Ki-Bok Min PhDBok Min, PhD - Seoul National University

Engineering Mathematics IChapter 6 Laplace Transforms- Chapter 6. Laplace Transforms

- 6장. 라플라스변환

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Ki-Bok Min PhDKi Bok Min, PhD서울대학교 에너지자원공학과 조교수Assistant Professor, Energy Resources Engineering, gy g g

schedule

• 2 May, 4 May (Quiz): Laplace Transform• 9 May 11 May : Laplace Transform• 9 May, 11 May : Laplace Transform• 16 May : 2nd Exam• 18 May ~ : Linear Algebra

Ch.5 Series Solutions of ODEs. Special Functions상미분방정식의급수해법 특수함수상미분방정식의급수해법. 특수함수

• 5.1 Power Series Method (거듭제곱급수해법)

• 5.2 Theory of the Power Series Method (거듭제곱급수해법의이론)• 5.3 Legendre’s Equation. Legendre Polynomials Pn(x) Legendre 방정식. Legendre다항식 Pn(x)

5 4 Frobenius Method (Frobenius해법) • 5.4 Frobenius Method (Frobenius해법) • 5.5 Bessel’s Equation. Bessel functions Jv(x). Bessel의방정식. Bessel 함수 Jv(x)

5 6 B l’ F ti f th S d Ki d Y ( ) 제2종 B l 함수 Y ( )• 5.6 Bessel’s Functions of the Second Kind Yv(x). 제2종 Bessel 함수 Yv(x)

• 5.7 Sturm-Liouville Problems. Orthogonal Functions. Sturm-Liouville문제. 직교함수

O f 직교고유함수의전개• 5.8 Orthogonal Eigenfunction Expansions. 직교고유함수의전개

Chapter 5. Laplace Transforms

• 6.1 Laplace Transforms. Inverse Transform. Linearity. s-Shifting

• 6.2 Transforms of Derivatives and Integrals. ODEsg• 6.3 Unit Step Function. t-Shifting

6 4 Short Impulses Dirac’s Delta function Partial Fractions• 6.4 Short Impulses. Dirac’s Delta function. Partial Fractions• 6.5 Convolution. Integral Equations• 6.6 Differentiation and Integration of Transforms• 6.7 Systems of ODEs6.7 Systems of ODEs• 6.8 Laplace Transforms. General Formulas• 6.9 Table of Laplace Transforms

Laplace TransformI t d tiIntroduction

• The Laplace transform method – a powerful method for solving linear ODEs and corresponding a powerful method for solving linear ODEs and corresponding

initial value problems

IVP AP S l i S l ti IVPInitial Value

Problem

APAlgebraic Problem

SolvingAP

by Algebra

Solution of the

IVP① ② ③

Step 1 The given ODE is transformed into an algebraic equation(“subsidiary equation”).p g g q ( y q )

Step 2 The subsidiary equation is solved by purely algebraic manipulations.

Step 3 The solution in Step 2 is transformed back, resulting in the solution of the given problem.p p , g g p

Laplace Transform. Inverse Transform. Li it ShiftiLinearity. s-Shifting

• Laplace Transform:

0

stF s f e f t dt

L

– 어떤함수 f(t) (적분이존재해야함)에 e-st를곱하여 0에서무한대까지적분한것

0

무한대까지적분한것

– Integral Transform 0

( , )F s k s t f t dt

k lkernel

( ) ( )f t or y t ( ) ( )F s or Y sLaplace

Transform

• Inverse Transform: 1 ( )f f 1 F f t L

1

1

( )

( )

f f

F F

L L

L L


• Example 1. Let when . Find . 1 0 f t t F s

00

1 11 0st stf e dt e ss s

L L

• Example 2. Let when where a is a constant. Find 0 atf t e t f L

1 1 0s a tat st ate e e dt e s aa s s a

L00 a s s a


• Need to learn by heart.


Theorem 1 Linearity of the Laplace Transform (라플라스연산은선형연산이다)

The Laplace transform is a linear operation ; that is, for any functions f (t) and g(t) whose

transforms exist and any constants a and b the transform of af (t)+ bg(t) exists, and

af t bg t a f t b g t L L L

• Ex.3 Find the transform of coshat and sinhat

1 1 1 1cosh , inh 2 2

at at at at at atat e e at e e e , es a s a

L L

2 2

1 1 1 1 cosh2 21

at at sat e es a s a s a

L L L

1 1 1 1 sinh2

atat e L L L 2 2

1 1 12

at aes a s a s a


• Ex.4 2 2cos st

s

L 2 2sin t

s

L


Theorem 2 First Shifting Theorem, s-shifting (제 1 이동정리)

If f (t) has the transform F(s) ( where s > k for some k ), then has the transform F(s - a) ate f t

(where s – a > k). In formulas,

ate f t F s a L

1at -e f t F s a L

or, if we take the inverse on both sides,

• Example 5. s-Shifting: Damped vibrations.

