中華大學 資訊工程系 fall 2002 chap 4 laplace transform. page 2 outline basic concepts...
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中華大學 資訊工程系
Fall 2002
Chap 4 Laplace Transform Chap 4 Laplace Transform
Page 2
Outline
Basic Concepts
Laplace Transform Definition, Theorems, Formula
Inverse Laplace Transform Definition, Theorems, Formula
Solving Differential Equation
Solving Integral Equation
Page 3
Basic Concepts
DifferentialEquation f(t)
Solution ofDifferential
Equation f(t)
AlgebraEquation F(s)
Solution ofAlgebra
Equation F(s)
Laplace Transform
Inverse Laplace Transform
L{ f(t)} = F(s)
L-1{F(s)} = f(t)
微分方程式 代數方程式
Page 4
Basic Concepts
1)0(,1)0(
423
yy
tyyy
Laplace Transform
Inverse Laplace Transform
L{ f(t)} = F(s)
L-1{F(s)} = f(t)
234
23
23
44)(
sss
sssF
2
1
1
123)(
2
sssssF
tt eet
tfy223
)(
Page 5
Laplace Transform
Definition
The Laplace transform of a function f(t)
is defined as
Converges: L{f(t)} exists
Diverges: L{f(t)} does not exist
dttfetfsF st )()}({)(0 L
Page 6
Laplace Transform
t
e-st
s=1
s=2
s=4s=8
s=0.5
s=0.25
s=0.125
Page 7
Laplace Transform
Example : Find L{ 1 }
Sol:
s
s
e
dte
dte
ts
ts
st
1
1}1{
0
)(
0
)(
0
L
Page 8
Laplace Transform
Example : Find L{ eat }
Sol:
as
as
e
dte
dteee
tas
tas
atstat
1
)(
}{
0
)(
0
)(
0L
Page 9
Laplace Transform
Example 4-2 : Find L{ tt }
Sol:
dttet tstt
0}{L
L{ tt } does not exist
Page 10
Laplace Transform
Exercise 4-1 :
Find
Find
Find
Find
}62{ tL
}{sin tL
}){( 2bat L
}{ bate L
Page 11
Laplace Transform
TheoremsTheorem Description
Definition of Laplace Transform
Linear Property
Derivatives
Integrals
First Shifting Property
Second Shifting Property
0)()()}({ sFdttfetf stL
)()()}()({ sbGsaFtbgtaf L
)0()0()()}({ )1(1)( nnnn ffssFstf L
)(1
})({0
sFs
df
L
)()}({ asFtfeat L
)()}()({ sFeatuatf asL
Page 12
Laplace Transform
TheoremsTheorem Description
Change of Scale Property
Multiplication by tn
Division by t
Unit Impulse Function
Periodic Function
Convolution Theorem
p st
psdttfe
etf
0)(
1
1)}({L
)()()}()({ sGsFtgtf L
aseat )}({L
)()1()}({ )( sFtft nnn L
duuFt
tfs
)(})(
{L
)(1
)}({a
sF
aatf L
Page 13
Linearity of Laplace Transform
Proof:
)()()}({)}(L{)}()(L{ sbGsaFtfbLtfatbgtaf
)()(
)()(
)()(
)()()}()({
00
00
0
sbGsaF
dttgebdttfea
dttbgedttafe
dttbgtafetbgtaf
stst
stst
st
L
Page 14
Application for Linearity of Laplace Transform
22
22
2222
L(sinwt)
L(coswt)
)L(
ws
wws
sws
wi
ws
seiwt
Page 15
First Shifting Theorem
If f(t) has the transform F(s) (where s > k), then eatf(t) has the transform F(s-a), (where s-a > k), in formulas,
or, if we take the inverse on both sides
)()}(L{ asFtfeat
)}({L)( 1 asFtfeat
Page 16
Examples for First Shifting Theorem
22
22
)(sinwt)L(
)(coswt)L(
was
we
was
ase
at
at
Page 17
Excises sec 5.1
#1, #7, #19, #24, #29,#35, #37,#39
Page 18
