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TitleThe numerical inversion of the Laplace transform
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Eindhoven University of Technology

MASTER

The numerical inversion of the Laplace transform

Egonmwan, A.O.

Award date:
2012

Link to publication

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Page 2

UNIVERSITÄT LINZ
JOHANNES KEPLER JKU

Technisch-Naturwissenschaftliche
Fakultät

The Numerical Inversion of the Laplace
Transform

MASTERARBEIT

zur Erlangung des akademischen Grades

Diplomingenieur

im Masterstudium

Industriemathematik

Eingereicht von:

Amos Otasowie Egonmwan

Angefertigt am:

Institut für Industriemathematik

Beurteilung:

Dr. Stefan Kindermann

Linz, June 2012

Page 52

See Boumenir and Al-Shuaibi [11].

Remark 5.24. We recall that the spectrum of compact operators have at
most countably many eigenvalues. Now, since the spectrum of the operator
L ≡

∫∞
0
· e−stdt not countable as it was shown in Al-Shuaibi [11], therefore

the Laplace transform operator is not compact. However, by the results of
Chapter 2, the Laplace transform is compact of it is restricted to a �nite
interval L : L2(0, T )→ L2(0, T ).

5.4 The inverse Laplace transform

Given a function f(t) with Laplace transform F (s), the inverse Laplace trans-
form corresponding to the Definition 5.4 is denoted as L−1{F (s)}. Thus, the
equivalent relationship existing between the direct Laplace transform and its
inverse is given as

F (s) = L{f(t)}, f(t) = L−1{F (s)}.

By definition, we note that the determination of the Laplace transform F (s)
for a given function f(t) is unique. In a similar way, using the complex in-
version integral, it can be shown also that for a given F (s), there exists a
unique f(t). This implies that there exists a one-to-one equivalence between
the Laplace transform and its inverse, justifying the notation L−1.

It should be noted that the study of the inverse Laplace transform is very
important because many solutions of practical problems usually provide a
known F (s) from which f(t) has to be reconstructed.

We now illustrate, by examples, the use of partial fractions in reconstructing
the original function f(t) from its Laplace transform F (s).

Example 5.25. Find the inverse Laplace transform of

F (s) =
s− 3

s2 + 5s+ 6
.

By partial fractions, we can write

F (s) =
s− 3

(s+ 2)(s+ 3)
=

A

s+ 2
+

B

s+ 3
. (5.4)

47

Page 53

To determine the constant A, we multiply (5.4) by (s + 2) and set s = −2.
This leads to

A = F (s)(s+ 2)
∣∣∣
s=−2

=
s− 3
s+ 3

∣∣∣
s=−2

= −5.

Similarly, to determine B, we multiply (5.4) by (s+ 3) and set s = −3

B = F (s)(s+ 3)
∣∣∣
s=−3

=
s− 3
s+ 2

∣∣∣
s=−2

= 6.

Using tables of Laplace transforms, we obtain

f(t) = L−1{F (s)} = −5L−1
{ 1
s+ 2

}
+ 6L−1

{ 1
s+ 3

}
= −5e−2t + 6e−3t.

Example 5.26. Find the inverse Laplace transform of

F (s) =
s+ 1

[(s+ 2)2 + 1](s+ 3)
.

Again, by partial fractions, we can write

F (s) =
s+ 1

[(s+ 2)2 + 1](s+ 3)
=

A

s+ 3
+

Bs+ C

[(s+ 2)2 + 1]
. (5.5)

To obtain the value of the constant A, we multiply (5.5) by (s + 3) and set
s = −3. This leads to

A = F (s)(s+ 3)
∣∣∣
s=−3

=
−3 + 1

(−3 + 2)2 + 1
= −1.

To determine the values of B and C, we proceed as follows: merge the frac-
tions in (5.5) and using the value of A = −1, we obtain

F (s) =
−1[(s+ 2)2 + 1] + (s+ 3)(Bs+ C)

[(s+ 2)2 + 1](s+ 3)
=

−3 + 1
(−3 + 2)2 + 1

= −1.

Rearranging in terms of the powers of s, we obtain

−(s2 + 4s+ 5) +Bs2 + (C + 3B)s+ 2C = s+ 1,

48

Page 103

[60] A.N. TIKHONOV and V.Y. ARSENIN, Solution of ill-posed problems.
John Wiley, New York, (1977).

[61] J.M. VARAH, Pitfalls in the numerical solution of linear ill-posed prob-
lems. SIAM J. Sci. Stat. Comput. 4, 2, (1983).

[62] G. WAHBA, On the approximate solution of Fredholm integral equations
of the �rst kind. Mathematics Research Center, U.S. Army, University
of Wisconsin, Madison, (1969).

[63] Y. WANG, A.G. YAGOLA and C. YANG, Optimization and regular-
ization for computational inverse problems and applications. Springer,
Berlin, (2010).

[64] D.V. WIDDER, The Laplace transform. Princeton University Press,
New Jersey, (1946).

[65] W.Y. YANG, W. CAO, T. CHUNG, and J. MORRIS, Applied numerical
methods using MATLAB. John Wiley and Sons Inc., New Jersey, (2005).

[66] M.S. ZHDANOV, Geophysical inverse theory and regularization prob-
lems. Elsevier, Amsterdam, (2002).

98

Page 104

Sworn Declaration
I, Amos Otasowie Egonmwan, hereby declare under oath that the submit-

ted Master’s thesis has been written solely by me without any third-party as-
sistance. Information other than provided sources or aids have not been used
and those used have been fully documented. Sources for literal, paraphrased
and cited quotes have been accurately credited. The submitted document
here present is identical to the electronically submitted text document.

Linz, June 2012

————————————————
Amos Otasowie Egonmwan

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