Geoscience Reference
In-Depth Information
follows
t
y
(
t
)
=
x
(
τ
)
u
1
(
t
−
τ
)
d
τ
t
−
m
t
t
1
2!
+
τ
1
)
x
(
τ
2
)
u
2
(
t
−
τ
1
,
−
τ
2
)
d
τ
1
d
τ
2
x
(
t
t
−
m
t
−
m
t
t
t
1
3!
+
x
(
τ
1
)
x
(
τ
2
)
x
(
τ
3
)
t
−
m
t
−
m
t
−
m
×
u
3
(
t
−
τ
1
,
t
−
τ
2
,
t
−
τ
3
)
d
τ
1
d
τ
2
d
τ
3
+···
(A30)
To facilitate numerical computations, one can replace
τ
by (
t
−
τ
), etc., in (A30) to
obtain, as in (A15) (after absorbing the factorials in the kernels),
m
m
m
y
(
t
)
=
u
1
(
τ
)
x
(
t
−
τ
)
d
τ
+
u
2
(
τ
1
,τ
2
)
x
(
t
−
τ
1
)
x
(
t
−
τ
2
)
d
τ
1
d
τ
2
0
0
0
m
m
m
+
u
3
(
τ
1
,τ
2
,τ
3
)
x
(
t
−
τ
1
)
x
(
t
−
τ
2
)
x
(
t
−
τ
3
)
d
τ
1
d
τ
2
d
τ
3
+···
(A31)
0
0
0
REFERENCES
Barrett, J. F. (1963). The use of functionals in the analysis of non-linear physical systems.
J. Electron.
Contr.
,
15
, 567-615.
Greenberg, M. D. (1971).
Applications of Green's Functions in Science and Engineering
. Englewood
Cliffs, NJ: Prentice Hall.
Nash, J. E. (1959). Systematic determination of unit hydrograph parameters.
J. Geophys. Res.
,
64
,
111-115.
Volterra, V. (1913).
Le¸ons sur les equations integrales et les equations integro-differentielles
. Paris:
Gauthier-Villars.
(1959).
Theory of Functionals and Integral and Integro-Differential Equations
. New York: Dover.
