Heaviside’s Operational Calculus

I have found a fine introduction to Heaviside’s methods in Spiegel’s “Applied Differential Equations”, 3rd ed, 1981, Prentice Hall. To be specific, pages 204-211. I can’t very well say, “Don’t buy the book just for this” – because that’s exactly what I did!

Let me emphasize that, as far as I know, Heaviside’s methods are now of primarily historical interest. I would not say that Laplace transforms make Heaviside’s methods rigorous – but that Laplace transforms provide a rigorous alternative which, like Heaviside’s, lets us do algebra instead of calculus.

Like Laplace transforms, the quick use of Heaviside’s methods takes advantage of shifting properties, linearity, and tables of known results to speed up calculations. I’m not going to take them that far. With Mathematica® or another symbolic system, I see no need to go beyond the introduction to Heaviside’s methods. What I wanted to see was: just what was Heaviside’s fundamental idea? It turns out that his fundamental idea suffices, given other tools available today.

Of course – as we have seen and as we’ll see below – Mathematica can quickly solve the differential equations to which Heaviside’s methods apply (linear, with constant coefficients). We don’t need Heaviside…

…but it turns out there’s at least one question Heaviside’s methods can answer very, very quickly: find a particular solution (rather than the general solution). I don’t know that I will ever use Heaviside for that, but I know that I could.

One last thing. For my present purposes, it suffices that I will find solutions to a differential equation – and I will confirm that my answers are solutions… but I’m not going to try to prove anything at all. I don’t know what the limitations of this method are – instead of confirming that the differential equation under consideration satisfies some set of conditions, I’ll simply confirm that the answer works.

With that, let me roll up my sleeves and show this to you.
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Happenings – 2012 Aug 25

In some respects, life continues to move slowly, as it has for the past few weeks.

I’ve been focused on Heaviside’s operational calculus for differential equations. For some reason, I failed to find this material in the book I bought for the purpose… but I found it on a 2nd look, and it has been just delightful. I expect that Monday’s post will be the promised follow-up on differential equations. It’s been fun, and has given me at least one new insight.

On the other hand, my trusty laptop is beginning to fail. So far, I’ve been able to work around the problem, but at the very least I need to try to have it repaired. But that would leave me unable to do Mathematica!

And I’ve been thinking about buying a desktop… I’ve been lusting after a 27″ display.

Now I own one.

I have to say that it’s great for putting out posts: the “edit post” window is on my left, and the “preview” is on my right, and it is _so_ nice to see both at once.

Unfortunately, the latest Mac OS X breaks 2 major programs: Rosetta Stone and Dragon Dictate. I am strongly considering erasing the operating system and installing Snow Leopard, that is, OS X 10.6 instead of 10.8.

Oh, and I need new activation codes to run Mathematica on the new machine. Maybe I should wait until I decide if I’m going to change the operating system.

In other words, in some respects my life is moving far faster than I want it to.

And with that, let me get going today – on the old failing machine where Mathematica lives for now.

Ordinary Differential Equations and the Laplace Transform

Introduction

Down the road, I expect to be using Laplace transforms to set up and solve electric circuits, and for transfer functions in control theory. An obvious starting point is to remind you just what a Laplace transform is.

So I should show you at least one example of solving a differential equation using Laplace transforms.

But if I do that, I really should remind you of the alternative solution, the one you almost certainly learned 1st.

On top of that, I really should show you what Mathematica® can do.

As if all that weren’t enough – though it really won’t take very long – I have seen a nice approach to Heaviside’s operational calculus, and I want to show that to you, too. Ah, by the time I explain it, this will justify a post of its own.

