Group Abstract Group Abstract

Message Boards Message Boards

5
|
4.4K Views
|
19 Replies
|
21 Total Likes
View groups...
Share
Share this post:
GROUPS:
Wolfram U

[WSG25] Daily Study Group: Introduction to Laplace Transforms

Juan Ortiz, Wolfram Research INC
Posted 6 months ago
POSTED BY: Juan Ortiz
19 Replies
Sort By:
POSTED BY: Mitchell Sandlin

Dear Mitch:

We used 'analytical' essentially meaning that there was a closed formula for it, particularly in the numerical evaluation lesson to distinguish between the two.

The parameter s, in the integral definition of the Laplace transform is complex, and this is the variable in the Laplace transform. That is where complex numbers come from.
POSTED BY: Juan Ortiz
Phil Earnhardt
Phil Earnhardt
Posted 5 months ago
POSTED BY: Phil Earnhardt

Thank you Phil. This is a very nice real LIVE example.
POSTED BY: Juan Ortiz
Phil Earnhardt
Phil Earnhardt
Posted 5 months ago

I'm a bit late to this DSG; I joined yesterday. Last night, I viewed the first session from the BigMarker video sessions. The "Download Materials" link is NOT visible in the BigMarker archive videos -- but the other links that @Cassidy inserted in the moderator's chat are visible. This appears to be a limitation of the BigMarker archive/repaly mechanism and has nothing to do with Wolfram Research. I was able to get the amoeba.wolfram.com URL by pausing Juan's presentation and taking a picture of the address bar in his browser on my iPad and parsing the URL there like an animal. ;)

I suggest the Wolfram U team be mindful of the limitations of Big Marker's video playback system and put all pertinent URLs for a course as moderator chat messages. That may mean putting whatever URL is put in the "download" area a second time in that chat window. That will help truant students get access to all the course materials on replay of the classes. Thank you, Cassidy.

I'm a bit overwhelmed with the math right now, but I appreciate where it is used in the world. That's a start. Reviewing the conversations here has been quite helpful. Thank you.
POSTED BY: Phil Earnhardt

I asked in today's lecture but I stepped away for a moment and when I came back the lecture was over and never saw if I got a reply:

Why do you choose to use the 2D version of ComplexPlot? I find the rainbow colors to be difficult to interpret. The 3D version of the plot more clearly shows poles and zeroes and seems to provide a better understanding of the behavior of the functions. Specially with the ability to rotate it around and look at if from all angles.

Do you have a special reason for that? Or is it just a question of taste?
POSTED BY: Carl Hahn
POSTED BY: Juan Ortiz

Thanks Juan,

But there is a ComplexPlot3D. For example:

ComplexPlot3D[(z^2 + 1)/(z^2 - 1), {z, -2 - 2 I, 2 + 2 I}]

Carl
POSTED BY: Carl Hahn

Sorry Carl, I understood you question wrongly. Yes, is more a matter of taste but since as for now we are more interested in looking for places with zeros and poles, once you get comfortable with ComplexPlot, it is more direct, in my opinion anyways. Both work for our purposes. Thanks for clarifying.

POSTED BY: Juan Ortiz

I have signed up to take the quizzes for certification in the Laplace Transform DSG. I have the downloaded the textbook, the daily lessons, and the exercises (with solutions). But where are the quizzes? (I do not see any links to quizzes in the download section for the study group.)
POSTED BY: Michael Ulrey

Dear Michael:

The quizzes are in the course framework: https://www.wolframcloud.com/obj/online-courses/introduction-to-laplace-transforms/quiz-1.html
POSTED BY: Juan Ortiz

Thanks!
POSTED BY: Michael Ulrey
Graham Gyatt
Graham Gyatt
Posted 5 months ago

Without giving too much away, I'm wondering if there might be a typo in problem 6 and 10 of Quiz 1. For #6 I get a slightly different answer from one of them, but 'None of the Above' is not given as correct. For #10, I also get a slightly different answer from one of them, but not an exact match. Has anyone else given the Quiz a try?
POSTED BY: Graham Gyatt

