Chemistry Reference
In-Depth Information
Appendix D
Continuously Distributed Time Lags
Continuous time dynamical systems with continuously distributed time lags are
frequently modeled with Volterra-type integro-differential equations, when some
or all state variables in the usual differential equation model are replaced by certain
averages of past values. If x.t/ is such a variable then its weighted average is
Z
t
x.t/
D
w
.t
s;T;m/x.s/ds;
(D.1)
0
where the weighting function is of the form
(
T
e
t
s
if m
D
0
T
w
.t
s;T;m/
D
(D.2)
mŠ
T
mC1
.t
s/
m
e
m.t
s/
1
if m
1:
T
Here m is a non-negative integer and T is a positive real parameter.
First we will examine some fundamental properties of this special weighting
function.
(a) The area under the weighting function converges to 1 as t
!1
.
For m
D
0 we have
"
1
T
#
t
0
D
1
e
Z
t
e
t
s
1
T
T
e
t
T
ds
D
t
T
;
1
T
0
and for m
1 by introducing the new integration variable x
D
m.t
s/=T we
have
Z
t
Z
mt
T
mC1
Tx
m
m
m
T
mC1
m
T
1
mŠ
1
mŠ
e
x
Tdx
m
.t
s/
m
e
m.t
s
T
ds
D
0
0
Z
m
T
1
mŠ
x
m
e
x
dx:
D
0
305
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