Differential Equations
© 2007 Paul Dawkins 55 http://tutorial.math.lamar.edu/terms.aspx
Dirac Delta Function
When we first introduced Heaviside functions we noted that we could think of them as switches
changing the forcing function, g(t), at specified times. However, Heaviside functions are really
not suited to forcing functions that exert a “large” force over a “small” time frame.
Examples of this kind of forcing function would be a hammer striking an object or a short in an
electrical system. In both of these cases a large force (or voltage) would be exerted on the system
over a very short time frame. The Dirac Delta function is used to deal with these kinds of forcing
function.
Dirac Delta Function
There are many ways to actually define the Dirac Delta function. To see some of these
definitions visit Wolframs MathWorld. There are three main properties of the Dirac Delta
function that we need to be aware of. These are,
1.
(
)
0,
tata
d
-=¹
2.
( )
1,0
a
atadt
e
ede
+
-
-=>
ò
3.
() ( ) ()
,0
a
afttadtfa
e
ede
+
-
-=>
ò
At
ta
=
the Dirac Delta function is sometimes thought of has having an “infinite” value. So, the
Dirac Delta function is a function that is zero everywhere except one point and at that point it can
be thought of as either undefined or as having an “infinite” value.
Note that the integrals in the second and third property are actually true for any interval
containing
ta
=
, provided it’s not one of the endpoints. The limits given here are needed to
prove the properties and so they are also given in the properties. We will however use the fact
that they are true provided we are integrating over an interval containing
ta
=
.
This is a very strange function. It is zero everywhere except one point and yet the integral of any
interval containing that one point has a value of 1. The Dirac Delta function is not a real function
as we think of them. It is instead an example of something called a generalized function or
distribution.
Despite the strangeness of this “function” it does a very nice job of modeling sudden shocks or
large forces to a system.
Before solving an IVP we will need the transform of the Dirac Delta function. We can use the
third property above to get this.
( )
{ }
( )
0
provided 0
stas
tatadta
dd
¥
--
-=-=>
òeeL
Note that often the second and third properties are given with limits of infinity and negative
infinity, but they are valid for any interval in which
ta
=
is in the interior of the interval.
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