Basic ConceptsDifferential Equation ft Solution of Differential Equation ft Algebra Equation Fs Solution of Algebra Equation Fs... Laplace transorm of the derivative of any order n 0 .
Trang 1Chap 4 Laplace Transform
Trang 2■ Basic Concepts
■ Laplace Transform
■ Definition, Theorems, Formula
■ Inverse Laplace Transform
■ Definition, Theorems, Formula
■ Solving Differential Equation
Trang 3Basic Concepts
Differential Equation f(t)
Solution of Differential Equation f(t)
Algebra Equation F(s)
Solution of Algebra Equation F(s)
Trang 4Basic Concepts
1 )
0 ( ,1 )
0
(
4 2
′
−
′′
y y
t y
y y
4
23
2 3
4
4 )
(
s s
s
s
s s
2
3 + + − + −
=
t f
y = ( )
Trang 5f e
t f s
Trang 7Laplace Transform
■ Example : Find L { 1 }
Sol:
s e
dt e
dt e
t s
t s st
1
1 }
1 {
0
)(0
)(0
Trang 8Laplace Transform
■ Example : Find L { eat }
Sol:
a s
e
dt e
dt e
e e
t a s
t a s
at st at
} {
)(
0
)(
0L
Trang 9∴ L { tt } does not exist
Trang 10{ t + L
}
{sin t π
L
} ) {( at + b 2
L
Trang 11( )}
( )
( { af t + bg t = aF s + bG s
L
) 0 ( )
0 ( )
( )}
( { f (n) t = snF s − sn−1f − − f (n−1)
L
) (
1 }
) (
)}
( { eat f t = F s − a
L
Trang 121 ( )}
(
L
du u
F t
)
( { L
) (
1 )}
(
{
a
s F a
at
L
Trang 13Linearity of Laplace
Transform
Proof:
) ( )
( )}
( { )}
( L{
)}
( )
(
L{ af t + bg t = a f t + bL f t = aF s + bG s
) ( )
(
) ( )
(
) ( )
(
) ( )
( )}
( )
( {
0 0
0 0
0
s bG s
aF
dt t g e
b dt
t f e a
dt t bg e
dt t af e
dt t
bg t
af e
t bg t
af
st st
st st
Trang 14Application for Linearity of Laplace
Transform
2 2
2 2
2 2
2 2
L(sinwt)
L(coswt)
) L(
w s
w
w s
s
w s
w i
w s
=
Trang 15First 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 sidesL{ eat f ( t )} = F ( s − a )
)}
( { L )
f
Trang 16Examples for First Shifting Theorem
2 2
2 2
) (
sinwt) L(
) (
coswt) L(
w a
s
w e
w a
s
a
s e
Trang 17Excises sec 5.1
■ #1, #7, #19, #24, #29,#35, #37,#39
Trang 18Laplace of Transform the
Derivative of f(t)
■ Prove
Proof:
) 0 ( )
( )}
( { f ′ t = sF s − f
L
) ( )
( )
(
) ( )}
( {
00
0
dt t
f e
s t
f e
dt t
f e
t f
st st
Trang 19Laplace transorm of the derivative of
any order n
) 0 (
) 0 ( ' )
0 ( )
( )
(
) 0 ( ' )
0 ( )
( )
0 ( ' )
' ( )
"
(
) 1 ( 2
1 )
n n
f
L
f sf
f L s f
f sL f
L
Trang 21Differential Equations, Initial Value
Problem
■ How to use Laplace transform and Laplace inverse to solve the
differential equations with given initial values
1
0, ' ( 0 ) )
y a
1)
(
)(
)
()
(
)0()
0()(
)()
0()
0()(
)(
)()
0()
0()
0(
)}
({
)(
2 2
2 2
s Q Assume
b as
s
s
R b
as s
y y
a s
Y
s R y
y a s
Y b as
s
s R bY
y sY
a y
sy Y
s
t y Y
Let
t r by
y a y
=
++
++
+
′+
+
=+
+
⇒
=+
−+
′+
′
L
Trang 22Example : Explanation of the Basic
Concept
■ Examples
1 )
0 ( ' ,
1 )
0 ( ,
y
1 )
0 ( ' , 1 )
0 ( ,
' 2
" + y + y = e − y = − y =
Trang 23Laplace 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
Trang 24An Application of Integral Theorem
■ Examples
) (
, ) (
1 )
w s
, ) (
1 )
w s
s
f
+
=
Trang 25t a
t
u
, 1
,
0 )
(
Trang 26) 2 (
) 2 (
), 2 (
)
(
, sin
5 )
f t
u t
f
t t
f
)) 6 (
) 4 (
2 )
1 (
(
, )
(
− +
u t
u
k
k t
f
Trang 27Second Shifting Theorem; t-shifting
