Laplace TransformsLaplace Transforms

Math IIMath IIMrs Suchitra PattnaikMrs Suchitra Pattnaik

Mr parsuram SahuMr parsuram SahuGIFTGIFT

Mathematics DepartmentMathematics Department

What Are Laplace What Are Laplace Transforms?Transforms?

The The Laplace transformLaplace transform is a widely is a widely used used integral transformintegral transform. It has many . It has many

important applications in mathematics, important applications in mathematics, physics, engineering, and probability physics, engineering, and probability

theory. theory.

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 A Laplace transform is an example of an an improper integralimproper integral : one of its limits : one of its limits is infinite.is infinite.

0 0

( ) lim ( )h

s t s t

he f t dt e f t dt

Define

A CalculationA Calculation

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

To What End Does One To What End Does One Use Laplace Transforms?Use Laplace Transforms?

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

Then What?Then What?

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

How Do You Transform an How Do You Transform an Differential Equation?Differential Equation?

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...

How Do You Find Inverse How Do You Find Inverse Laplace Transforms? Laplace Transforms?

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.

Inverse Laplace Inverse Laplace TransformsTransforms

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.

Why Use Such Dangerous Why Use Such Dangerous Machines?Machines?

Don’t use them...

unless you really have to.

When Might You 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?Impulse?

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.

The Dirac Delta FunctionThe Dirac Delta Function

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.

The Laplace Transform of The Laplace Transform of thetheDirac Delta FunctionDirac Delta Function

{ ( )} 1/ a sL t a e

Beware!Beware!

Use it Only when you need to Use it Only when you need to be expertisebe expertise

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.

What Do We Expect You to What Do We Expect You to Be Able to Do?Be Able to Do?

• Know the Know the definitiondefinition of the Laplace of the Laplace transformtransform

• Know the Know the propertiesproperties of the Laplace of the Laplace transform transform

• Know that the Know that the inverseinverse Laplace Laplace transform is an improper integraltransform is an improper integral

• Know Know whenwhen you should use a Laplace you should use a Laplace transform on a differential equationtransform on a differential equation

• Know Know whenwhen you should you should notnot use a use a Laplace transform on a differential Laplace transform on a differential equationequation

Be able to solve IVPs using Be able to solve IVPs using Laplace transforms…Laplace transforms…

When Appropriate