Laplace Transform & its Application

1. Gujarat Technological University
L.D. College of Engineering
Year: 2017-18
Subject: Advanced Engineering Maths(2130002)
Topic: Laplace Transform & its Application
Name of the Students:
Dudhagara Chirag 160280102011 Gondaliya Umang 160280102016
Dudhat Sumit 160280102012 Gopani Vardhaman 160280102017
Gajera Dhruvkumar 160280102013 Jadeja Meetrajsinh 160280102018
Gajjar Manan 160280102014 Kapadiya Darshan 160280102019
Gangani Keyur 160280102015 Lakhara Gopal 160280102020

2. Topics
› Definition of Laplace Transform
› Linearity of the Laplace Transform
› Laplace Transform of some Elementary Functions
› First Shifting Theorem
› Inverse Laplace Transform
› Laplace Transform of Derivatives & Integral
› Differentiation & Integration of Laplace Transform
› Evaluation of Integrals By Laplace Transform
› Convolution Theorem
› Application to Differential Equations
› Laplace Transform of Periodic Functions
› Unit Step Function
› Second Shifting Theorem

3. Definition of Laplace
Transform
or
› Let f(t) be a given function of t defined for all t  0 then
the Laplace Transform of f(t) denoted by L{f(t)}
f (s) or F(s) or (s) is defined as

L{ f (t)} f (s)  F(s)  (s)  est
f (t)dt
0
provided the integral exists, where s is a parameter real
or complex.

4. Linearity of the Laplace
Transform
› If L{f(t)}= f (s) and then for any constants
a and b
L[g(t)]  g (s)
L[af (t) bg(t)] aL[ f (t)] bL[g(t)]
Proof :-By Definition

L{af (t)  bg(t)} est
[af (t)  bg(t)]dt
0
 
 aest
f (t)dt  best
g(t)dt
0 0
L{af (t)  bg(t)} aL[ f (t)] bL[g(t)]
_

5. Laplace Transform of some
Elementary Functions
1
s - a
1
1
0
 if s  a
s - a
(2) L(eat
) 

(1) L(1) 
1
s
Proof : -By Definition



   (s  a) 
  0
 , (s  0)
s  s 0
 e ( sa )t
Proof : -By Definition
 
L(ea t
)   est
.eat
dt  e ( sa )t
dt
0 0
 est


L(1)   est
.1dt  

6. s
a
1
2
22
1
s2
- a2
s2
- a2
s2
- a2
,s |a |(5)Similarly, L[coshat] 
,s |a |


1  1 1
 
and coshat Proof :-Wehavesinh at 
By definition
(4)L[sinh at] 
,s  -a
a
(3) L[e-at
] 
2 s  a s  a
L(sinh at)  L  
2
[L(e )  L(e )]
s a
at at
 eat
 eat

eat
eat
eat
eat

7. a s
Equating realand imaginary parts, weget
as
1
1
sa
s2
s2
s2
s2
s2
s2
s2
and L[cosat]  ,s  0
 a2
 a2
L[sin at]
 a2
 i
 a2
 a2

s ia






,s  0and L[cosat]
 a2
 a2
(6) L[sin at] 
s  a 
s ia
L[cosat  isin at]  L[eiat
] 
Proof :-Weknow thateix
 cosx  isin x[Euler's Formula]
eiat
 cosat  isin at
L(eat
) 

8. n!
1
0
0
0

 





n 0,1,2... n 1  n!
n1 , n -1
Sn1
or L(tn
) 
 n  ex
xn1
dx, n  0
S
n 1
L(tn
) 
u
u(n1)1
dun1 e
S
s s 
n
 u  du
 eu


Proof :-L(tn
)  est
.tn
dt, putting st u
0
n!
Sn1
Sn1
(8)L(tn
) 
n 1
or

9. First Shifting
Theorem
Theorem- First shifting theorem,s - shifting
If L[f(t)] f(s), then L[eat
f(t)]  f (s  a)
Proof  ByDefinition

L[eat
f(t)] e-st
eat
f (t)dt
0

 e-(s-a)t
f (t)dt
0

 e-rt
f (t)dt wheres-a  r
0
 f (r)  f (s  a)
L[eat
f(t)]  f (s  a)
Similarly, L[e-at
f(t)]  f (s  a)

10. s-2
s
s
s2
s2
(s-2)2
 22
22
L(cosh2t)
(s 3)2
 42
s 3
42
L(cos4t)
 L(e2t
cos4t) 
(1) L(e2t
cosh2t)
 L(e-3t
cos4t) 
Eg: 
(1) L(e-3t
cos4t)

11. Inverse Laplace Transform
Definition - If L[f(t)]  f(s), then f(t) is called the
inverse laplace transform of f(s) and is denoted by
L-1
{ f (s)}  f (t)

12. 1
2
12
1
2


1 1 



1
 
(s 1)(s 2)(s)

1
(1) L1
2tt   1e  e 
2(s  2) s 
(s 1)
 L 
1  A(s  2)(s) B(s 1)(s) c(s1)(s 2)
If s  -1than
A  -1
If s  -2 than
B 
1
2
If s  0 than
C 
1
2
B

C1 A
(s 1)(s  2)(s) (s 1) (s  2) s
L1

13. Laplace Transform of Derivatives
& Integral
1
Laplace transformof theintegration of f(t)
t
0

Also L-1
 f(s)  f(u)du
1


0
t
Laplace transformof the derivative off(t)
 If f(t)is continousfor all t  0 and f (t)is piecewisecontinous,
thenL{f (t)}exists,providedlimest
f(t) 0 and
t
L{f (t)} sL{f(t)}-f(0)  sf(s)- f(0)
L{f n
(t)} sn
f(s)-sn-1
f(0)-sn-2
f  (0)....fn-1
(0)
s
 If L{f(t)} f(s),then Lf(u)du 
s
f(s)

