Environmental Engineering Reference
In-Depth Information
The corresponding M-file
'parest2a.m'
is included in the accompanying
software.
The main body of the M-file is omitted. In the f-function module we obtain two
solutions by calling the
bvp4c
command for the solution of boundary value
problems. Both differential equations are of Poisson type
@
2
c
@x
2
¼ const
; therefore,
it is possible to use the same module (
mat4ode
) with different formal parameters. It
is an alternative to use two different functions. The second order differential
equation is re-written as two first order differential equations with
c
1
¼ c
:
@
c
1
@x
¼ c
2
@
c
2
@x
¼ const
(10.34)
In the
mat4ode
sub-module the vector y has two elements representing
c
1
and
c
2
.
The
mat4bc
module specifies the boundary conditions (
10.31
) and (
10.33
), which in
terms of
c
1
and
c
2
are given by:
c
1
ðx ¼
0
Þ¼const
c
2
ðx ¼
1
Þ¼
0
(10.35)
The two functions
guess
and
guess1
deliver initial guesses for both functions
fulfilling the boundary conditions. The graphical output resulting from that algo-
rithm is identical to Fig.
10.8
.
It has been demonstrated that the procedure, using solutions of differential
equations within the
fzero
-module, is applicable to parameter estimation in
situations in which an explicit formula for the solution is not available. We chose a
one-dimensional set-up as demonstration example and had to solve boundary value
problems for ordinary differential equations. However, the same concept can be
applied to more general set-ups, steady or unsteady, in one or more space dimensions.
In general, the solutions of partial differential equations are required within the zero-
search-algorithm.
References
Bauer P, Attinger S, Kinzelbach W (2001) Transport of a decay chain in homogeneous porous
media: Analytical solutions. J Contaminant Transport 49:217-239
Grutzmacher G (2006) Umweltbundesamt (German Federal Environment Agency), Personal
communication
Guerrero JS, Skaggs TH, vam Genuchten MTh (2009) Analytical solutions for multi-species
contaminant transport subject to sequential first-order decay reactions in finir media. Transport
Porous Media 80:373-387
Pekdeger A (2006) Freie Universit
at Berlin, Germany, Personal communication
Rom N(2005) Welche Werkzeuge stellt FEMLAB f
€
ur die Problemoptimierung bereit.
COMSOL
€
News
(1): 15 (in German)
Yuan D, Kernan W (2007) Explicit solutions for exit-only radioactive decay chains. J Appl Phy
101:094907-1-12
