About fundamentals and basic question

1. HUMANITIES AND SCIENCE DEPARTMENT.
SARDAR PATEL COLLEGE OF
ENGINEERING,BAKROL

2. TOPIC NAME: LAPLACE TRANSFORM
ELECTRICAL DEPARTMENT
STUDENT’S NAME ENROLLMENT NUMBER
ANUJ VERMA 141240109003
KARNVEER CHAUHAN 141240109011
MACHHI NIRAV 141240109012
MALEK MUAJHIDHUSEN 141240109013
DHARIYA PARMAR 141240109014
JAYEN PARMAR 141240109015
PARTH YADAV 141240109016
HARSH PATEL 14124010901
Guided by: ASHISH DHOKIYA and NIKHIL PARMAR

3. Definition of Laplace Transform
• Let f(t) be a given function of t defined for all
then the Laplace Transform ot f(t) denoted by L{f(t)}
or F(s) or is defined as
provided the integral exists, where s is a parameter real or
complex.
0t
)(sf )(s
dttfessFsftfL st
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0

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
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4. Laplace Transform of some
Elementary Functions
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8. Example

9. First Shifting Theorem
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10. 22)-(s
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t
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t
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EXAMPLE

11. Second Shifting Theorem
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L(f(t))e
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then(s)fL(f(t))If
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as
as-
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12.  
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13. Differentiation & Integration of
Laplace Transform
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14. 3
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Example of Laplace multiplication

15. 
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Example of Laplace Integration

16. Evaluation of Integrals By Laplace
Transform
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17. Laplace Transform of Derivatives &
Integral
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f(u)du(s)f
1
LAlso
(s)f
1
f(u)duLthen(s),fL{f(t)}If
f(t)ofnintegratiotheoftransformLaplace
(0)(0)....ffs-f(0)s-(s)fs(t)}L{f
f(0)-(s)fsf(0)-sL{f(t)}(t)}fL{
and0f(t)elimprovidedexists,(t)}fL{then
continous,piecewiseis(t)fand0tallforcontinousisf(t)If
f(t)ofderivativetheoftransformLaplace
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t
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18. 22
2
22
3
22
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at)L(sin
at)L(sins
s
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19. Application to Differential Equations in
Laplace
04L(y))yL(
sidebothontranformLaplaceTaking
.
.
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(0)y-sy(0)-Y(s)s(t))yL(
y(0)-sY(s)(t))yL(
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20. tt
s
s
2sin
2
3
2cos
4s
6
4s
Y(s)
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Y(s)
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21. Laplace Transform of Periodic
Functions
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0
st
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f(t)functionperiodiccontinouspiecewiseaoftransformlaplaceThe
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22. 
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23. Unit Step Function
s
1
L{u(t)}
0aif
e
s
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s
e
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24. Express the following function in the terms of unit step
function and also find it’s Laplace transform
Example

25. Direct Delta function
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26. Find the Laplace transform by direct delta method
Example

27. Inverse Laplace Transform
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thecalledisf(t)then(s),fL[f(t)]If-Definition
1-
tfsf 
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28. Example of Laplace inverse
Example

29. =
=

30. Laplace inverse by partial differential form
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31. Convolution Theorem
g(t)*f(t)
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t
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32.  
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Example of convolution theorem in Inverse Laplace

33. Example of convolution theorem

34. Thank You