> Some thirty or so years ago, Bessel functions were included in
the syllabus, but in our day they are out of the question.
> Teaching a subject of which no honest examples can be given is, in my opinion, demoralizing.
I don't get this. Differential equations theory is about proving existence and uniqueness of solutions. If you have to use numerical techniques to actually compute the solution, then that's perfectly fine. After all, even if the solution is explicit, like sin(x), or especially a special function, then we still need to use numerical techniques to actually evaluate that explicit solution.
The article is not questioning the "theory", the identified problem is the teaching at undergraduate level. As it was/is commonly taught, it is neither pure nor applied. Rather it is categorisations and tricks, little of which has any practical value.
As a body of work Differential equations are so messy that theorists landed on so many disparate results. As such Differential equations courses are commonly taught as a "survey of the land" type of courses, so they tend to be incoherent. On the other hand if the teaching focused on practicality there is a lot of commonality among the practical cases.
I recently saw them in a grad eng class, but I agree with the article - from what I saw there is no need to give them the math professor treatment. You can use them as a piece of trivia - ie pde of type x has this set of basis functions - now apply the principles of basis functions to solve your your problem.
But are they useful now, other than as nomenclature? Bessel functions are defined as the solutions of Bessel's differential equation. It's all a bit circular. (there's the series expansion, but it doesn't gain you much)
30 years ago, if I wanted to plot the result of solving an equation like this, Bessel functions were useful as I'd just reach for Abramowitz&Stegun and look at the tabulated values. But now I have a computer, tabulated special functions don't matter nearly so much.
It's a long time since I had to use Bessel functions, so I could be very wrong, but this might be one of the reasons Rota said that.
I don't get this. Differential equations theory is about proving existence and uniqueness of solutions. If you have to use numerical techniques to actually compute the solution, then that's perfectly fine. After all, even if the solution is explicit, like sin(x), or especially a special function, then we still need to use numerical techniques to actually evaluate that explicit solution.