14. 2
s2
a
s2
-a3
 a2
L(sin at)
 a  s L(sin at)
 a2
Example Derivelaplace transformof sin at
Soln
: Let f(t)  sinat thenf(t) a cosat and f(t)  -a2
sinat
Also f(0) 0,f(0) a from this
L{-a2
sinat}  s2
L(sin at)-a
1
1



0

0 
s(s2
1)
Lcosudu 
s
f (s)
Sol:-Heref(u)  cosu
Eg  Lcosudu
t
t

15. Differentiation & Integration
of Laplace Transform

  0
f(t)
then L
t
  f(s)ds
t
If L{f(t)} f (s)and
f(t)
has Laplace transform,
Integration of Laplace Transforms
dsn
If L{f(t)} f (s) then L[tn
f(t)] (-1)n
[f(s)], wheren 1,2,3,..
Differentiation of Laplace Tranform
dn

16. 2
2
22 at 1
ds
ds
d2
(s a)3





 (s a)
2
d  1
 






s a
Sol:- L(t e )  (1)
Example: L(t2
eat
)

17. 
 
 

  
s

s 

ds
t
  s
1  s 1 s 
s s 22
  cot  tan

2
 tan1
 tan1
tan1

1 
 .L(sin t) Sol:-
Example L sint 

18. Evaluation of Integrals By Laplace
Transform



1
2
0
s
ds  s 1
L(t cost)  (1)
d 
s2
s
L(cost) 

Example:- te3t
costdt
0

L{f (t)} est
f (t)dt
0
s  3 f (t)  t cost

L(t cost)  est
t costdt
0
2

(91)2
9 1 8

2
100 25





 1


0

(s2
1)2
s 1
L(t cost) 
(s2
1)2
 (s2
1)  2s2

e3t
t costdt 
est
tcostdt

19. Convolution Theorem
If L-1
{f(s)} f(t)andL-1
{g(s)} g(t) then
t
L-1
{f(s)g(s)} f(u)g(t - u)  f *g
0
 f(t)*g(t)

20. 1
1
1
0
0
0
2
2
2

1 
  








  1
 et
 (t 1)
s (s 1)
L1


s (s 1)


 L11
.
s (s 1)
L1
s 1s2
by convolution theorem
Sol : Here wehave f(s) 
1
 L(t) and g(s) 
s (s 1)
Example:L1 
tu
 et
ueu
 e 
t
 et
u.eu
du
t
  u.etu
du
 L(et
)

21. Application to Differential
Equations
eg :  y  4y  0 y(0) 1 y(0) 6
L(y(t)) Y(s)
L(y(t))  sY(s)- y(0)
L(y(t))  s2
Y(s)-sy(0)- y(0)
L(y(t))  s3
Y(s)-s2
y(0)- sy(0)- y(0)
.
.
Taking Laplace tranformon both side
L(y)  4L(y)  0

22. s 6
Taking inverselaplace transform
s2
s2
s2
 cos2t 
3
sin 2t
2

 4  4
Y(s) 
 4
Y(s) 
s 6
s2
Y(s)-sy(0)- y(0) 4(Y(s)) 0
(s2
 4)Y(s)-s -6  0

23. Laplace Transform of Periodic Functions
est
f(t)dt (s 0)
0
1
L{f(t)}
Definition – A function f(t) is said to be periodic function with
period p( 0)if
f(t p)  f(t)for all t  0
The laplace transform of a piecewise continuous periodic function
f(t) with period p is
p
1- e-ps

24.  














 

2w
cot h sπ 
1e
w
.
1 e
1e
w
.
1e
1
L[F(t)]
1 e
(ssinwt wcoswt)
s w
Nowe sinwtdt

w
and ft 
π   f(t)for all t
 f(t)  sinwt for 0  t 
π
w
 Find thelaplace transformof thefull- waverectification of
f(t)|sinwt| t  0
s2
 w2

sπ sπ
w e2w
w e2w
.

sπ sπ
s2
 w2
w
w 

sπ

s2
 w 2 
w

sπ
w 

sπ

s2
 w 2 
w2
π
w
0
w
π
0
22
est
st

25. Unit Step
Function
e-as
if a  0
L{u(t)}
1
s

1
 s a
s
 
 e-st


u(t-a)  0,t  a
1, t a

L{u(t-a)} e-st
u(t -a)dt
0
a 
 e-st
(0)dte-st
(1)dt
0 a

26. Second Shifting Theorem
If L(f(t)) f(s) then
L(f(t- a) u(t -a))  e-as
f(s)
 eas
L(f(t))
Corr.- L(f(t)u(t -a))  e-as
L(f(t a))

27. L
(ii)L
1
2222
2 2
s
 u(t  2)cos(t)
(t  2) u(t  2)cos



 s 
 u(t  2)L



s 



 s 

s  3 s 3
 e
1-1 se2s
 By secondshifting theroem
L-1
eas
f (s) f (t  a)u(t  a)
-1  se2s
e2(s3)
(2s6)
Example (i)L[e-3t
u(t  2)]
 f(t)  e-3t
,a  2
L[e-3t
u(t  2)]  e2s
L{e3(t2)
}
 e2s
.e6
L{e3t
}

28. Thanks…