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Year: 2016-17
Subject: Advanced Engineering Maths(2130002)
Topic: Laplace Transform & its Application
Name of the Students:
Gujarat Technological University L.D. College of Engineering
Agnihotri Aparna 160283105001Agnihotri Shivam 160283105002Kansara Sagar 160283105004Makvana Yogesh 160283105005Padhiyar Shambhu 160283105006
Patil Dipak 160283105008Patil Mayur 160283105009Rohit Chetan 160283105010Sindhav Jaydrath 160283105011Vasava Yogesh 160283105012
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› Application in Chemical Engineering
Definition of Laplace Transform› Let f(t) be a given function of t defined for all then the Laplace Transform of 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
Linearity of the Laplace Transform
› If L{f(t)}= and then for any constants a and b
)(sf )()]([ sgtgL
)]([)]([)]()([ tgbLtfaLtbgtafL
)]([)]([)}()({
)()(
)]()([)}()({
Definition-By :Proof
00
0
tgbLtfaLtbgtafL
dttgebdttfea
dttbgtafetbgtafL
stst
st
Laplace Transform of some Elementary Functions
as if a-s
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Definition-By :Proof a-s
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Definition-By :Proof s1L(1) (1)
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)(
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)(
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atat
at
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ase
dtedteL
ss
edteL
tas
tasst
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|a|s, a-s
sat]L[cosh ly,(5)Similar
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1121
)]()([21
2eLat)L(sinh
definitionBy 2
eatcosh and 2
eatsinh have -We:Proof
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at
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at-
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ee
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get weparts,imaginary and real Equating as
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Formula] s[Euler' sincose that know -We:Proof
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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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)2coshL(e (1)
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4ssL(cos4t)
)4cosL(e (1)
:
222t
22
2t
223t-
22
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t
t
t
t
Eg
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
21
2112
1
)2(21
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than0s 21B
than-2s -1A
than-1s 2)1)(sc(s1)(s)B(s2)(s)A(s1
)2()1())(2)(1(1
2)(s)1)(s(s1L )1(
21
1
1
tt eesss
L
If
If
If
sC
sB
sA
sssL
Laplace Transform of Derivatives & Integral
f(u)du(s)f1L Also
(s)f1f(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
t
0
1-
t
0
1-n2-n1-nnn
st
t
s
s
22
222
3
22
2n
saat)L(sin
at)L(sin ss
a-
a-at)L(sin ssinat}L{-a
thisfrom a(0)f0,f(0) Also sinat-a(t)f andat cos a(t)f sinat thenf(t)Let :Sol
atsin of transformlaplace DeriveExample
a
aa
)1(1
)(1cos
cosf(u) -Here:Sol
cos
2
0
0
ss
sfs
uduL
u
uduLEg
t
t
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[dsd(-1)f(t)]L[t then (s) fL{f(t)} If
Tranform Laplace ofation Differenti
3
2
2
22at2
at2
)(2
)(1
1)1()e ( -:Sol
)e (:
as
asdsd
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tLExample
ss
s
ssds
ttLExample
s
11
11
1
s22
cottan2
tantan
tan1.t)L(sin -:Sol
sin
Evaluation of Integrals By Laplace Transform
1)1()cos(
1)(cos
cos)cos(
cos)( 3
)()}({
cos -:Example
2
2
0
0
0
3
ss
dsdttL
sstL
tdttettL
tttfs
dttfetfL
tdtte
st
st
t
252
1008
)19(19cos
cos)1(1)cos(
)1(2)1(1
20
3
022
2
22
22
tdtte
tdttessttL
sss
t
st
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 Ift
0
1-
-1-1
)1(e
e
.e
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1
)1(1.1
)1(1
n theoremconvolutioby
)(1
1(s)g and )(1(s)f have we:
)1(1:
t
0t
0
t
02
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1
2
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t
eue
dueu
dueuss
L
ssL
ssL
eLs
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HereSol
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tuu
tu
tut
t
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
tts
s
2sin232cos
4s6
4sY(s)
transformlaplace inverse Taking4s6Y(s)
06-s-4)Y(s)(s
04(Y(s))(0)y-sy(0)-Y(s)s
22
2
2
2
Laplace Transform of Periodic Functions
p
0
st 0)(sf(t)dt ee-11L{f(t)}
is p periodwith f(t)function periodic continous piecewise a of transformlaplace The
0 tallfor f(t)p)f(tif 0)p(
periodith function w periodic be tosaid is f(t)Afunction -Definition
ps-
2wsπhcot
wsw
e
e.e1
e1.ws
w
e1ws
w.e1
1L[F(t)]
e1ws
w
wcoswt)ssinwt(ws
esinwtdteNow
tallfor f(t)wπtf and
wπt0for sinwt f(t)
0t|sinwt|f(t) ofion rectificat wave-full theof transformlaplace theFind
22
2wsπ
2wsπ
wsπ
wsπ
22
wsπ
22wsπ
wsπ
22
2
wπ
0
wπ
022
stst
Unit Step Function
s1L{u(t)}
0a if
es1
se
(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-
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-
)(cos)2(
)2(cos)2()2(L
)()()(L
theroemshifting secondBy
(ii)L
331
}{.
}{)]2(L[e
2,ef(t)
)]2((i)L[e
221
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21-
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)3(2)62(
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Application In Chemical Engineering › A fast numerical technique for the solution of P.D.E.
describing time-dependent two- or three-dimensional transport phenomena is developed. It is based on transforming the original time-domain equations into the Laplace domain where numerical integration is performed and by subsequent numerical inverse transformation the final solution can be obtained. The computation time is thus reduced by more than one order of magnitude in comparison with the conventional finite-difference techniques.
Continue…› Application of Laplace transforms for the solution of
transient mass- and heat-transfer problems in flow systems
› Application to mass-transfer in single and multi-stream laminar parallel-plate flow systems
Thanks…