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A Laplace transform is a type of integral transform.
Plug one function in0
s te dt
( )f t
Get another function out
( )F s
The new function is in a different domain.
( )F s is the Laplace transform of ( ).f t
Write ( ) ( ),f t F sL
0 s te dt
( )f t ( )F sWhen
( ) ( ),
( ) ( ), etc.
y t Y s
x t X s
L
L
A Laplace transform is an example of an improper integral : one of its limits is infinite.
0 0
( ) lim ( )h
s t s t
he f t dt e f t dt
Define
Let0 if
( )1 if
t cu t c
t c
This is called the unit step function orthe Heaviside function.
It’s handy for describing functions that turn on and off.
c
1
t
0 if ( )
1 if
t cu t c
t c
The Heaviside Function
0
1 1
( ) ( ) lim
lim lim ( )
hs t s t
hc
h s cs t s h s cs sch h
u t c e u t c dt e dt
ee e e s
L
Calculating the Laplace transform of theHeaviside function is almost trivial.
Remember that ( )u t c is zero untilthen it’s one.
,t c
We can use Laplace transforms to turn an initial value problem
" 3 ' 4 ( 1)
(0) 1, '(0) 2
y y y t u t
y y
into an algebraic problem
2
2 1( )*( 3 4) ( 1) ss
s eY s s s s
Solve for y(t)
Solve for Y(s)
1
1
A sawtooth function
t
Laplace transforms are particularly effectiveon differential equations with forcing functionsthat are piecewise, like the Heaviside function,and other functions that turn on and off.
I.V.P.
Laplace transform
Algebraic Eqn
If you solve the algebraic equation
2
2 2
( 1) ( 1)( )
( 3 4)
s ss s e eY s
s s s
and find the inverse Laplace transform of the solution, Y(s), you have the solution to the I.V.P.
Algebraic Expression
Soln. to IVP
Inverse Laplace transform
The inverse Laplace transform of
is
4 43 32 15 80 4 16
4325 5
( ) ( 1)( + ( ) )
( )( ( ) )
t tee
t t
y t u t e e t
u t e e
2
2 2
( 1) ( 1)( )
( 3 4)
s ss s e eY s
s s s
4 43 32 15 80 4 16
4325 5
( ) ( 1)( + ( ) )
( )( ( ) )
t tee
t t
y t u t e e t
u t e e
is the solution to the I.V.P.
" 3 ' 4 ( 1)
(0) 1, '(0) 2
y y y t u t
y y
Thus
You need several nice properties of Laplace transforms that may not be readily apparent.
First, Laplace transforms, and inversetransforms, are linear :
1 1 -1
( ) ( ) ( ) ( ) ,
( ) ( ) ( ) ( )
cf t g t c f t g t
cF s G s c F s G s
L = L +L
L = L +L
for functions f(t), g(t), constant c, andtransforms F(s), G(s).
there is a very simple relationshipbetween the Laplace transform of a given function and the Laplace transform of that function’s derivative.
2
'( ) ( ) (0),
''( ) ( ) (0) '(0)
f t s f t f
f t s f t s f f
L = L
L = L
and
These show when we apply differentiationby parts to the integral defining the transform.
Second,
Now we know there are rules that letus determine the Laplace transformof an initial value problem, but...
First you must know that Laplace transforms are one-to-one on continuous functions.
In symbols
( ) ( ) ( ) ( )f t g t f t g t L = L
when f and g are continuous.
That means that Laplace transforms are invertible.
If ( ) ( ),f t F sL
1 12( ) ( )
c i s ti c i
F s e F s ds
L
then -1 ( ) ( ),F s f tL where
An inverse Laplace transform is an impropercontour integral, a creature from the worldof 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 inverseLaplace transforms are obtained using tablesand computer algebra systems.
Don’t use them...
unless you really 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.
An impulse is the effect of a force that acts over a very short time interval.
Engineers and physicists use the Dirac delta function to model impulses.
A lightning strike creates an electricalimpulse.The force of a major leaguer’s bat
striking a baseball creates a mechanicalimpulse.
This so-called quasi-function was createdby P.A.M. Dirac, the inventor of quantummechanics.
0( ) 0 ( ) 1t a t a t a dt
when and
People use this thing all the time. Youneed to be familiar with it.
{ ( )} 1/ a sL t a e
Laplace transforms have limited appeal.
You cannot use them to find general solutionsto differential equations.
You cannot use them on initial value problemswith initial conditions different from
1 2(0) , '(0) ,y c y c etc.
Initial conditions at a point other than zerowill not 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
When Appropriate
