Environmental Engineering Reference
In-Depth Information
(Maas and Olsthoorn
1997
) where
K
0
is the Bessel function, already
discussed in Chap. 12.
h
0
here is the element wise product of pumping rate
with the reciprocal of conductivity and thickness.
The MATLAB
®
command is simply:
More details are given by Maas (
1986
) and Maas and Olsthoorn (
1997
).
For an extension of the method for unsteady flow towards wells see Hemker
(
1985
) and Hemker and Maas (
1987
).
18.3 Eigenvalues and Phase Space
Eigenvalues of a matrix allow a much deeper insight in the behavior of a system of
differential equations. For every square matrix
C
, the eigenvalues
l
are values for
which the system of linear equations:
Cx
¼
l
x
(18.13)
has a non-zero solution vector
x
. The vector
x
is called eigenvector for the
eigenvalue
.
Eigenvalues and eigenvectors as basic characteristics of matrices are discussed
in every textbook on linear algebra or matrix algebra (for example: Robbin
1995
).
It is not the place here to recall properties of eigenvalues and eigenvectors; the
reader who is not yet familiar with these terms should refer to a textbook on linear
algebra. As MATLAB
l
's origin is numerical linear algebra, the determination of
eigenvalues is one of the most basic tasks for which this software can be used.
In MATLAB
®
®
eigenvalues are calculated using the
eig
command, for example:
has the eigenvalues
i.e. the matrix
12
34
l
1
¼
0.3723 and
l
2
¼
5.3723. The
eigenvalues of a diagonal matrix are the elements in the diagonal: