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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
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dttgebdttfea
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stst
st
Laplace Transform of some Elementary Functions
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Inverse Laplace Transform
)()}({L
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21
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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
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1-n2-n1-nnn
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saat)L(sin
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a-
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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
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Differentiation & Integration of Laplace Transform
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n
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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
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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
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st
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)1(2)1(1
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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 Ift
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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
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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-
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e
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w
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w.e1
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e1ws
w
wcoswt)ssinwt(ws
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tallfor f(t)wπtf and
wπt0for sinwt f(t)
0t|sinwt|f(t) ofion rectificat wave-full theof transformlaplace theFind
22
2wsπ
2wsπ
wsπ
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22
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22wsπ
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22
2
wπ
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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-
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)2(cos)2()2(L
)()()(L
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)]2((i)L[e
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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…
