Numerical Inversion of the Laplace Transform: The Fourier Series Approximation
Numerical Inversion of the Laplace Transform: The Fourier Series Approximation
This Demonstration shows how you can numerically compute the inverse of the Laplace transform of a simple function : and . 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)
ℒ(f(t))(p)=F(p)=+1
1
2
p
f(t)=sin(t)
F(p)
f(t)=F(σ)+ReFσ+cos-ImFσ+sin
σt
e
T
1
2
∞
Σ
k=1
kπi
T
kπt
T
kπi
T
kπt
T
You can select the appropriate values of and that give the correct inverse. This choice must be such that and , where is a measure of the maximum relative error and is the exponential order of .
σ
T
2T>
t
max
σ=-
σ
0
ln(E)
2T
E
σ
0
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 and are not correctly selected. The first snapshot presents a correct inversion result. The next two snapshots show situations where the method gives erroneous data.
σ
T