Transcript Laplace Transforms

Laplace Transforms
Math II
Mrs Suchitra Pattnaik
Mr parsuram Sahu
GIFT
Mathematics Department
What Are Laplace Transforms?
The Laplace transform is a widely used
integral transform. It has many important
applications in mathematics, physics,
engineering, and probability theory.
A Laplace transform is a type of
integral transform.

 s t
e

0
f (t ) dt  F ( s )
Plug one function in
Get another function out
The new function is in a different domain.
When


0
e
 st
f (t ) dt  F ( s )
F ( s ) is the Laplace transform of f (t ).
Write
L  f (t )  F (s),
L  y (t )  Y ( s),
L  x(t )  X ( s), etc.
A Laplace transform is an example of an
improper integral : one of its limits is
infinite.
Define

e

0
 s t
h
f (t )dt  lim  e
h 
0
 s t
f (t )dt
A Calculation
0 if t  c
Let u (t  c)  
1 if t  c
This is called the unit step function or
the Heaviside function.
It’s handy for describing functions that
turn on and off.
0 if t  c
u (t  c)  
1 if t  c
1
c
The Heaviside Function
t
Calculating the Laplace transform of the
Heaviside function is almost trivial.

h
L u (t  c)   e u (t  c)dt  lim  e
 s t
h 
0
lim
h 
1
s
e
 s t h
c
 lim
h 
1
s
(e
 s h
e
 s c
 s t
dt 
c
)e
Remember that u (t  c) is zero until
then it’s one.
 s c
s
t  c,
To What End Does One Use
Laplace Transforms?
We can use Laplace transforms to turn an
initial value problem
y " 3 y ' 4 y  t  u (t  1)
y (0)  1, y '(0)  2
Solve for y(t)
into an algebraic problem
Y (s)*( s  3s  4)  ( s  1) 
2
Solve for Y(s)
s 1
s 2 es
Laplace transforms are particularly effective
on differential equations with forcing functions
that are piecewise, like the Heaviside function,
and other functions that turn on and off.
1
1
t
A sawtooth function
I.V.P.
Laplace transform
Algebraic Eqn
Then What?
If you solve the algebraic equation
( s  1)  ( s  e  1)  e
Y ( s) 
2
2
s  ( s  3s  4)
2
s
s
and find the inverse Laplace transform of
the solution, Y(s), you have the solution to
the I.V.P.
Algebraic Expression
Inverse
Laplace
transform
Soln. to IVP
The inverse Laplace transform of
( s  1)  ( s  e  1)  e
Y ( s) 
2
2
s  ( s  3s  4)
2
s
y (t )  u (t  1)(  e +
t
2
5e
3e4
80
s
t 4
 (e )  t  )
t 4
 u (t )(  e   (e ) )
2
5
t
is
3
5
1
4
3
16
Thus
y (t )  u (t  1)(  e +
t
2
5e
3e4
80
t 4
 (e )  t  )
t 4
 u (t )(  e   (e ) )
2
5
t
3
5
is the solution to the I.V.P.
y " 3 y ' 4 y  t  u (t  1)
y (0)  1, y '(0)  2
1
4
3
16
How Do You Transform an
Differential Equation?
You need several nice properties of Laplace
transforms that may not be readily apparent.
First, Laplace transforms, and inverse
transforms, are linear :
L cf (t )  g (t ) = c L  f (t )+ L  g (t ) ,
L1 cF ( s)  G ( s) = c L1  F ( s)+ L-1 G ( s)
for functions f(t), g(t), constant c, and
transforms F(s), G(s).
Second, there is a very simple relationship
between the Laplace transform of a given
function and the Laplace transform of that
function’s derivative.
L  f '(t ) = s  L  f (t )  f (0), and
L  f ''(t ) = s  L  f (t )  s  f (0)  f '(0)
2
These show when we apply differentiation
by parts to the integral defining the transform.
Now we know there are rules that let
us determine the Laplace transform
of an initial value problem, but...
How Do You Find Inverse
Laplace Transforms?
First you must know that Laplace transforms
are one-to-one on continuous functions.
In symbols
L  f (t ) = L  g (t )  f (t )  g (t )
when f and g are continuous.
That means that Laplace transforms are
invertible.
Inverse Laplace Transforms
If
then
1
L
L  f (t )  F (s),
L F (s)  f (t ),
-1
F (s) 
1
2 i

c i
c  i
st
where
e F (s) ds
An inverse Laplace transform is an improper
contour integral, a creature from the world
of complex variables.
That’s why you don’t see them naked very
often. You usually just see what they yield,
the output.
In practice, Laplace transforms and inverse
Laplace transforms are obtained using tables
and computer algebra systems.
Why Use Such Dangerous
Machines?
Don’t use them...
unless you really have to.
When Might You Have To?
When your forcing function is a piecewise,
periodic function, like the sawtooth function...
Or when your forcing function is an impulse,
like an electrical surge.
Impulse?
An impulse is the effect of a force that acts
over a very short time interval.
A lightning strike creates an electrical
impulse.
The force of a major leaguer’s bat
striking a baseball creates a mechanical
impulse.
Engineers and physicists use the Dirac
delta function to model impulses.
The Dirac Delta Function
This so-called quasi-function was created
by P.A.M. Dirac, the inventor of quantum
mechanics.

 (t  a)  0 when t  a and   (t  a)dt  1
0
People use this thing all the time. You
need to be familiar with it.
The Laplace Transform of the
Dirac Delta Function
L{ (t  a)}  1/ e
a s
Beware!
Use it Only when you need to be
expertise
Laplace transforms have limited appeal.
You cannot use them to find general solutions
to differential equations.
You cannot use them on initial value problems
with initial conditions different from
y(0)  c1 , y '(0)  c2 , etc.
Initial conditions at a point other than zero
will not do.
What Do We Expect You to Be
Able to Do?
• Know the definition of the Laplace
transform
• Know the properties of the Laplace
transform
• Know that the inverse Laplace
transform is an improper integral
• Know when you should use a Laplace
transform on a differential equation
• Know when you should not use a
Laplace transform on a differential
equation
Be able to solve IVPs using
Laplace transforms…
When Appropriate