WOLFRAM|DEMONSTRATIONS PROJECT

Numerical Inversion of the Laplace Transform: The Fourier Series Approximation

​
T
50
ο
0.2
This Demonstration shows how you can numerically compute the inverse of the Laplace transform
F(p)
of a simple function
f(t)
:
ℒ(f(t))(p)=F(p)=
1
2
p
+1
and
f(t)=sin(t)
. The selected method is the Fourier series approximation. This method uses the following formula in order to perform the inversion of
F(p)
:
f(t)=
σt
e
T

1
2
F(σ)+
∞
Σ
k=1
ReFσ+
kπi
T
cos
kπt
T
-ImFσ+
kπi
T
sin
kπt
T

.
You can select the appropriate values of
σ
and
T
that give the correct inverse. This choice must be such that
2T>
t
max
and
σ=
σ
0
-
ln(E)
2T
, where
E
is a measure of the maximum relative error and
σ
0
is the exponential order of
f(t)
.
The red curve is the sine function and the blue dots are the selected numerical values of the inverse of
F(p)
.
You can clearly see how this method may fail to give an accurate inverse if the values of
σ
and
T
are not correctly selected. The first snapshot presents a correct inversion result. The next two snapshots show situations where the method gives erroneous data.