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.