Transcript Chapter 4: The Laplace Transform

Chapter 4: The Laplace Transform
4.1
4.2
Definition of the Laplace transform
Inverse Transform, Transforms of Derivatives
4.3
4.4
4.5
Translation Theorems
Additional Properties
Dirac Delta Function
Introduction
The Laplace transform studied in this chapter is
an invaluable tool in solving problems such as
these ( section 4.4 )
4.1 Definition of the Laplace Transform
f ( x, y )
Leads to

2
1
Function of two variables
f ( x, y)dx  g ( y)
Function of one variable
Improper integral


0
b
k ( s, t ) f (t )dt  lim [  k ( s, t ) f (t )dt ]
b
0
4.1 Definition of the Laplace Transform
Note:
Definition 4.1 Laplace Transform
1) L[f(t)] is a function of s
Let f be a function defined for t  0. Then the integral 2) Improper integral

L[f(t)]   est f(t)dt
0
is said to be the Laplace transform of f . provided
the integral converges.
L[..]
f (t )
Example 1: Using Definition 4.1
Evaluate
L[1]
F (s )
4.1 Definition of the Laplace Transform

L[f(t)]   e f(t)dt
st
0
Example 2: Using Definition 4.1
Evaluate
L[t]
Example 3: Using Definition 4.1
Evaluate
Example 4: Using Definition 4.1
Evaluate
L[ sin( 2t ) ]
3t
L[e ]
4.1 Definition of the Laplace Transform
L[..]
is a Linear Transform
d
d
d
[f ( x)  βg(x)]  α
[f(x)]  β
[g(x)]
dx
dx
dx
1) derivative
2) integral
 [f ( x)  βg(x)]dx  α  f(x)dx  β  g(x)dx
3) Laplace L[f (t )  βg(t)]  αL[f(t)]  βL[g(t)]
Example :
Evaluate
L[ 1  5t]
4.1 Definition of the Laplace Transform
Theorem 4.1 Transforms of Some Basic Functions
1
a) L[1] 
s
n!
b) L[t ]  n1 ,
s
n  1,2,3,...
n
k
s2  k 2
k
f) L[sinh (kt)]  2
s  k2
d) L[sin (kt)] 
19) f (t )  2t 4
23) f (t )  t  6t  3
2
25) f (t )  (t  1)
HW
3
29) f (t )  (1  e 2t ) 2
1
c) L[e ] 
sa
at
s
s2  k 2
s
f) L[cosh(kt)]  2
s  k2
e) L[cos(kt)] 
35) f (t )  et sinh( t )
36) f (t )  et cosh( t )
4.1 Definition of the Laplace Transform
Example 5: Piecewise defined function
Evaluate
L[ f (t )]
0,
f (t )  
2,
0t 3
t 3
4.1 Definition of the Laplace Transform
Definition 4.2 Exponential Order
A function f is said to be of exponentia l order c
if there exists constants c, M  0, and T  0
such that
f (t )  Me ct
g (t )  Mect
for all t  T
t
Example : f (t )  t , f (t )  e ,
are all of exponentia l order c  1.
Example : f (t )  et
is not of exponentia l order
2
f (t )
f (t )  2 cos t
T
4.1 Definition of the Laplace Transform
Theorem 4.2 Sufficient Condition for Existence
1) f (t ) is piecewise continuous on [0, )
2) f (t ) is of exponentia l order c for t  T

L[ f (t )] exists for s  0
4.1 Definition of the Laplace Transform
39) One definition of the gamma function is given by
the improper integral

( )   t  1et dt ,   0
0
a )(  1)  ( )
1
c ) ( )  
2
(  1)
b) L[t ] 
s 1

  -1
40) Use Problem 39 to find the Laplace transform of
a) f (t )  t
1

2
b) f (t )  t
1
2
HW 3
c) f (t )  t 2
Matlab and Mathematica
Laplace
Transofrm
f (t )  t
4
L[t ]
Laplace Transofrm
L[sin( t )]
4
Matlab
Command
syms t
f = t^4
laplace(f)
Return
s24/s^5
Mathematica Command
LaplaceTransform[Sin[t],t,s]