Environmental Engineering Reference
In-Depth Information
Chapter 4
Transport Solutions
4.1
1D Transient Solution for the Infinite Domain
An analytical solution for the transport (3.18) with constant coefficients and
q ¼
0
was given by Ogata and Banks (
1961
). It reads:
þ
D
x
erfc
c
in
2
x
vt
2
v
x
þ
vt
2
p
p
cðx; tÞ¼
erfc
exp
(4.1)
Dt
Dt
'erfc' denotes the complementary error-function, which is defined as follows:
x
2
dB
2
p
p
erfc
ðxÞ :¼
1
exp
B
0
'erfc' is related to the error-function 'erf':
x
2
dB
2
erfc
ðxÞ :¼
1
erf
ðxÞ
mit erf
ðxÞ :¼
p
exp
B
0
Both functions can directly by called by a MATLAB
®
command, equal to the
abbreviation used above. The sequence of commands
illustrates the graphs of the two functions (Fig.
4.1
).
In MATLAB
other functions relevant in connection with the error-function
are implemented. Details can be obtained by using the keyword 'error function' in
the MATLAB
®
help system.
®