■ IF f(t) has the transform F(s), then the “shifted function”
has the transform e-asF(s) That is
a t
f
a
t a
t u a t
f t
f
), (
,
0 )
( ) (
) (
~
) ( )}
( ) (
{ f t − a u t − a = e−asF s
L
Trang 28The Proof of the T-shifting
Theorem
■ Prove
Proof:
) ( )}
( ) (
{ f t − a u t − a = e−asF s
L
, ) (
) 1 )(
( )
0 )(
(
) (
) (
)}
( ) (
{
) ( 0
0
t-a v
Let dv
v f e
dt a
t f e dt
a t f e
dt a t u a t f e a
t u a t f
a a a
a v s
a
st
a st st
Trang 29Application of Unit Step Functions
■ Note
■ Find the transform of the function
) ( )}
( ) (
t u
π
2 2
0 , sin 0
2 )
(
t t
t t
t f
Trang 30■ Example :
Find the inverse Laplace transform f(t) of
1
4 2
2 )
2 2
e s
e s
s F
s s
Trang 31Short Impulses Dirac’s Delta
, 0
, /
1 )
f k
t a
Trang 32t a
t
, 0
, )
(
) (
lim )
(
0 f t a a
Trang 33t − )} = −
( { δ L
ks
k k
k k
k k
ks as
s k a
as k
k
e
a t
f a
t f a
t
a t
f a
t
ks
e e
e
e ks
a t
f
k a
t u a
t
u k
a t
lim )}
( lim
L{
)}
( L{
) (
lim )
(
1 ]
[
1 )}
( L{
))]
( (
) (
[
1 )
(
0 0
0
) (
δ δ
Trang 34■ Example
) 1 (
) (
) 2 (
) 1 (
) (
0 )
0 ( ' , 0 )
0 ( ),
( 2
' 3
+
t t
r B
case
t u t
u t
r A
case
y y
t r y
y y
δ
Trang 36Differentiation and Integration of
Transforms
■ Differentiation of transforms
) ( ' )}
( {
)]
( [
) ( '
) ( )}
( {
) (
0
0
s F
t tf
dt t
f t e
s F
dt t
f e
t f
s F
Trang 37■ Example
? }
Trang 38Integration of Transform
t
t
f s
d s
F
s d s
F t
t f
s
s
)
( }
~ )
~ ( {
~ )
~ ( }
)
( {
Trang 39■ Example
Find the inverse transform of the function
) 1
Trang 40f t
g t
g
21
2
Trang 42Laplace Transform
■ Example 4-7 : Prove
Proof:
) ( ) ( )}
( )
( { f t ∗ g t = F s G s
L
, )
( ) (
) (
) (
) (
) ( )}
( )
( {
0
) (
0
0 0
0 0
t v
Let dv
d v g f
e
dt d
t g f
e
dt d
t g f
e t
g t
f
v s
t st
t st
τ
τ τ
τ
τ τ
τ
τ
L
Trang 43Differential Equation
)
(t
r by
y a
+
=
= +
+
t
d r
t q t
y
s R s
Q y
r s
R b
as s
s Q let
r y
b as
s
0
2 2
) ( ) (
) (
) ( ).
( )
L(
) L(
) ( ),
/(
1 )
(
) L(
) L(
) (
τ τ
τ
Trang 44Integration Equations
0 ( ) sin( ) )
1 1
1
1
1
1
} sin {
)}
( {
sin )
sin(
) ( )
(
2
2 2
0
s Y
s
Y s
t y
t t
y Y
t y
t d
t y
t t
=
=
∗ +
=
− +
= ∫
L L
τ τ
τ
Trang 461 , p > −
1
)1
a
s −
1
Trang 47)( − +ω
−
a s
a s
, 2 , 1 , n =
Γ p
Trang 48Inverse Laplace Transform
■ Definition
The Inverse Laplace Transform of a function F(s) is defined as
ds s
F
e i
s F t
i a
2
1 )}
( {
)
π
1-
L
Trang 49Inverse Laplace Transform
( )}
( )
( { aF s + bG s = af t + bg t
-1L
) ( )}
0 ( )
0 ( )
( { snF s − sn−1f − − f (n−1) = f (n) t
-1L
∫
= t f d s
L
) ( )}
( { F s − a = eat f t
-1
L
ds s F
e i
s F t
i a
st ( ) 2
1 )}
( { )
( = = ∫ −+∞∞
π
1 -
L
Trang 50Inverse Laplace Transform
1 ( )}
u
Fs
)
( }
) (
1 -
L
) ( )}
L
Trang 51Inverse Laplace Transform
s
Trang 52Inverse Laplace Transform
■ Formula
F(s) f(t) = L -1{F(s)}
2 2
)( − +ω
−
a s
a s
2
2 −ω
s s
Trang 53Solving Differential Equation
1 )
(
) (
)
( )
(
) 0 ( )
0 ( ) (
) ( )
0 ( )
0 ( ) (
) (
) ( )
0 ( )
0 ( )
0 (
)}
( {
) (
2
2 2
2 2
b as
s
s Q Assume
b as
s
s
R b
as s
y y
a
s Y
s R y
y a s
Y b as
s
s R bY
y sY
a y
sy Y
s
t y Y
Let
t r by
y a y
+
′ +
+
=
⇒
+ +
=
+ +
+ +
+
′ +
+
= +
+
⇒
= +
− +
′ +
′′
L