Download - Laplace transform and its applications
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S.Y. M-2
Shah Nisarg (130410119098)
Shah Kushal(130410119094)
Shah Maulin(130410119095)
Shah Meet(130410119096)
Shah Mirang(130410119097)
Laplace Transform And Its
Applications
![Page 2: Laplace transform and its applications](https://reader030.vdocuments.mx/reader030/viewer/2022020208/55a645a81a28abed148b4692/html5/thumbnails/2.jpg)
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
Dirac Delta Function
![Page 3: Laplace transform and its applications](https://reader030.vdocuments.mx/reader030/viewer/2022020208/55a645a81a28abed148b4692/html5/thumbnails/3.jpg)
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 or F(s) or is defined as
provided the integral exists,where s is a parameter real
or complex.
0t
)(sf )(s
dttfessFsftfL st )()()()()}({0
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Linearity of the Laplace Transform
If L{f(t)}= and then for any
constants a and b
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Laplace Transform of some Elementary
Functions
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First Shifting Theorem
)(f]f(t)L[e ,
)(f]f(t)L[e
)(f)(f
ra-s where)(e
)(e
)(ef(t)]L[e
DefinitionBy Proof
)(f]f(t)L[ethen , (s)fL[f(t)] If
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as
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2-s)4cosL(e
2s
sL(cosh2t)
)2coshL(e (1)
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4s
sL(cos4t)
)4cosL(e (1)
:
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t
t
Eg
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Inverse Laplace Transform
)()}({L
by denoted is and (s)f of transformlaplace inverse
thecalled is f(t) then (s),fL[f(t)] If-Definition
1- tfsf
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Laplace Transform of Derivatives &
Integral
f(u)du(s)f1
L Also
(s)f1
f(u)duLthen (s),fL{f(t)} If
f(t) ofn integratio theof transformLaplace
(0)(0)....f fs-f(0)s-(s)fs(t)}L{f
f(0)-(s)fsf(0)-sL{f(t)}(t)} fL{
and 0f(t)elim provided exists, (t)} fL{ then
continous, piecewise is (t) f and 0 tallfor continous is f(t) If
f(t) of derivative theof transformLaplace
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t
s
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at)L(sin ss
a-
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thisfrom a(0)f0,f(0) Also
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atsin of transformlaplace DeriveExample
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t
t
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Differentiation & Integration of Laplace
Transform
0
n
nnn
ds (s)ft
f(t)Lthen
, transformLaplace has t
f(t) and (s) fL{f(t)} If
Transforms Laplace ofn Integratio
1,2,3,...n where, (s)]f[ds
d(-1)f(t)]L[t then (s) fL{f(t)} If
Tranform Laplace ofation Differenti
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Evaluation of Integrals By Laplace
Transform
1)1()cos(
1)(cos
cos)cos(
cos)( 3
)()}({
cos -:Example
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3
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s
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Convolution Theorem
g(t)*f(t)
g*fu)-g(t f(u)(s)}g (s)f{L
theng(t)(s)}g{L and f(t)(s)}f{L If
t
0
1-
-1-1
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Application to Differential Equations
04L(y))yL(
sideboth on tranformLaplace Taking
.
.
(0)y-(0)ys-y(0)s-Y(s)s(t))yL(
(0)y-sy(0)-Y(s)s(t))yL(
y(0)-sY(s)(t))yL(
Y(s)L(y(t))
6(0)y 1y(0) 04yy :
23
2
eg
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tts
s
2sin2
32cos
4s
6
4sY(s)
transformlaplace inverse Taking
4s
6Y(s)
06-s-4)Y(s)(s
04(Y(s))(0)y-sy(0)-Y(s)s
22
2
2
2
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Laplace Transform of Periodic Functions
p
0
st 0)(sf(t)dt ee-1
1L{f(t)}
is p periodwith
f(t)function periodic continous piecewise a of transformlaplace The
0 tallfor f(t)p)f(t
if 0)p(
periodith function w periodic be tosaid is f(t)Afunction -Definition
ps-
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2w
sπhcot
ws
w
e
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e1.
ws
w
e1ws
w.
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w
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esinwtdteNow
tallfor f(t)w
πtf and
w
πt0for sinwt f(t)
0t|sinwt|f(t)
ofion rectificat wave-full theof transformlaplace theFind
22
2w
sπ
2w
sπ
w
sπ
w
sπ
22
w
sπ
22
w
sπ
w
sπ
22
2
w
π
0
w
π
0
22
stst
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Unit Step Function
s
1L{u(t)}
0a if
es
1
s
e
(1)dte(0)dte
a)dt-u(tea)}-L{u(t
at1,
at0,a)-u(t
as-
a
st-
a
st-
a
0
st-
0
st-
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Second Shifting Theorem
a))L(f(tea))-u(t L(f(t)-Corr.
L(f(t))e
(s)fea))-u(t a)-L(f(t
then(s)fL(f(t)) If
as-
as
as-
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)(cos)2(
)2(cos)2()2(L
)()()(L
theroemshifting secondBy
(ii)L
33
1
}{.
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)]2((i)L[e
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atuatfsfe
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s
as
s
ss
ts
ts
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Dirac Delta function
1))((
))((
0
1
0lim0
tL
eatL
tε , a
ε at , aε
at , - a)δ(t
as
ε
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sin3tcos3t2ex(t)
sin3t2ecos3t2ex
inversion on
92)(s
6
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2)2(s
134ss
102sx
2(1)x13x(0)]-x4[s(0)]x-sx(0)x[s
have weTransform, Laplace Taking
0(0)x and 2x(0)0,t(t),213xx4x
0(0)x and 2x(0)0,at t here w
(t)213xx4xequation the-Solve:Example
2t
2t-2t-
222
2
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