Biomedical Engineering Reference
In-Depth Information
The Laplace Transform is a function transformation commonly used in the so-
lution of complicated differential equations. With the Laplace transform it is fre-
quently possible to avoid working with equations of different differential order di-
rectly by translating the problem into a domain where the solution presents itself
algebraically. The formal definition of the Laplace transform is given as
∞
L
f(t)
=
e
−
st
f(t)dt
=
f(s)
(A.5)
0
The Laplace Transform of the function
f(t)
is said to exist if the above definition is
a convergent integral. The requirement for this is that
f(t)
does not grow at a rate
higher than the rate at which the exponential term
e
−
st
decreases.
Another commonly used function is the Laplace convolution:
t
f(t)
∗
g(t)
=
f(t
−
τ)g(τ)dτ
=
g(t)
∗
f(t)
(A.6)
0
The convolution of two function in the domain of
t
is sometimes complicated to
resolve; however, in the Laplace domain (
s
), the convolution results in the simple
function multiplication:
L
f(t)
g(t)
=
f(s)
∗
g(s)
˜
(A.7)
One final important property of the Laplace transform that should be addressed
is the Laplace transform of a derivative of integer order
n
of the function
f(t)
,given
by
n
−
1
n
−
1
L
f
n
(t)
=
s
n
f(s)
−
s
n
−
k
−
1
f
(
0
)
=
s
n
f(s)
−
s
k
f
(n
−
k
−
1
)
(
0
)
(A.8)
k
=
0
k
=
0
The Mittag-Leffer function is an important function that finds widespread use in
the world of fractional calculus. Just as the exponential naturally arises out of the
solution to integer order differential equations, the Mittag-Leffer function plays an
analogous role in the solution of non-integer order differential equations. In fact,
the exponential function itself is a very specific form, one of an infinite set, of this
seemingly ubiquitous function. The standard definition of the Mittag-Leffer is
∞
z
k
(αk
=
E
α
(z)
1
)
,α>
0
(A.9)
+
k
=
0
The exponential function corresponds to
α
= 1. It is also common to represent the
Mittag-Leffer function in two arguments,
α
and
β
, hence:
∞
z
k
(αk
E
αβ
(z)
=
β)
,α>
0
,β>
0
(A.10)
+
k
=
0