2222 cos cos

astest atLL

2222 sin sin tet atLL

Use these formulas to find the inverse of the transform

2222 ass

2222 ass

13732

sfL 40122 ss

f

ttess

ss

sf t 20sin720cos34001

2074001

13400114013

21

21

21

LLL


Theorem 3 Existence Theorem for Laplace Transform

If f (t) is defined and piecewise continuous on every finite interval on the semi-axis and 0t

satisfies for all and some constants M and k, then the Laplace transform

exists for all s > k.

ktf t Me

fL

0t

Existence Theorem for Laplace Transforms (라플라스 변환의 존재정리)

함수 가 영역 상의 모든 유한구간에서 구분적 연속(piecewise tf 0t함수 가 영역 상의 든 유한구간에서 구분적 연속(pcontinuous)인 함수. 어떤 상수 와 에 대해 (너무 빠른 속도로 값이커지지 않음) 모든 에 대해 의 라플라스 변환 가 존재

ktMetf f

k Mks fL tf

Uniqueness (라플라스 변환의 유일성) – 주어진 변환의 역변환은 유일하다.

연속인 두 함수가 같은 변환값을 가지면 두 함수는 동일.값

구간연속인 두 함수가 같은 변환값을 가지면 일부 고립된 점에서 다른 값을 가질지언정 구간내에서는 다를 수 없다

Transform of Derivatives and Integrals. ODEODEs

• Laplace Transform of derivatives

2

0

0 0

f s f f

f s f sf f

L L

L L

00'0 121 nnnnn ffsfsfsf LL

– Ex.1 Let Find tttf sin fL

ttttffttttff sincos2 ,00 ,cossin' ,00 2

2ss 222

2222

2sin 2

ssttffsf

ssf LLLLL


• Ex2. Derive the following formulas cos 22

stL 22 s

22sin

s

tLs


• Laplace Transform of Integral (적분의 라플라스 변환):

sFdfsFdfsFtf -tt 11 1LLL

sFs

df, sFs

dfsFtf 00

LLL

tgsgtgstgtftftg LLLL )0( ),()(

Ex 3 Find the inverse of and 1

1

ggggffg )(),()(

If f(t) satisfy the growth restriction, so does the g(t)

– Ex.3. Find the inverse of and 22 ss

tdss

ts

t

cos11sin1 sin11

2221

221

LL

222 ss

sss 0

3222221 sincos111

ttd

ss

t

L 0 ss


• Differential Equations. Initial Value Problems

10 0' ,0 ,''' KyKytrbyayy – Step 1. Setting up the subsidiary equation (보조방정식의 도출):

rRyY LL sRbYysYaysyYs 00'02

St 2 S l ti f th b idi ti b l b

rRyY LL , sRyyasYbass 0'02

– Step 2. Solution of the subsidiary equation by algebra:Transfer Function (전달함수):

222

41

21

11

abasbass

sQ

solution of the subsidiary equation: 42

sQsRsQyyassY 0'0

– Step 3. Inversion of Y to obtain Yy -1L


• Ex.4 Solve the initial value problem (IVP) 10' ,10 ,'' yytyy

– Step 1 subsidiary equation

,, yyyy

22

22 111 10'0 sYsYysyYs 22 ss

1– Step 2 transfer function1

12

s

Q

1111111 sQQsY

– Step 3 inversion

222222 11111

ssssssQ

sQsY

ttesss

Yty t

sinh1

11

11

21

2111 LLLL


• Laplace Transform Method


• Ex.5 Comparison with the usual method '' 9 , 0 0.16, ' 0 0y y y t y y , ,y y y y y

• Ex.6 Shifted Data Problems

'' 2 , , 2 24 2 4

y y t y y 4 2 4


• Advantages of the Laplace Method

– Solving a nonhomogeneous ODE does not requireSolving a nonhomogeneous ODE does not require first solving the homogeneous ODE.

– Initial values are automatically taken care of.Initial values are automatically taken care of.

– Complicated inputs r(t) (right sides of linear ODEs) can be handled very efficiently as we show in thecan be handled very efficiently, as we show in the next sections.

