dasar transformasi laplace - masud.lecture.ub.ac.id
TRANSCRIPT
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Matematika Industri II
DASAR TRANSFORMASI
LAPLACE
Matematika Industri II
![Page 2: DASAR TRANSFORMASI LAPLACE - masud.lecture.ub.ac.id](https://reader030.vdocuments.site/reader030/viewer/2022012423/61782b0a9ee62248fd0825f6/html5/thumbnails/2.jpg)
Matematika Industri II
Bahasan
• Transformasi Laplace
• Transformasi Laplace Invers
• Tabel Transformasi Laplace
• Transformasi Lplace dari Suatu Turunan
• Dua Sifat Transformasi Laplace
• Membuat Transformasi Baru
• Transformasi Laplace dari Turunan yang Lebih Tinggi
![Page 3: DASAR TRANSFORMASI LAPLACE - masud.lecture.ub.ac.id](https://reader030.vdocuments.site/reader030/viewer/2022012423/61782b0a9ee62248fd0825f6/html5/thumbnails/3.jpg)
Matematika Industri II
Bahasan
• Transformasi Laplace
• Transformasi Laplace Invers
• Tabel Transformasi Laplace
• Transformasi Lplace dari Suatu Turunan
• Dua Sifat Transformasi Laplace
• Membuat Transformasi Baru
• Transformasi Laplace dari Turunan yang Lebih Tinggi
![Page 4: DASAR TRANSFORMASI LAPLACE - masud.lecture.ub.ac.id](https://reader030.vdocuments.site/reader030/viewer/2022012423/61782b0a9ee62248fd0825f6/html5/thumbnails/4.jpg)
Matematika Industri II
Transformasi Laplace
• A differential equation involving an unknown function f ( t ) and its derivatives is said to be initial-valued if the values f ( t ) and its derivatives are given for t = 0. these initial values are used to evaluate the integration constants that appear in the solution to the differential equation.
• The Laplace transform is employed to solve certain initial-valued differential equations. The method uses algebra rather than the calculus and incorporates the values of the integration constants from the beginning.
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Matematika Industri II
Introduction to Laplace transforms
The Laplace transform
Given a function f ( t ) defined for values of the variable t > 0 then the Laplace
transform of f ( t ), denoted by:
is defined to be:
Where s is a variable whose values are chosen so as to ensure that the semi-
infinite integral converges.
( )L f t
0
( ) ( )st
t
L f t e f t dt
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Matematika Industri II
Introduction to Laplace transforms
The Laplace transform
For example the Laplace transform of f ( t ) = 2 for t ≥ 0 is:
0
0
0
( ) ( )
2
2
2(0 ( 1/ ))
2 provided > 0
st
t
st
t
st
t
L f t e f t dt
e dt
e
s
s
ss
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Matematika Industri II
Bahasan
• Transformasi Laplace
• Transformasi Laplace Invers
• Tabel Transformasi Laplace
• Transformasi Lplace dari Suatu Turunan
• Dua Sifat Transformasi Laplace
• Membuat Transformasi Baru
• Transformasi Laplace dari Turunan yang Lebih Tinggi
![Page 8: DASAR TRANSFORMASI LAPLACE - masud.lecture.ub.ac.id](https://reader030.vdocuments.site/reader030/viewer/2022012423/61782b0a9ee62248fd0825f6/html5/thumbnails/8.jpg)
Matematika Industri II
Introduction to Laplace transforms
The inverse Laplace transform
The Laplace transform is an expression involving variable s and can be
denoted as such by F (s). That is:
It is said that f (t) and F (s) form a transform pair.
This means that if F (s) is the Laplace transform of f (t) then f (t) is the inverse
Laplace transform of F (s).
