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1. LAPLACE
TRANSFORM
This is the first part of presentation . Look
into further ppts for more specific topics
on laplace transform
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2. PROJECT 1
Make a short draft of properties of Laplace transform from
memory. Then compare your notes with the text and write a
report of 2-3 pages on these operations and their significance
in applications.
Laplace Transformations was
introduced by
Pierre Simmon Marquis De
Laplace (1749-1827), a French
Mathematician known as a
Newton of French.
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3. CONTENTS
• Definition of Laplace Transform:
• FORMULAS
• PROPERTIES OF LAPLACE
TRANSFORM: LINEARITY PROPERTY
• Problem 01
• Change of Scale Property:
• Problem 2.
• First Shifting Property:
• Application of first shifting property
• Problem 3.
• Unit Step Function:
• Application of Laplace
Transform
• conclusion
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4. Definition of Laplace Transform:
Let 𝑓(𝑡)be a given function defined for all 𝑡≥0,
• then the Laplace transform of 𝑓(𝑡)is denoted by L{𝑓(𝑡
)} or𝐹(𝑠) and is defined by
0
{ ( )} ( )
st
L f at e f at dt


 
The symbol L , which transforms 𝑓(𝑡)into 𝐹(𝑠),is called
Laplace Transformation Operator.
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5. FORMULAS 5
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6. PROPERTIES OF LAPLACE
TRANSFORM:
If a, b, c be any three constants and f , g , h be
any function of t whose Laplace transform
exist, then
𝐿{ 𝑎𝑓(𝑡) - 𝑏𝑔(𝑡) + 𝑐ℎ(𝑡) }
= 𝑎 𝐿{𝑓(𝑡)} - 𝑏 𝐿{𝑔(𝑡)} + 𝑐 𝐿{ℎ(𝑡)}
 Linearity Property:
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7. Problem 01
Find the Laplace transform of
f(t)=5t−2
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8. if 𝐿{𝑓(𝑡)}=𝐹(𝑠) ,
then 𝐿{𝑓(𝑎𝑡)}=1/𝑎{𝑓(𝑆𝑎)} where a is positive constant.
Proof: By definition of Laplace transform, we have
Change of Scale Property:
0
{ ( )} ( )
st
L f at e f at dt


 
Let 𝑎𝑡=𝑥 ⟹ t = ⟹
x
a
dx
dt
a

Also if 𝑡→ ∞ ⟹ 𝑥→ ∞ 𝑎𝑛𝑑 if 𝑡→0⟹𝑥→0
therefore the above integral becomes
since
( )
0
1 1
{ ( )} ( ) ( )
s
x
a
s
L f x e f x dx F
a a a


 
 0
( ) ( )
st
e f t dt F s




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9. Problem 2.
Find the Laplace transform of f(t)=cos4t
using the change of scale property.
thus
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10. First Shifting Property:
if 𝐿{𝑓(𝑡)}=𝐹(𝑠) , then
where a be any real constant.
{ ( )} ( )
at
L e f t F s a
 
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12. Find the Laplace transform of
f(t)= e-5t sin3t. 12
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13. Unit Step Function:
Unit step function is denoted as 𝑢(𝑡−𝑎) and is
defined as
, where (𝑎≥0)
0
1
( ) { ift a
ift a
u t a 

 
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14. Application of Laplace
Transform
• To determine structure of astronomical object from
spectrum
• To find moment generating function in statistics
• To determine the present value of consol or bond in
economics
• To solve the problem related to communication and
network analysis.
• To make a equation in simple form from hard
equation like vibration of
• spring.
• To solve Mixing Problem Involving Two Tanks
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15. CONCLUSION
Laplace Transformation is powerful tool using in
different areas of mathematics, physics and
engineering. With the ease of application of
Laplace transforms in many applications, many
research software have made it possible to
simulate the Laplace transformable equations
directly which has made a good advancement in
the research field.
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16. THANK YOU 16
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