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

1)(

e.)e(

Definition-By :Proof a-s

1)L(e (2)

)0(,s11.)1(

Definition-By :Proof s1L(1) (1)

0

)(

0

)(

0

atat

at

00

ase

dtedteL

ss

edteL

tas

tasst

stst

|a|s, a-s

sat]L[cosh ly,(5)Similar

|a|s, a-s

a

1121

)]()([21

2eLat)L(sinh

definitionBy 2

eatcosh and 2

eatsinh have -We:Proof

a-saat]L[sinh (4)

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22

22

at

atat

22

at-

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eLeLe

ee

as

atatat

atat

0s, as

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aat]L[sin

get weparts,imaginary and real Equating as

ai as

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1)L(e 1]e[]sin[cos

sincose

Formula] s[Euler' sincose that know -We:Proof

0s, as

sat] L[cos and as

aat]L[sin (6)

2222

222222

atiat

iat

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2222

asias

LatiatL

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n!1n 0,1,2...n n!)(or

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n!or 1)()8(

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1)1(1

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nn

nxn

n

nun

nu

nstn

nnn

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ndxxeSntL

duueS

sdu

sue

puttingdttet

SSntL

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

shifting-s theorem,shiftingFirst -Theorem

at-

at

0

rt-

0

a)t-(s-

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st-at

at

asSimilarly

as

asr

dttf

dttf

dttfe

as

at

22)-(s

2-s)4cosL(e

2ssL(cosh2t)

)2coshL(e (1)

43)(s

3s)4cosL(e

4ssL(cos4t)

)4cosL(e (1)

:

222t

22

2t

223t-

22

3t-

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

)1(1

21C

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

asdsdtL

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

.)1(

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

1

12

1

2

21

t

eue

dueu

dueuss

L

ssL

ssL

eLs

tLs

HereSol

ssLExample

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

22

21-

1-

22

21-

)3(2)62(

362

)2(323t-

3t-

-3t

ttu

ttussLtu

sse

atuatfsfe

sse

se

se

eLee

eLetu

a

tuExample

s

as

s

ss

ts

ts

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…