving
+1
k
Eq.(26) easily yields 0
;M
u
k;M
l
; v
k;M
j;l
M
Corollary 3.
x;
z
J q
!9 )
k;M
M
! 0 :
! 0. Hence, one gets that
k;M
! 0 which leads to thereby leading to J q
and leads to corollary 3 below.
t;
;M
k;M
! (0; 0) ) u ; v
! J t;
! J k;M
k;M
l
j;l
Follows from proofs of Theorems 1 and 2 above.
t;
x;
k
k
l
j;l
(q ) ! J (q ) when t; x; z ! 0 and q
Proof.
z
! u ;v ;
z
! 1.
x;
(q ) which concludes the proof of this theorem
x;
Q as M
t;
;M
z
M
! q in
10
Robert R. Ferdinand
Corollary 3 above shows J ton be oa continuous functional over each of the compact subsets Q of Q in the sequence Q and leads to J having a minimizer q 2 Q over each
Q . Using the abstractnleast-squares
theory presented in Banks and Kunisch (1989), this
o
sequence of minimizers q has a subsequence converging to a minimizer q 2 Q of J (q )
over set Q in the topology of D. Hence one arrives at the existence of a solution to the
inverse problem stated earlier.
M
M
M
M
M
M
In the next and nal section, a numerical example is presented to illustrate accuracy of
the theoretical results proved in this section.
3. Numerical Experiment
To begin with, observed data is generated computationally. In order to accomplish this,
parameters of the model equation Eqs.(1-2) are given the following known values:
xmax = zmax = 1:0; Tmax = 2:5 10 2
x = z = 1:0 10 1 , t = 10 4
= = = 1:0
a (u) = u2 , b (u; u ) = (uu )2 , e (v; v ) = (vv )2
(z ) = (z 1)2 , (t) = e , (x; z ) = (x 1)2 (z
z
z
x
x
t
1)2
It would be worthwhile to mention that nonlinear a; b and e in the list above represent nonhomogeneity in the composition of liquid and contaminant. It should also be noted that
these parameter values are chosen for the sole purpose of validating the theoretical results
proved earlier and may not necessarily represent a real-life phonemenon in particular.
The parameter to be estimated numerically is the function a. The other unknown parameters b; e; ; ; can be approximated in a similar fashion. To accomplish this, the model
equation Eqs.(1-2) is solved numerically to obtain the following data:
Z 1
8
>
>
=
u (t ; z ) dz
>
>
<
0
>
Z 1Z 1
>
>
>
: =
v (t ; x; z ) dz dx;
0 0
where
t = 0:0005i;
i = 0; ; 50:
Now choose the innite-dimensional parameter space Q for this experiment as the D-closure
of the set
i
i
i
i
i
fja (u)j L; ja (u1) a (u2)j L ju1 u2j ; 8u1; u2 2 [0; 1) ; a = L for u umaxg ;
(31)
AAM: Intern. J., Vol. 2, No. 1 (June 2007)
11
L and umax being xed constants. Hence, Q follows as a compact subset of D from the
Arzela-Ascoli theorem.
n
o
To implement Step II, Q is approximated numerically by a sequence Q
of nitedimensional
compact
subsets in the topology of D. Each Q uses the set of linear splines
o
n
(u; umax) =0 on a uniform partition of [0; umax] as its approximating elements. Thus,
the unknown parameter a (u) is approximated over each Q in the form of the interpolant
M
M
M
M
j
j
M
umax I (a) = a j
M
=0
M
X
M
M
j
u 2 [0; 1)
(u; umax) ;
(32)
j
with a extended to a continuous function on [0; 1) by letting
when u umax, for j = 0; ; M .
M
j
(u; umax) =
M
j
(umax; umax)
The Peano kernel theorem in Schultz (1973) leads to lim !1 I (a) = a. Hence, if a
is represented as
M
M
a (u) =
M
X
M
2Q
M
(u; umax) ;
=0
then the numerical parameter estimation problem involves the identication
of (M + 2) un
known constants : j = 0; ; M and umax so that J a is minimized. This is
performed computationally using Eq.(4) and the FORTRAN subroutine LMDIF1, obtained
from NETLIB, that implements the Levenberg-Marquardt algorithm with all integrals computed numerically using Simpson's rule of integration. Next,
M
M
M
j
j
j
j
t;
x;
z
M
= 0:5 : j = 0; ; M
and
umax = 1
are taken as initial guesses and the following results are obtained.
(i) Figure 2 below shows a comparison between exact and estimated function a (u) =
u2 when M = 9. Exact function here is given by the straight line graph while the
estimated function is given by the dots. umax is estimated here as 0.98 and the value
of J (a9 ) at the end of the experiment is of the order of 10 5 . This estimation
took about 2 hours of computing time on a UNIX machine.
M
j
t;
x;
z
aHuL
1
0.8
0.6
0.4
0.2
0.2
0.4
0.6
0.8
1
u
Figure 2: Exact Function - Solid Line; Estimated Function - Dots
12
Robert R. Ferdinand
(ii) Figure 3 below shows the same for M = 11. Estimation of umax was once again 0.98
and J (a11 ) at the end of the experiment came out to be of the order of 10 6 .
Computing time was about 2 hours on a UNIX machine.
t;
x;
z
aHuL
1.4
1.2
1
0.8
0.6
0.4
0.2
0.2
0.4
0.6
0.8
1
u
Figure 3: Exact Function - Solid Line; Estimated Function - Dots
4. Closing Comments
It would be worthwhile to mention that the computational estimation was performed in
the FORTRAN programming language executed on a UNIX machine at East Central University in Ada, OK, USA. This facility is an SCO UNIX 5.0.5 machine, consisting of two 550
MHz Xeon processors in parallel. Both the above graphs were plotted using MATHEMATICA which proved to be extremely eÆcient for this purpose.
Further, it may also be noted by looking at Figures 2 and 3, that the numerical estimation of a (u) = u2 is more accurate towards the interior of [0; umax]. This is owing to a much
higher concentration of observed data present in the interior as opposed to the boundaries of
this interval. Hence one concludes this section and also this paper by stating that numerical
results obtained herein illustrate the accuracy of the theoretical results proved in section 2.
Acknowledgement
The author wishes to express gratitude to several anonymous referees who provided the most
constructive comments and suggestions thereby enabling the author to revise and make this
manuscript most appealing to readers of the AAM journal.
References
A. S. Ackleh, Parameter Identication in Size-Structured Population Models with Nonlinear Individual Rates, Mathematics Computation and Modeling, 30, pp. 81-93, (1999).
H. T. Banks and K. Kunisch, Estimation Techniques for Distributed Parameter Systems,
Birkhauser, Boston, (1989).
R. R. Ferdinand, Numerical Solution of a Nonlinear Coupled Advection-Diusion Equation.
To Appear in Journal of Computational Analysis and Applications (2007).
AAM: Intern. J., Vol. 2, No. 1 (June 2007)
13
R. R. Ferdinand, Parameter Estimation in a Nonlinear Two-Dimensional Population Diusion Model, Numerical Functional Analysis and Optimization, 25, pp. 57-67, (2004).
E. A. Sudicky and E. O. Frind, Contaminant Transport in Fractured Porous Media; Analytic
Solutions for a System of Parallel Fractures, Water Resources Research, 18(6), pp. 16341642, (1982).
M. H. Schultz, Spline Analysis, Englewood Clis, NJ, (1973).