Laplace_1.pptx

The Laplace Transform
Major Tahmina Sultana, PhD
Military Institute of Science and Technology
Department of Mathematic
Bangladesh
Prepared by Maj Tahmina 1

The French Newton
Pierre-Simon Laplace
• Developed mathematics in
astronomy, physics, and statistics
• Began work in calculus which led to
the Laplace Transform
• Focused later on celestial mechanics
• One of the first scientists to suggest
the existence of black holes

History of the Transform
• Euler began looking at integrals as solutions to differential equations in
the mid 1700’s:
• Lagrange took this a step further while working on probability density
functions and looked at forms of the following equation:
• Finally, in 1785, Laplace began using a transformation to solve equations
of finite differences which eventually lead to the current transform

The Laplace Transform of a function, F(t), is defined as;
Definition

Laplace Transform of
*notes
1
)
( 
t
F

*notes
t
t
F 
)
(

*notes
at
t
F sin
)
( 

at
e
t
F 
)
(

Laplace Transform of )
cosh(
)
( at
t
F 

1. Find Laplace Transform of )
sinh(
)
( at
t
F 
2. Find Laplace Transform of )
cos(
)
( at
t
F 
3. Find Laplace Transform of
F(t)

F(t)=

F(t)
F(t) F(t)
f
Example
Example

Restrictions
• There are two governing factors that
determine whether Laplace transforms can be
used:
– F(t) must be at least piecewise continuous for
t ≥ 0
– |F(t)| ≤ Meαt where M and α are constants

Linearity Property
Let and
Then if c1 and c2 are any constants,
Proof:
(Proved)

Example

Example
Find
Solution :

Evaluate
1.
2.
3. Find the Laplace transform of
Home Work

First Shifting/Translation Property
If ,then
Solution :
(Proved)

Example
Find the Laplace transform of F(t)=e−5tsin3t.

Second Shifting/Translation Property
If , and
Then

Proof:

Find where
Example
and
Solution : From second shifting property, we can write

Change of Scale Property
If , then
Proof:

Example
Taking Laplace transform on both sides
(Change of scale property and linearity)

Example
[ ]
From the change of scale property , we have
From the First shifting property , we have

Multiplication By power of t
Where n=1, 2, 3,……
Proof:

Example
Solution :
_ _ _ _ _

C.W
H.W

Example
Solution :
.

Example
Solution :

Example
Solution :
.
.

The Laplace Transform of Derivative
First Derivative
Second Derivative
Third Derivative
nth Derivative

( )
( )

Division By t
Proof:
Hence Prepared by Maj Tahmina 41

The Laplace Transform of Integral
Proof:

Initial Value
If the function F(t) and its first derivative are Laplace transformable and f(t)
Has the Laplace transform f(s), and the exists, then
0
)
0
(
F
)
(
F
lim
)
(
lim





t
s
t
s
sf
The utility of this theorem lies in not having to take the inverse of f(s)
in order to find out the initial condition in the time domain. This is
particularly useful in circuits and systems.
Theorem:
Initial Value
Theorem


s
s
sf )
(
lim

Initial Value Theorem:
Example:
Given;
2
5
2
)
1
(
)
2
(
)
(




s
s
s
f
Find F(0)
1
)
2
26
(
2
2
2
2
2
2
2
2
lim
25
1
2
2
2
2
lim
2
5
2
)
1
(
)
2
(
lim
)
(
lim
)
0
(























s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
sf
F


s


s 

s


s

Final Value Theorem:
If the function F(t) and its first derivative are Laplace transformable and f(t)
has the Laplace transform f(s), and the exists, then
)
(
lim s
sF


s
)
(
)
(
lim
)
(
lim 

 F
t
F
s
sf
0

s 

t
Again, the utility of this theorem lies in not having to take the inverse
of f(s) in order to find out the final value of F(t) in the time domain.
This is particularly useful in circuits and systems.
Final Value
Theorem

Final Value Theorem:
Example:
Given:
t
te
s
f
note
s
s
s
f t
3
cos
)
(
1
2
3
2
)
2
(
2
3
2
)
2
(
)
( 2








 




Find )
(
F .
0
2
3
2
)
2
(
2
3
2
)
2
(
lim
)
(
lim
)
( 





 






s
s
s
s
sf
F
0

s
0

s

The Laplace Transform of Sine Integral
Multiplication by power of t
Laplace transform of 1st Derivative
By Integrating
-
Differentiation under integration
Proof:

The Laplace Transform of cosine Integral
Proof:
Differentiation under integration
Multiplication by power of t
Laplace transform of 1st Derivative
By Integrating

Example
Proof:

Laplace Transformation of Error Function
Error Function:
Solution :

Laplace Transform of Step Function

Proof:
F(t)
F(t)
F(t) F(t)
F(t) F
F(t) F F
F
F
Laplace Transformation of Periodic Function
F F
……..1
1

Example

Laplace_1.pptx

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