Comments on: Approximating a solution that doesn’t exist

By: Jan Van lent

Jan Van lent — Thu, 15 Dec 2011 20:05:21 +0000

Note that the location of the singularity is given by t(1), where t(w) is the solution of the initial value problem t'(w) = 1/((1-w)^2*t(w)^2+1), t(0) = 0. The value of t(1) = 0.96981072 can easily be found numerically. The solution to the original problem can be written using Bessel functions.

By: Carnival of Mathematics #56 « Reasonable Deviations

Carnival of Mathematics #56 « Reasonable Deviations — Fri, 28 Aug 2009 07:22:35 +0000

[...] blogger John Cook writes about an intriguing nonlinear differential equation , with initial value . Suppose we would like to [...]

By: ekzept

ekzept — Wed, 12 Aug 2009 05:41:57 +0000

Very nice illustration! Just to finish off the scholarship, the Boyce and DiPrima (first) edition I have (INTRODUCTION TO DIFFERENTIAL EQUATIONS, 1970, SBN 471-09338-6) puts the illustration of this problem in section 7.7, out at pages 277-278. It is interesting it is dumped in a section introducing numerical methods rather than being placed far ahead, in the Introduction. That Introduction uses Newton’s Second Law as a start, and follows with the general form of ODEs. I think this shows, in part, how heavily computation has influenced our collective view of this material.

To balance, however, I’d say there are some algorithms and processes which give answers with phenomena and data theory is quite hopeless at dicing. (Think Navier-Stokes.) Sure, they have and need parts to be robustly built, and existence solutions are critical. And, sure, people — including me — could do with more careful attention to theoretical underpinnings in nearly everything I do. I try. We try. But, for example, there’s a lot of good engineering that can be done with Laplace transforms that doesn’t really need an understanding of the proof of Lerch’s Theorem. I’d say that’s good!

By: Daniel Lemire

Daniel Lemire — Wed, 12 Aug 2009 02:25:08 +0000

While the term thermodynamics came about after the first engines, we already had a lot of theory worked out before. Pressure was a well known concept. So, I wouldn’t go so far as to say that “practical application” predates theory.

I suspect it is more of an entanglement… theory without practice is barren… practice without theory is crude…

By: Keith

Keith — Tue, 11 Aug 2009 17:10:33 +0000

I thought the article below had an interesting comment about theory vs. practice from an engineering standpoint.

http://www.americanheritage.com/articles/magazine/it/1997/3/1997_3_20.shtml

Q: You were developing transonic theory after the sound barrier had already been broken. Hasn’t much of your historical study also involved engineering problems that were “solved” in a practical sense before they were understood theoretically?

A: Yes, and I think that’s a typical situation in technology. You have to look hard to find cases in which the theory is well worked out before the practice. Look at the steam engine and thermodynamics; that whole vast science got started because people were trying to explain and calculate the performance of the reciprocating steam engines that had been built.