Unit Step Function. t-shifting

• e.g., nonhomogeneous ODE with periodic external force0'' ' cosmy cy ky F t

짧은 충격!

• Complicated driving force:

0y y y

Unit Step Function or Dirac Delta Function

짧은 충격!

– Single wave, discontinuous input, impulsive force (hammerblows)


• Unit Step Function (단위계단함수) or Heaviside function:

atat

atu10

• Laplace Transform of Unit Step Function:

• Laplace Transform of Unit Step Function:

eatuas

L s

atuL


• On and off of functions ( ) ( )f t u t a


• Time shifting (t-Shifting) –제 2 이동정리 f t F sL

asf t a u t a e F s L

f t F sL

f t a u t a e F sL

1- asf t a u t a e F s L f t a u t a e F s L

dfedfeesFe assasas )(

dtatfesFe

dfedfeesFe

a

stas

00

dtatuatfesFe stas )(0


• Ex. 1 Write the following function using unit step function and find its transform.

2

2 0 1

1 2 2

ttf t t

2 2

cos 2t t

111

21cos

211

21112 2 tuttututtutf


2322

21111

2111

211

21 se

ssstutttut

LL

22

23

222

821

21

821

221

21

21

21 s

esss

tutttut

LL

22 11

21

21sin

21cos

se

stuttut

LL

222 1111122 ssss eeeefL

22323 1822

es

esss

esss

ess

fL


32• Ex.2 Find the inverse transform of 2

3

22

2

22 2

se

se

sesF

sss

t

ssin1

221

L

tte

st

s2

21

21

21 1

LL (제 1이동정리) s-shifting

3322sin111sin1 32 tuettuttuttf t

1t00 (제 2이동정리) t-shifting

333t2 02t1 sin1t0 0

32 t

t

(제 2이동정리) t-shifting

3t 3 32 tet

Short Impulses. Dirac’s Delta Function

• Dirac’s Delta Function or unit impulse function (단위충격함수)

0

t at a

otherwise

1t a dt

0 otherwise 0

( ) ( )g t t a dt g a

atfatkatakatf kkk

0lim

0

1

0

( ) ( )g g

k 00

111

dddfka

1 100

dtatdtk

dtatfa

k


• Laplace Transform of Dirac’s Delta Function

ast a e L

katuatuk

atfk 1k

eeeeatfks

asskaas

11L Put k 0

kseee

ksatfk L Put k 0


• Ex.1 Mass-Spring System Under a Square Wave

00' ,00 ),2(1)(2'3'' yytututryyy

11

100)1(2)1(

ee

ttt

)2(

21

21

21

22)2(2)1(2)2()1(

)()(

tt

eeee

eey

tttt

22



• Ex.2 Hammerblow Response of a Mass-Spring System 00' ,00 ,12'3'' yytyyy ,, yyyyy

seYsYYs 232

s

se

ssssesY

21

11

21

1t1t0 0

1211

tt eeYty L

1t ee

sFeatuatf as- 1 L f



• Ex.4 Damped Forced Vibrations

'' 2 ' 2 ( ) ( ) 10sin 2 0 0 0 1 ' 0 5y y y r t r t t if t and if t y y 2 2 ( ), ( ) 10sin 2 0 0 , 0 1, 0 5y y y r t r t t if t and if t y y

Convolution. Integral Equation

• Convolution (합성곱) ? gfgffg LLLLLL -1

1 , ex) gefgfgffg

t


• Convolution 0

( )t

h t f g t f g t d

• Properties of convolution

0

– Commutative law– Distributive law

fggf

fffDistributive law– Associative law

2121 gfgfggf

vgfvgf

000 ff

• Convolution Theorem( ) 0f f

000 ffUnusual Properties of convolution (특이성질)ff 1

• Convolution Theorem gfgf LLL H FG


• Ex.1 Let Find sassH 1 th

11 ,1 11

at

se

asLL

1111 0

att

aat ea

deeth


• Ex.2 Let Find 2221

wssH

1

th

12 2

1 (sin ) /

sin sin 1 i i ( )t

wt ws w

wt wth t t d

L )cos()cos(21sinsin yxyxyx

20

2

sin sin ( )

1 cos cos(2 )2

t

h t w w t dw w w

wt w wt dw

0

2

2

1 sin( cos2

w

wtw

0

2 )2

tw wtw

2

1 sin cos2

wtt wtw w


• Ex.4 Resonance. – In an undamped mass-spring system, resonance occurs if the In an undamped mass spring system, resonance occurs if the

frequency of the driving force equals the natural frequency of the system (Sec. 2.8).

