Biomedical Engineering Reference
In-Depth Information
at each eigenvalue. To reiterate, the eigenslope has a simple inverse rela-
tion to the Fourier coecients (C), diagrammatically depicted as
C
i
=
k
i
f
0
(
i
)
where k is independent of
i
. Note that the Fourier coecients can be
obtained numerically, analytically, or even graphically | by measurement
of the plotted slope with ruler and protractor.
The intractability of Fourier coecients in all but the most simple
boundary conditions is probably the reason why the simple relation be-
tween Fourier coecients and the eigenslope has not been heretofore rec-
ognized, or at least widely recognized. One of the authors (L.G.) reviewed
45 PDE textbooks, 28 research compendia on 2nd order PDEs, as well as
several hundred articles on PDEs in mathematics, physics, engineering, and
mathematical biology. No previous work could be found that mentioned the
relation between the eigenslope or the relation of the rst derivative of the
eigenvalue function to Fourier coecients. Accordingly, our work on this
topic is described here.
3. Derivation of the Eigenslope Method
Let
i
be a sequence of distinct positive eigenvalues for the Fourier series
X
1
A
i
i
(x) e
j
t
V (x; t) =
i=1
where the A
i
are the Fourier coecients and the
i
are trigonometric or
exponential eigenfunctions. As is typical in neuroscience applications, we
assume that
P
P
j
k
j
2
<1: Although separation of
variables with modied orthogonality conditions can be used when the
boundary conditions are suciently simple, more realistic models usually
require solution by Laplace transforms and the method of residues
6;7;11
.
The Laplace transform of V (x; t) is
j
k
j
1
=1and
X
1
A
i
i
(x)
p +
j
V (x; p) =
i=1
It follows that there exists g (p; x) and h (p) that are analytic in p such that
g (x; p)
h (p)
V (x; p) =
(1)
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