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.
Plug one function in
Get another function out
The new function is in a different domain.
is the Laplace transform of
This is called the unit step function or
the Heaviside function.
It’s handy for describing functions that
turn on and off.
The Heaviside Function
is zero until
Calculating the Laplace transform of the
Heaviside function is almost trivial.
then it’s one.
initial value problem
Solve for y(t)
into an algebraic problem
Solve for Y(s)
on differential equations with forcing functions
that are piecewise, like the Heaviside function,
and other functions that turn on and off.
and find the inverse Laplace transform of the solution, Y(s), you have the solution to the I.V.P.
Soln. to IVP
is the solution to the I.V.P.
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).
between the Laplace transform of a given
function and the Laplace transform of that
These show when we apply differentiation
by parts to the integral defining the transform.
us determine the Laplace transform
of an initial value problem, but...
are one-to-one on continuous functions.
when f and g are continuous.
That means that Laplace transforms are
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.
unless you really have to.
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.
A lightning strike creates an electrical
The force of a major leaguer’s bat
striking a baseball creates a mechanical
Engineers and physicists use the Dirac delta function to model impulses.
This so-called quasi-function was created
by P.A.M. Dirac, the inventor of quantum
People use this thing all the time. You
need to be familiar with it.
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.