So, I propose to take a typical equation for these methods – linear, with constant coefficients – and I am going to

  1. let Mathematica solve it symbolically
  2. check the symbolic answer
  3. let Mathematica solve it numerically
  4. solve it using Laplace transforms
  5. solve the homogeneous equation and then find a particular solution to the inhomogeneous equation

and in a subsequent post I will

  • solve it using Heavisisde’s operational calculus

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Happenings – 2012 Aug 18

I was out of town on business for most of the week… and I didn’t take any math books with me. It feels like forever since I’ve even looked at math. We’ll see what happens today. I’d like to have some fun.

I don’t sleep well in hotels in general, nor in other time zones… back in my own bed finally, I got 9 hours sleep last night (6.5 is usual)… so I’m running late today.

I think I will take it easy today:

Try to take Monday’s post thru stage 4 – written but not on the blog. It’s already close, I think.

Turn all three young alter-egos loose: the kid can do whatever, the grad student was doing arithmetic (number theoretic) functions, and the undergraduate is supposed to do Laplace transforms and circuit theory. All easy stuff.

As for last Monday’s rather terse post… it turned out that most of my regression texts were already listed. I’m just glad my alter ego the librarian got the mechanics right: it had been 2 years and 5 months since I edited the bibliography page.

And that’s all I’ve got today.

Books Added – Regression and Statistics

The following books have been added to the bibliography.

Atkinson, A.C. Plots, Transformations and Regression. Oxford Science Publications, reprinted 1988.
ISBN 0 19 853359 4.
[regression; 13 Aug 2012]
This is devoted to detecting outliers (i.e. using single deletion statistics) and to transformations of the variables. It looks like an excellent supplement to Draper and Smith. Text. Epilog.

Mendenhall, William and Scheaffer, Richard L. Mathematical Statistics with Applications. Duxbury Press, 1973.
ISBN 0 87872 047 2.
[statistics; 13 Aug 2012]
While I know there is a 7th edition, I can only hope that it is as clear and useful as my first edition was. Text. Answers.

McQuarrie, Allan D. R. and Tsai, Chih-Ling, Regression and Time Series Model Selection. World Scientific Press, 1978.
ISBN 981 02 3242 X.
[statistics, regression, time series; 13 Aug 2012]
This book has a whole lot more information than what I extracted for my selection criteria; and it covers tests for more than just OLS regression (e.g. robust regression, wavelets, and time series). Monograph. Epilog.

Ryan, Thomas P. Modern Regression Methods. Wiley Interscience, 1997.
ISBN 0 471 52912 5.
[regression; 13 Aug 2012]
A fine second look at regression. I find it a little too terse for learning new things from, but it provides additional insight for things I already understand. Text. Answers.

Wine, R. Lowell Statistics for Scientists and Engineers. Prentice Hall, 1964.
[statistics; 13 Aug 2012]
This is my reference – although it was, in fact, the text for my very first statistics course. It is thorough and precise… and, I hate to say, I’m not surprised that such a rigorous book is out of print. Text.

Happenings – 2012 Aug 11

It’s been about as slow a week as I’ve ever had… I couldn’t even justify a headlines ticker-tape today.

My alter ego the kid has been looking at surfaces again… my alter ego the grad student has been looking at so-called “arithmetic functions”.

My main personality is thinking I should elaborate on the latest regression post. Sorry, but I can be somewhat more precise than I was… and I have another educational example.

And the landing of Curiosity – officially the Mars Scientific Laboratory – was a glorious success. Just how did it control its landing? (From 14 light-minutes away, JPL was not involved in real-time.) Wiki, as usual, is informative.

And that’s it for now.

Regression 1 – Inapplicability of the Fundamental Theorem

Introduction

Aug 12. Edit: I’ve added a few remarks in one place. As usual, search on “edit”.

I want to look at the t-statistics for two regressions in particular. I will refresh our memories very soon, but what we had was two regressions that we could not particularly decide between. Let’s go back and look at them.

Let me get the Hald data. I set the file path…

I set the usual uninformative names – I wouldn’t dare change them after all the time I’ve spent getting used to them!… and I might as well display the data matrix…


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