Laplace of Transform the Derivative of f(t)
Prove
Proof:
)0()()}({ fssFtf L
)0()(
)0()(
)()()(
)()}({
0
00
0
fssF
fdttfes
dttfestfe
dttfetf
st
stst
st
L
Page 19
Laplace transorm of the derivative of any order n
)0(....)0(')0()()(
)0(')0()()0(')'()"()1(21)(
2
nnnnn ffsfsfLsfL
fsffLsffsLfL
Page 20
Examples
Example 1:
Let f(t)=t2, Derive L(f) from L(1) Example 2:
Derive the Laplace transform of cos wt
Page 21
Differential Equations, Initial Value Problem
How to use Laplace transform and Laplace inverse to solve the differential equations with given initial values
10 )0(',)0( kyky )(trbyyay
)()()(0()0()(
)(
1)(
)(
)(
)(
)0()0()(
)()0()0()()(
)()0()0()0(
)}({
)(
2
22
2
2
sQsRsQyyasY
basssQAssume
bass
sR
bass
yyasY
sRyyasYbass
sRbYysYaysyYs
tyYLet
trbyyay
L
Page 22
Example : Explanation of the Basic Concept
Examples
1)0(',1)0(," yytyy
1)0(',1)0(,'2" yyeyyy t
Page 23
Laplace Transform of the Integral of a Function
Theorem : Integration of f(t) Let F(s) be the Laplace transform of f(t). If f(t) is
piecewise continuous and satisfies an inequality of the form (2), Sec. 5.1 , then
or, if we take the inverse transform on both sides of above form
t
sFs
df0
)(1
})(L{
t
sFs
df0
1 )}(1
{L)(
Page 24
An Application of Integral Theorem
Examples
)(,)(
1)L(
22tffind
wssf
)(,)(
1)L(
222tffind
wssf
Page 25
Laplace Transform
Unit Step Function (also called Heaviside’s
Function)
at
atatu
,1
,0)(
Page 26
)2()2(),2()(
,sin5)(
tutftutf
ttf
))6()4(2)1((
,)(
tututuk
ktf
Page 27
Second Shifting Theorem; t-shifting
IF f(t) has the transform F(s), then the “shifted function”
has the transform e-asF(s). That is
atatf
atatuatftf
),(
,0)()()(
~
)()}()({ sFeatuatf asL
Page 28
The Proof of the T-shifting Theorem
Prove
Proof:
)()}()({ sFeatuatf asL
)(
)(
,)(
)1)(()0)((
)()()}()({
0
)(
0
0
sFe
dvvfee
t-avLetdvvfe
dtatfedtatfe
dtatuatfeatuatf
as
svas
a
aa
avs
a
sta st
st
L
Page 29
Application of Unit Step Functions
Note
Find the transform of the function
)()}()({ sFeatuatf asL
s
eatu
as
)}({L
2
2
0
,
sin
0
2
)(
t
t
t
t
tf
Page 30
Example : Find the inverse Laplace transform f(t) of
1
422)(
2
2
2
2
2
s
se
s
e
s
e
ssF
sss
Page 31
Short Impulses. Dirac’s Delta Function
otherwise ,0
,/1)(
katakatfk
ta
Area = 1
ka
k
1
Page 32
Laplace Transform
Unit Impulse Function (also called Dirac Delta
Function)
at
atat
,0
,)(
)( at
ta
Area = 1
aseat )}({L
)(lim)(0
atfat kk
Page 33
Laplace Transform
Example 4-6 : Prove
Proof
aseat )}({L
asks
as
k
kk
kk
kk
ksasskaas
k
k
eks
ee
atfatfat
atfatks
eeee
ksatf
katuatuk
atf
1lim
)}(L{lim)}(limL{)}(L{
)(lim)(
1][
1)}(L{
))](()([1
)(
0
00
0
)(
Page 34
Example
)1()(
)2()1()(
0)0(',0)0(),(2'3"
ttrBcase
tututrAcase
yytryyy
Page 35
Homework
section 5-2
#4, #7, #9, #18, #19 Section 5-3
#3, #6, #17, #28, #29
Page 36
