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Laplace Transforms. Math 2306 Dr. Dillon SPSU Mathematics Department. What Are Laplace Transforms?. A Laplace transform is a type of integral transform. Plug one function in. Get another function out. The new function is in a different domain. When. is the Laplace transform of. Write.

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laplace transforms

Laplace Transforms

Math 2306

Dr. Dillon

SPSU

Mathematics Department

slide3

A Laplace transform is a type of

integral transform.

Plug one function in

Get another function out

The new function is in a different domain.

slide4

When

is the Laplace transform of

Write

a calculation
A Calculation

Let

This is called the unit step function or

the Heaviside function.

It’s handy for describing functions that

turn on and off.

slide7

1

c

t

The Heaviside Function

slide8

Remember that

is zero until

Calculating the Laplace transform of the

Heaviside function is almost trivial.

then it’s one.

slide10

We can use Laplace transforms to turn an

initial value problem

Solve for y(t)

into an algebraic problem

Solve for Y(s)

slide11

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

Asawtooth function

slide12

Laplace transform

I.V.P.

Algebraic Eqn

slide14

If you solve the algebraic equation

and find the inverse Laplace transform of the solution, Y(s), you have the solution to the I.V.P.

slide15

Inverse

Laplace

transform

Algebraic Expression

Soln. to IVP

slide17

Thus

is the solution to the I.V.P.

slide19

You need several nice properties of Laplace

transforms that may not be readily apparent.

First, Laplace transforms, and inverse

transforms, are linear :

for functions f(t), g(t), constant c, and

transforms F(s), G(s).

slide20

there is a very simple relationship

between the Laplace transform of a given

function and the Laplace transform of that

function’s derivative.

Second,

These show when we apply differentiation

by parts to the integral defining the transform.

slide21

Now we know there are rules that let

us determine the Laplace transform

of an initial value problem, but...

slide23

First you must know that Laplace transforms

are one-to-one on continuous functions.

In symbols

when f and g are continuous.

That means that Laplace transforms are

invertible.

slide25

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.

slide27

Don’t use them...

unless you really have to.

slide29

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.

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
The Dirac Delta Function

This so-called quasi-function was created

by P.A.M. Dirac, the inventor of quantum

mechanics.

People use this thing all the time. You

need to be familiar with it.

slide34

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

Initial conditions at a point other than zero

will not do.

slide36
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