Page 6 - Laplace transform numerical inversion
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Bruno Josso & Leif Larsen: Laplace transform numerical inversion - June 2012 -
p 4
2 The Laplace transform
2.1 Direct transform
Let
f
(
t
)
be a function with a real argument
t
∈
R
. The bilateral Laplace transform of
f
(
t
)
is
L
[
f
(
t
)] = ˆ
F
(
p
)
, with
p
∈
C
being the Laplace complex argument. The Laplace transform is defined
as follows:
ˆ
F
(
p
) =
Z
+
∞
−∞
f
(
t
)
e
−
pt
dt
(1)
In practice, one more commonly uses the unilateral Laplace transform expressed as follow
ˆ
F
(
p
) =
Z
+
∞
0
f
(
t
)
e
−
pt
dt
(3)
The Laplace argument
p
can explicitly be written:
p
=
σ
+
i
2
πν
with
(
σ, ν
)
∈
R
2
. Now, if
FT
[
.
]
is the
Fourier transform operator, one can then write that the Fourier transform of the function
f
(
t
)
e
−
σt
is equal to the Laplace transform
L
[
f
(
t
)]
:
L
[
f
(
t
)] = ˆ
F
(
σ
+
i
2
πν
) =
Z
+
∞
−∞
f
(
t
)
e
−
σt
e
−
2
iπνt
dt
=
FT
f
(
t
)
e
−
σt
(4)
Equation
, is the key point when considering the implementation of the Laplace transform
and its inverse. Indeed, very fast algorithms exist that very efficiently perform Discrete Fourier
Transform (
D
FT
[
.
]
).
2.2 Inverse transform
The inverse Laplace transform can easily be expressed by referring to the Fourier transform as
seen in
. Hence, if
FT
−
1
[
.
]
is the inverse Fourier transform, one can successively write:
FT
f
(
t
)
e
−
σt
= ˆ
F
(
σ
+ 2
iπν
)
(5)
f
(
t
) =
e
σt
FT
−
1
h
ˆ
F
(
σ
+ 2
iπν
)
i
(6)
=
e
σt
Z
+
∞
−∞
ˆ
F
(
σ
+ 2
iπν
)
e
2
iπνt
dν
(7)
One can therefore deduce from expression
that the inverse Laplace transform written
L
−
1
[
.
]
can be expressed as follows:
L
−
1
h
ˆ
F
(
p
)
i
=
f
(
t
) =
Z
+
∞
−∞
ˆ
F
(
σ
+ 2
iπν
)
e
(
σ
+2
iπν
)
t
dν
(8)
=
1
2
iπ
Z
σ
+
i
∞
σ
−
i
∞
ˆ
F
(
p
)
e
pt
dp
(9)
From expressions
and
, it can be said that the inverse Laplace transform can be calculated
by integrating in the complex plane along a vertical line whose abscissa equals
σ >
0
. One
assumes that there are no singularities on this line.
3 Laplace numerical inversion algorithms testing
3.1 Algorithms
Numerous algorithms can be found in the literature to perform Laplace transform numerical in-
version. According to reference
, a main list of
14
inversion method families can be retained.
1
If the function
f
(
t
)
to be processed by unilateral Laplace transform is not equal to
0
for
t <
0
one therefore calculates
the Laplace transform of the product
f
(
t
)
u
(
t
)
, with
u
(
t
)
being the Heaviside step function:
∀
t
∈
R
, u
(
t
) =
0 if
t <
0
1 if
t
≥
0
.
(2)