Differentiation and Integration of Transforms
Differentiation of transforms
)()}('{L
)(')}({
)]([)('
)()}({)(
1
0
0
ttfsF
sFttf
dttftesF
dttfetfsF
st
st
L
L
Page 37
Example
?}{ g Lt,sin2t
g(t)
Page 38
Integration of Transform
t
tfsdsF
sdsFt
tf
s
s
)(}~)~({
~)~(})(
{
1-L
L
Page 39
ExampleFind the inverse transform of the function
)1ln(2s
2w
Page 40
Convolution. Integration Equation
Convolution
Properties
dtgftgtft
)()()()(0
fggf
2121 )( gfgfggf
)()( hgfhgf
000 ff
Page 41
Example1
Using the convolution, find the inverse h(t) of
Example 2
Example 3
22 )1()(
ssH
1
3)(
ssH
1
)(,)(
)(2
thfindass
sH
1
Page 42
Laplace Transform
Example 4-7 : Prove
Proof:
)()()}()({ sGsFtgtf L
)()(
)()(
,)()(
)()(
)()()}()({
00
0
)(
0
00
00
sGsF
dvvgedfe
tvLetdvdvgfe
dtdtgfe
dtdtgfetgtf
svs
vs
t st
tst
L
Page 43
Differential Equation
)(trbyyay 0)0(',0)0( yy
tdrtqty
sRsQy
rsRbasssQlet
rybass
0
2
2
)()()(
)().()L(
)L()(),/(1)(
)L()L()(
Page 44
Integration Equations
Example t
dtytty0
)sin()()(
6
}1
{}1
{}{)(
111
1
11
}sin{)}({
sin )sin()()(
3
42
424
2
22
0
tt
ssYty
sss
sY
sY
s
tyttyY
tytdtyttyt
1-1-1- LLL
LL
Page 45
Homeworks
Section 5-4 #1,#13
Section 5-5 #7, #14, #27
Page 46
Laplace Transform
Formulaf(t) F(s) = L {f(t)}
1
,3,2,1, ntn
1, pt p
ate
tcos
tsin
s
1
1
!ns
n
1
)1(
ps
p
as 1
22 s
s
22 s
Page 47
Laplace Transform
Formulaf(t) F(s) = L {f(t)}
teat cos
teat sin
22)(
as
as
tcosh
tsinh
22 s
s
22 s
22)(
as
,2,1, net atn
1, pet atp
1)(
! nas
n
1)(
)1(
pas
p
Page 48
Inverse Laplace Transform
Definition
The Inverse Laplace Transform of a
function F(s) is defined as
dssFei
sFtfia
ia
st )(2
1)}({)(
1-L
Page 49
Inverse Laplace Transform
TheoremsTheorem Description
Inverse Laplace Transform
Linear Property
Derivatives
Integrals
First Shifting Property
Second Shifting Property
)()()}()({ tbgtafsbGsaF -1L
)()}0()0()({ )()1(1 tfffssFs nnnn -1L
t
dfsFs 0
)()}(1
{ 1-L
)()}({ tfeasF at-1L
)()()}({ atuatfsFe as -1L
dssFei
sFtfia
ia
st )(2
1)}({)(
1-L
Page 50
Inverse Laplace Transform
TheoremsTheorem Description
Change of Scale Property
Multiplication by tn
Division by t
Unit Impulse Function
Unit Step Function
Convolution Theorem
)(}{ ats
e as
UL 1-
)()()}()({ tgtfsGsF -1L
)(}{ ate as -1L
)()1()}({ )( tftsF nnn -1L
t
tfduuF
s
)(})({
1-L
)()}({ atafa
sF 1-L
Page 51
Inverse Laplace Transform
FormulaF(s) f(t) = L -1{F(s)}
1
ate
tcos
tsin
s
1
as 1
22 s
s
22 s
!n
tn
,2,1,0,1
1 n
sn
t2
1
s
Page 52
Inverse Laplace Transform
FormulaF(s) f(t) = L -1{F(s)}
22)(
as
as
22 s
s
22 s
22)(
as
1)(
! nas
n
1)(
)1(
pas
p
teat cos
teat sin
tcosh
tsinh
,2,1, net atn
1, pet atp
Page 53
Solving Differential Equation
)()()(0()0()(
)(
1)(
)(
)(
)(
)0()0()(
)()0()0()()(
)()0()0()0(
)}({
)(
2
22
2
2
sQsRsQyyasY
basssQAssume
bass
sR
bass
yyasY
sRyyasYbass
sRbYysYaysyYs
tyYLet
trbyyay
L