Understand Heaviside Functions & Laplace Transforms

[DmytriE]

Good evening everyone,

I have a final exam where one of the questions will most likely be a heaviside function and using Laplase transforms since we just studied it. I am having trouble understanding how the equations are created using piece-wise data.

1. How do you create the heaviside function using basic piecewise data.
2. How does one test it and determine that it works?

I have tried to justify the step down function to myself inputing values but I can't seem to figure out how to test whether the function works. Maybe I'm inputing incorrect values?

I am going to see my professor tomorrow to see if he can help further but it seems very easy when I'm sitting next to him and he's explaining it. Once I leave to do it on my own I get confused. Thanks for any and all help!

 

[S_Happens]

The Heaviside function (unit step function) is a specific piecewise function, but that's not important to what you're asking.

I am taking the same final, but have another final to study for in the meantime. I have to struggle with MathJax (or whatever) here to get the equations right, so I don't have time to write it all out tonight. If you still don't understand what your professor says and your final is later than tomorrow, I might come back and type it out tomorrow evening.

The text version is that you need to add and subtract over the ranges that the functions are valid. If the function is equal to 3 between say 0 and 4, and x when greater than four then you need to write it as so.

$f(x) = 3 - 3u(x-4) + xu(x-4)$

You add the three in first, then subtract it from the range of x greater than 4, then add in the x on the range of x greater than 4. Then you need to write x in terms of (x-4).

So you get

$f(x) = 3 - 3u(x-4) + (x-4+4)u(x-4)$

Factor out the +4 to get

$f(x) = 3 - 3u(x-4) + (x-4)u(x-4) + 4u(x-4)$

Combine the +4 and -3

$f(x) = 3 + (x-4)u(x-4) + u(x-4)$

Then apply the Laplace transform, where u(x-4) is e^-4t. It becomes.

$L(s) = 3/s + e^{-4t}/s^2 + e^{-4t}/s$


As far as testing it, I don't know. Does that help?