00' ,00 ,sin'' 020 yytKyy

2' 0 '' 0 ' 0f s f f , f s f sf f L L L L

K

Ks

KY 220

20

)(

1

2 22 2

1 sin1 cos2

wtt wtw ws w

L

tttKty 00020

sincos2


• Integral Equation– Unknown function y(t) appear in an integralUnknown function y(t) appear in an integral

• Ex. 6 A Volterra Integral Equation of the Second Kind

tdtytyt

sin0

ttyy sin

2211sYsY 22 1 ss

6

111 3

424

2 tttyssY

6424 sss

Differentiation and Integration of T f Transforms.

• Differentiation of F(s)

dttffF stL

0

'

fdfdF

dttfefsF

st

st

L

L

0

' tfdtttfedsdFsF st L

1' , 'tf t F s F s tf t L L


• Ex. 1 Derive the following three formulas

Ff 'L

2s미분에 의하여

sFttf 'L

22sin

s

tL 222

2sin

s

sttL미분에 의하여

sin

sttL 222

sin2

s

tL

s 22222 2 sss미분에 의하여 22cos

sstL

222222

2cos

s

s

s

ssttL

222222

미분에 의하여

222

2222

22222

22 1sin1cos

s

ssss

stttL


• Integration of Transform

ttfsdsFsdsF

ttf

ss

~~ ~~ 1LL


222 • Ex.2 Find the inverse transform of 2

22

2

2ln1ln

ss

s

22d222 미분 222222 22lnln

ss

ssss

dsd

2

22

2

2ln1ln

ss

s

미분

Case 1) 변환의 미분이용

ttftss

sssF

sfsF

2cos222' 1ln 222

112

2

LLL

sFttf 'L

sss

tt

tf cos12 sdsFttf

s

~~

L

Case 2) 적분이용

22 1

s

2

1 12

2ln 1 1 coss

g tG s ds t

s t t

L L

1cos2 22 122

tGtg

ssssG

L


• Special Linear ODEs with variable coefficients 0'

dsdYsYysY

dsdty L

sFttf 'L

020'0'' 22 ydsdYssYysyYs

dsdty

dsds

L sdsFttf

s

~~

L

• Ex.3 Laguerre’s Equation. Laguerre Polynomials nnyytty 2100'1'' , , , nnyytty 210 01

22 01 0002

Ysn

dsdYs-snY

dsdYsYysYy

dsdYssY

121 1

11

n

n

ssYds

sn

snds

s-ssn

YdY

,2 ,1 ,!

0 11

netdtd

ne

n, Ytl tn

n

ntn L

Systems of ODEs

• Ex.1 Mixing Problem Involving Two Tanks– Balance Law (mass conservation)Balance Law (mass conservation)

Time rate of change = Inflow/min – Outflow/min8 2' 6y y y 61 1 2

2 1 2

6100 1008 (2 6)'

100 100

y y y

y y y

1 26( 0.08 ) 0.02

0 08 ( 0 08 ) 150

s Y Ys

Y s Y

라플라스 변환

1 lb/gal

2 1 2100 100y y y

1 20.08 ( 0.08 ) 150Y s Y

100 gal water with 150 lb salt at t = 0

100 gal pure water at t = 0

Systems of ODEs

1

2

9 0.48 100 62.5 37.5( 0.12)( 0.04) 0.12 0.04150 12 0 48 100 125 75

sYs s s s s s

2

2150 12 0.48 100 125 75( 0.12)( 0.04) 0.12 0.04

s sYs s s s s s

0.12 0.04100 62 5 37 5t t 0.12 0.041

0.12 0.042

100 62.5 37.5

100 125 75

t t

t t

y e e

y e e

Systems of ODEs

• Ex.3 Model of Two Masses on Springs kyyyy 300100 ''

2 3 YYkkYksYs 라플라스 변환

kyyyy 300 ,100 2121

지배방정식:

21222

1211

3

3

kYYYkksYs

YYkkYksYs

라플라스 변환

2122

1211

''

''

kyyyky

yykkyy

Cramer의 법칙

또는 소거법 적용

ks

kks

s

kks

kskksksY3

3

2

32322222

2

1

tktkYty 3sincos11

1 L 역변환

ks

kks

s

kks

kskksksY3

3

2

32322222

2

2

tktkYty 3sincos2

12 L

Laplace Transform. General Formulas



Table of Laplace Transforms






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Ki-Bok Min PhDBok Min, PhD - Seoul National University

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