Dear Graham:

Both seem fine to me. For problem 6, I will suggest comparing your answer with the answers provided considering that s>0 in this case. For example, you can try the command FullSimplify[yourAnswer==answerCompared, s>0] or you can cheat a bit to very using InverseLaplaceTransform on each of the posible aswers provided. For 10, make sure to set the WorkingPrecision to 10 as an option in your call to LaplaceTransform.
POSTED BY: Juan Ortiz

Is there a formal explanation of when one should be using Laplace Transforms and when one should be using Fourier Transforms? It seems that when doing signal analysis one uses Fourier Analysis and when analyzing physical systems one uses Laplace Transforms. But then when you want to run a signal through a system you start describing the system using Fourier Transforms. But I've never read or heard someone say what exactly are the formal guidelines. When and why? Can you provide some insight into that question?

POSTED BY: Carl Hahn

Dear Carl:

I believe that in signal analysis the Fourier transform is the way to go. Note that in the Laplace transform s is complex and in fact if h(t) is the Heaviside function and s=c+i y, then the Laplace transform of f(t) is equal to the Fourier transform of exp(-c t) h(t)f(t) depending on y. This is actually used to get the inversion formula for the Laplace transform using the Fourier transform. For more on this I can refer you to the documentation page of the FourierTransform function in the Wolfram language, where I included a subsection on signals and systems in the Applications section.
POSTED BY: Juan Ortiz

I noticed in the exercises for Part 6 that when WorkingPrecision (WP) was increased, the resulting numerical answer differed before the last digit. For example, in exercise 2, when WP->10, the result was 0.1979119699 and when WP was increased to 20, the answer was 0.19791196966502245964 yielding a difference of 3x10^10. Granted, not much of a difference, but I expected a difference of no greater than 1x10^-10. I suspect the difference is due to the accumulation of rounding "errors". There is discussion in the WP documentation that final results from internal calculations done to n-digit precision may have much lower precision. So I wonder if there are examples where a long series of calculations resulted in significant differences. If so, it would be useful to know what to watch out for. I need to review the Precision & Accuracy Control Guide in the WL documentation to see if any guidelines are suggested.

POSTED BY: James Kralik

Dear James:

WorkingPrecision->n causes all internal computations to be done to at most n-digit precision. This does not imply that the answer will be correct to that number of digits. The final results you get may have much lower precision.

Instead of comparing WorkingPrecision->10 with WorkingPrecision->20, we should compare each to infinite precision. This will require to have an analytic representation for the transform. Consider exercise 1 in the same lesson:

exact = LaplaceTransform[
   Cos[Sqrt[t^2 - 4]]/Sqrt[t^2 - 4] UnitStep[t - 2], t, s] /. 
  s -> 17/10

approximate = 
 LaplaceTransform[Cos[Sqrt[t^2 - 4]]/Sqrt[t^2 - 4] UnitStep[t - 2], t,
   1.7]

approximateWP10 = 
 LaplaceTransform[Cos[Sqrt[t^2 - 4]]/Sqrt[t^2 - 4] UnitStep[t - 2], t,
   1.7, WorkingPrecision -> 10]

approximateWP20 = 
 LaplaceTransform[Cos[Sqrt[t^2 - 4]]/Sqrt[t^2 - 4] UnitStep[t - 2], t,
   1.7, WorkingPrecision -> 20]

Now:

exact - approximate=4.50937*10^-10

exact - approximateWP10=-1.14043*10^-12

exact - approximateWP20=-3.46945*10^-18
POSTED BY: Juan Ortiz

Juan is a highly-experienced instructor and has worked hard to create this wonderful introduction to Laplace transforms and their many applications.

I strongly recommend this study group to everyone!
POSTED BY: Devendra Kapadia
Reply to this discussion
Community posts can be styled and formatted using the Markdown syntax.
Tag limit exceeded
Note: Only the first five people you tag will receive an email notification; the other tagged names will appear as links to their profiles.
Reply Preview
Attachments
Remove
or Discard