**The Dirac Delta Function in Differential Equations**

The delta ”function” is often introduced into differential equations courses during the section on Laplace transforms. Of course the delta

”function” isn’t a function at all but rather what is known as a ”distribution” (more on this later)

A typical introduction is as follows: if one is working in classical mechanics and one applies a force to a constant mass at time then one can define the impulse of over an interval by where is the velocity. So we can do a translation to set and then consider a unit impulse and vary

according to where is; that is, define

Then is the force function that produces unit impulse for a given

Then we wave our hands and say (this is a great reason to introduce the concept of the limit of functions in a later course) and then argue that for all functions that are continuous over an interval containing 0,

.

The (hand waving) argument at this stage goes something like: ”the mean value theorem for integrals says that there is a

between and such that Therefore as by continuity. Therefore we can define the Laplace transform ”

**Illustrating what the delta ”function” does**.

I came across this example by accident; I was holding a review session for students and asked for them to give me a problem to solve.

They chose (I can remember what and were but they aren’t important here as we will see) with initial conditions

So using the Laplace transform, we obtained:

But with this reduces to

In other words, we have the ”same solution” as if we had with .

So that might be a way to talk about the delta ”function”; it is exactly the ”impulse” one needs to ”cancel out” an initial velocity of or,

equivalently, to give an initial velocity of and to do so instantly.

**Another approach to the delta function **

Though it is true that for all and

by design, note that fails to be continuous at and at .

So, can we obtain the delta ”function” as a limit of other functions that are everywhere continuous and differentiable?

In an attempt to find such a family of functions, It is a fun exercise to look at a limit of normal density functions with mean zero:

. Clearly for all

and .

Here is the graph of some of these functions: we use , and respectively.

**Calculating the Laplace transform**

Do some algebra to combine the exponentials, complete the square and do some algebra to obtain:

Now do the usual transformation to the standard normal random variable via

And we obtain:

for all . Note: assume and that is shorthand for the usual probability distribution function.

Now if we take a limit as we get on the right hand side.

Hence, one way to define is as . This means that while

is off by a factor of 2,

as desired.

Since we now have derivatives of the functions to examine, why don’t we?

which is zero at for all But the behavior of the derivative is interesting: the derivative is at its minimum at and at its maximum at (as we tell our probability students: the standard deviation is the distance from the origin to the inflection points) and as the inflection points get closer together and the second derivative at the

origin approaches which can be thought of as an instant drop from a positive velocity at .

Here are the graphs of the derivatives of the density functions that were plotted above; note how the part of the graph through the origin becomes more vertical as the standard deviation approaches zero.

[…] wrote this post for my college math teaching blog. I had fun. Now to proof read an article that I am planning to submit; that isn’t nearly as […]

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[…] Dirac Delta “function” in differential equations […]

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[…] and eigenvectors for the position and momentum operators. Yes, this means dusting off the dirac delta distribution. Eco World Content From Across The Internet. Featured on EcoPressed 12 Reasons to Recycle […]

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[…] to do that is to allow for distributions to be eigenvectors; the distribution we need here is the dirac delta. In the reference I linked to, one can see how the dirac delta can be thought of as a sort of limit […]

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[…] Remember the metric that we put on continuous functions over ? . Here, let be the zero function and hen the unit “circle” consists of the set of ALL functions where . Needless to say, there are two many to visualize or even organize in some “nice” manner. Sure, is one POINT in that circle, as is for . But there are many, many, many more. If you have had any experience with the Dirac Delta (which isn’t a function but a distribution), think of all of the continuous appro… […]

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[…] one can see the Dirac delta as the limit of Gaussian density functions as the standard deviation goes to zero. What I will attempt here will be the first approach with a […]

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