That is:
( ) ( )F s L f t
1( ) ( )f t L F s
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Matematika Industri II
Introduction to Laplace transforms
The inverse Laplace transform
For example, if f (t) = 4 then:
So, if:
Then the inverse Laplace transform of F (s) is:
4( )F s
s
1 ( ) ( ) 4L F s f t
4( )F s
s
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Matematika Industri II
Bahasan
• Transformasi Laplace
• Transformasi Laplace Invers
• Tabel Transformasi Laplace
• Transformasi Lplace dari Suatu Turunan
• Dua Sifat Transformasi Laplace
• Membuat Transformasi Baru
• Transformasi Laplace dari Turunan yang Lebih Tinggi
![Page 11: DASAR TRANSFORMASI LAPLACE - masud.lecture.ub.ac.id](https://reader030.vdocuments.site/reader030/viewer/2022012423/61782b0a9ee62248fd0825f6/html5/thumbnails/11.jpg)
Matematika Industri II
Introduction to Laplace transforms
Table of Laplace transforms
To assist in the process of finding Laplace transforms and their inverses a table
is used. For example:
1
2
( ) { ( )} ( ) { ( )}
0
1
1
( )
kt
kt
f t L F s F s L f t
kk s
s
e s ks k
te s ks k
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Matematika Industri II
Bahasan
• Transformasi Laplace
• Transformasi Laplace Invers
• Tabel Transformasi Laplace
• Transformasi Laplace dari Suatu Turunan
• Dua Sifat Transformasi Laplace
• Membuat Transformasi Baru
• Transformasi Laplace dari Turunan yang Lebih Tinggi
![Page 13: DASAR TRANSFORMASI LAPLACE - masud.lecture.ub.ac.id](https://reader030.vdocuments.site/reader030/viewer/2022012423/61782b0a9ee62248fd0825f6/html5/thumbnails/13.jpg)
Matematika Industri II
Introduction to Laplace transforms
Laplace transform of a derivative
Given some expression f (t) and its Laplace transform F (s) where:
then:
That is:
0
( ) ( ) ( )st
t
F s L f t e f t dt
0
00
( ) ( )
( ) ( )
(0 (0)) ( )
st
t
st st
tt
L f t e f t dt
e f t s e f t dt
f sF s
( ) ( ) (0)L f t sF s f
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Matematika Industri II
Bahasan
• Transformasi Laplace
• Transformasi Laplace Invers
• Tabel Transformasi Laplace
• Transformasi Laplace dari Suatu Turunan
• Dua Sifat Transformasi Laplace
• Membuat Transformasi Baru
• Transformasi Laplace dari Turunan yang Lebih Tinggi
![Page 15: DASAR TRANSFORMASI LAPLACE - masud.lecture.ub.ac.id](https://reader030.vdocuments.site/reader030/viewer/2022012423/61782b0a9ee62248fd0825f6/html5/thumbnails/15.jpg)
Matematika Industri II
Introduction to Laplace transforms
Two properties of Laplace transforms
The Laplace transform and its inverse are linear transforms. That is:
(1) The transform of a sum (or difference) of expressions is the sum (or
difference) of the transforms. That is:
(2) The transform of an expression that is multiplied by a constant is the
constant multiplied by the transform. That is:
1 1 1
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
L f t g t L f t L g t
L F s G s L F s L G s
1 1( ) ( ) and ( ) ( )L kf t kL f t L kF s kL F s
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Matematika Industri II
Introduction to Laplace transforms
Two properties of Laplace transforms
For example, to solve the differential equation:
take the Laplace transform of both sides of the differential equation to yield:
That is:
Resulting in:
( ) ( ) 1 where (0) 0f t f t f
( ) ( ) 1 so that ( )} { ( ) 1L f t f t L L f t L f t L
1 1
( ) (0) ( ) which means that ( 1) ( )sF s f F s s F ss s
1( )
( 1)F s
s s
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Matematika Industri II
Introduction to Laplace transforms
Two properties of Laplace transforms
Given that:
The right-hand side can be separated into its partial fractions to give:
From the table of transforms it is then seen that:
1( )
( 1)F s
s s
1 1( )
1F s
s s
1
1 1
1 1( )
1
1 1
1
1 t
f t Ls s
L Ls s
e
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Matematika Industri II
Introduction to Laplace transforms
Two properties of Laplace transforms
Thus, using the Laplace transform and its properties it is found that the
solution to the differential equation:
is:
( ) 1 tf t e
( ) ( ) 1 where (0) 0f t f t f
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Matematika Industri II
Bahasan
• Transformasi Laplace
• Transformasi Laplace Invers
• Tabel Transformasi Laplace
• Transformasi Lplace dari Suatu Turunan
• Dua Sifat Transformasi Laplace
• Membuat Transformasi Baru
• Transformasi Laplace dari Turunan yang Lebih Tinggi
![Page 20: DASAR TRANSFORMASI LAPLACE - masud.lecture.ub.ac.id](https://reader030.vdocuments.site/reader030/viewer/2022012423/61782b0a9ee62248fd0825f6/html5/thumbnails/20.jpg)
Matematika Industri II
Introduction to Laplace transforms
Generating new transforms
Deriving the Laplace transform of f (t ) often requires integration by parts.
