5.5, or more of piecewise-ness

Section 5.5 is entitled “Discontinuous Forcing Terms.”

We’ve already looked at sinusoidal forcing terms. It’s high time we look at piecewise continuous forcing terms.

Specifically, we’ll be looking at two kinds of functions: the interval function and the Heaviside function.

Some properties about the Heaviside function:

Also, the interval function H_ab(t) can be described using the Heaviside function:

You can also describe piecewise functions in terms of the Heaviside function; for example, if we consider

We have the intervals 0 ≤ t < 4, 4 ≤ t < 6, and 6 ≤ t, so we’ll use the interval functions H₀₄, H₄₆, and H₆:

Now, for the Laplace transform of the Heaviside function:

In particular, we have ℒ(H)(s) = 1/s. Also, for the interval function H_ab = H_a – H_b:

A proposition for you: “Suppose f(t) is piecewise continuous and is of exponential order. Let F(s) be the Laplace transform of f. Then, for c ≥ 0, the Laplace transform of H(t – c)f(t – c) is given by

Back to our original example: We had translated that piecewise function to 3H(t) – 8H(t – 4) + 5H(t – 6) + e^7-tH(t – 6) and now we want to find the Laplace transform. The first three terms can be handled with that handy proposition, and the last one can be rewritten as H(t – 6)e^-(t-6)×e, and we can use that table from one of the previous sections (http://differentialequationsjourney.blogspot.com/2013/11/52-54-where-you-refine-your-powers-of.html) and this proposition to take care of it. Thus we have

Another proposition for you: “Suppose that f(t) is piecewise continuous and is of exponential type. Suppose that F(s) = ℒ(f)(s). Then

Finally, let’s talk about periodic functions. A periodic function is one that has a repetitive pattern (as you can imagine). The formal definition says that “f is periodic with period T (or T-periodic) if f(t + T) = f(t) for all t” (222). Sinusoidal functions are periodic, just for an example. Another example of a periodic function is called a square wave. From a geometric perspective, if the function f is a T-periodic function, then it repeats every interval of T. For this type of function, we can define the window of f as

Finally (finally), here’s a proposition for you: “Suppose f is periodic with period T and piecewise continuous. Let F_T(s) be the Laplace transform of its window f_T. Then

That’s it for 5.5! I was planning on combining 5.5, 5.6, and 5.7 but I just don’t have the time to get them all written up. I’m not on a time crunch or anything (quite the opposite, actually), but I really do want to get them all up by Sunday. If I did that, I would only have those two sections of chapter 10 (if that) and I wouldn’t be thinking of them after my test.

Speaking of which, there shall be no summaries up next week. This is because of my differential equations test.

I’ll see you in 5.6.