##### Document Text Contents

Page 1

Eindhoven University of Technology

MASTER

The numerical inversion of the Laplace transform

Egonmwan, A.O.

Award date:

2012

Link to publication

Disclaimer

This document contains a student thesis (bachelor's or master's), as authored by a student at Eindhoven University of Technology. Student

theses are made available in the TU/e repository upon obtaining the required degree. The grade received is not published on the document

as presented in the repository. The required complexity or quality of research of student theses may vary by program, and the required

minimum study period may vary in duration.

General rights

Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners

and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.

• Users may download and print one copy of any publication from the public portal for the purpose of private study or research.

• You may not further distribute the material or use it for any profit-making activity or commercial gain

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

Eindhoven University of Technology

MASTER

The numerical inversion of the Laplace transform

Egonmwan, A.O.

Award date:

2012

Link to publication

Disclaimer

This document contains a student thesis (bachelor's or master's), as authored by a student at Eindhoven University of Technology. Student

theses are made available in the TU/e repository upon obtaining the required degree. The grade received is not published on the document

as presented in the repository. The required complexity or quality of research of student theses may vary by program, and the required

minimum study period may vary in duration.

General rights

Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners

and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.

• Users may download and print one copy of any publication from the public portal for the purpose of private study or research.

• You may not further distribute the material or use it for any profit-making activity or commercial gain

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