However, this process can sometimes be avoided if the transform of the
derivative is known:
For example, if f (t ) = t then f ′ (t ) = 1 and f (0) = 0 so that, since:
That is:
( ) { ( )} (0) then {1} { } 0L f t sL f t f L sL t
2
1 1{ } therefore { }sL t L t
s s
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Matematika Industri II
Bahasan
• Transformasi Laplace
• Transformasi Laplace Invers
• Tabel Transformasi Laplace
• Transformasi Lplace dari Suatu Turunan
• Dua Sifat Transformasi Laplace
• Membuat Transformasi Baru
• Transformasi Laplace dari Turunan yang Lebih Tinggi
![Page 22: DASAR TRANSFORMASI LAPLACE - masud.lecture.ub.ac.id](https://reader030.vdocuments.site/reader030/viewer/2022012423/61782b0a9ee62248fd0825f6/html5/thumbnails/22.jpg)
Matematika Industri II
Introduction to Laplace transforms
Laplace transforms of higher derivatives
It has already been established that if:
then:
Now let
so that:
Therefore:
( ) { ( )} and ( ) { ( )}F s L f t G s L g t
( ) ( ) so (0) (0) and ( ) ( ) (0)g t f t g f G s sF s f
{ ( )} { ( )} ( ) (0)
( ) (0) (0)
L g t L f t sG s g
s sF s f f
2{ ( )} ( ) (0) (0)L f t s F s sf f
𝐿 𝑓′ 𝑡 = 𝑠𝐹 𝑠 − 𝑓 0 𝑑𝑎𝑛 𝐿 𝑔′ 𝑡 = 𝑠𝐺 𝑠 − 𝑔(0)
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Matematika Industri II
Introduction to Laplace transforms
Laplace transforms of higher derivatives
For example, if:
then:
and
Similarly:
And so the pattern continues.
( ) { ( )}F s L f t
2{ ( )} ( ) (0) (0)L f t s F s sf f
3 2{ ( )} ( ) (0) (0) (0)L f t s F s s f sf f
𝐿 𝑓′ 𝑡 = 𝑠𝐹 𝑠 − 𝑓 0
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Matematika Industri II
Introduction to Laplace transforms
Laplace transforms of higher derivatives
Therefore if:
Then substituting in:
yields
So:
2( ) sin so that ( ) cos and ( ) sinf t kt f t k kt f t k kt
2{ ( )} ( ) (0) (0)L f t s F s sf f
2 2 2{ sin } {sin } {sin } .0L k kt k L kt s L kt s k
2 2{sin }
kL kt
s k
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Matematika Industri II
Introduction to Laplace transforms
Table of Laplace transforms
In this way the table of Laplace transforms grows:
1
2 2
2 2
2 2
2 2
( ) { ( )} ( ) { ( )}
sin 0
cos 0
f t L F s F s L f t
kkt s k
s k
skt s k
s k
1
2
( ) { ( )} ( ) { ( )}
0
1
1
( )
kt
kt
f t L F s F s L f t
kk s
s
e s ks k
te s ks k
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Matematika Industri II
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Matematika Industri II
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Matematika Industri II
Learning outcomes
Derive the Laplace transform of an expression by using the integral definition
Obtain inverse Laplace transforms with the help of a table of Laplace transforms
Derive the Laplace transform of the derivative of an expression
Solve linear, first-order, constant coefficient, inhomogeneous differential equations
using the Laplace transform
Derive further Laplace transforms from known transforms
Use the Laplace transform to obtain the solution to linear, constant-coefficient,
inhomogeneous differential equations of second and higher order.
Introduction to Laplace transforms
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Matematika Industri II
Reference
• Stroud, KA & DJ Booth. 2003. Matematika
Teknik. Erlangga. Jakarta