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:
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