Biomedical Engineering Reference
In-Depth Information
The factorial expression in (
A.22
) can be generalized for negative reals using the
gamma function, thus
−
α
m
1
)
m
(α
m)
(α)m
+
=
(
−
(A.23)
!
Using this relation we can now rewrite (
A.20
)for
−
α
and thus are left with the
Grunwald-Letnikov fractional integral:
t
−
a
h
(α
+
m)
d
−
α
f(x)
h
α
=
lim
h
f(x
−
mh)
(A.24)
m
!
(α)
→
0
m
=
0
The we discuss the form of the Fractional Integral Equation:
0
1
(α)
u(τ )
τ)
1
−
α
dτ
=
f(t)
(A.25)
(t
−
t
or equivalently:
J
α
u(t)
=
f(t)
(A.26)
The solution of this kind is
D
α
f(t)
u(t)
=
(A.27)
In the Laplace domain, integral equations of the first kind assume the form
L
Φ
α
(t)
u(t)
=
u(s)
s
α
˜
J
α
u(t)
=
Φ
α
(t)
∗
u(t)
=⇒
∗
(A.28)
which can be rewritten as
f(s)
=
s
f(s)
s
1
−
α
u(s)
=
s
α
(A.29)
or as:
a
1
−
α
s
f(
0
)
+
1
f(
0
)
s
1
−
α
f(s)
f(s)
s
α
u(s)
˜
=
=⇒
−
(A.30)
Inverting the first form back into the time domain, we get
t
1
d
dt
f(τ)
u(t)
=
τ)
α
dτ
=
f(t)
(A.31)
(
1
−
α)
(t
−
0
which is equivalent to solution of the equation with the Left Hand Definition. The
second form can be similarly inverted to yield
t
f
(τ )
(t
−
τ)
α
dτ
=
f(t)
+
f(
0
)
t
−
α
(
1
1
u(t)
=
(A.32)
(
1
−
α)
−
α)
0
