Could someone give some examples of applications of numerical methods for solving differential equations that are relevant to a HN crowd? Also, where might I find some introductory material that teaches it well, according to the the suggestions in the OP?
Ten lessons I wish I had learned before teaching differential equations (1997) [pdf]
31–40 of 122 posts
Re: Ten lessons I wish I had learned before teaching differential equations (1997) [pdf]
#32My major is computational math, from 15 years ago, from leading Russian university, so it is just anecdata, and by no means should be generalized. I absolutely love mathematics, for me it is the embodiment of pure beauty. Still, I positively, absolutely hated the sophomore course of ODEs. The way it was taught was extremely abstract: here is the equation, this is integration, this is separation, this is your SLP, now…
Strongly agree. The way ODEs are taught at the sophomore level violates the beauty of math by teaching a plug-and-pray method of solution. For this form of equation, try this form of solution, if it doesn't work, try this one, then this. If none work, oops. At least once the Laplace method is taught things get a little better.
Re: Ten lessons I wish I had learned before teaching differential equations (1997) [pdf]
#33> 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 f…
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.
Re: Ten lessons I wish I had learned before teaching differential equations (1997) [pdf]
#34I once told a math teacher at a Big Ten university, that I thought their undergrad math instruction for engineers was weak. As an example, I said that I didn't think students learned any engineering applications of differential equations. He looked at me with a straight face and said: "There are no engineering applications of differential equations."
My undergrad was at a Big Ten university. The analog signal processing course in EE teaches how to use differential equations to solve circuits. Right after they taught it, we learned laplace transform and never looked at a differential equation again.
Same with my EE undergrad. Then I got to grad school and decided to take a course in Linear Systems, which is when I realized my ODEs course taught me nothing.
Re: Ten lessons I wish I had learned before teaching differential equations (1997) [pdf]
#35I couldn't agree with the last point more.
> A course taught as a bag of tricks is devoid of educational value. One year later, the students will forget the tricks, most of which are useless anyway. The bag of tricks mentality is, in my opinion, a defeatist mentality...In an elementary course in differential equations, students should learn a few basic concepts that they will remember for the rest of their lives...
I hated the DE cleass I took in college and it was largely because I felt like it was nothing but a bag of tricks. I very distinctly remember one problem that seemed unsolvable until the teacher showed that you had to substitute a "2" with "1/2 + 3/2". And then, to make matters worse, he put the exact same problem on the test. So we were being rewarded, not for really understanding the core basic concepts, but for memorizing the tricks needed to solve specific problems.
Re: Ten lessons I wish I had learned before teaching differential equations (1997) [pdf]
#36"FORGET ABOUT EXISTENCE AND UNIQUENESS OF SOLUTIONS" what? most important thing about differential equations.
As an engineer, I promise you, I give not one single solitary conscious fuck about the uniqueness of solutions to differential equations. Mostly I just hit things with Fourier or Laplace transforms as appropriate until they stop moving.
Re: Ten lessons I wish I had learned before teaching differential equations (1997) [pdf]
#37My major is computational math, from 15 years ago, from leading Russian university, so it is just anecdata, and by no means should be generalized. I absolutely love mathematics, for me it is the embodiment of pure beauty. Still, I positively, absolutely hated the sophomore course of ODEs. The way it was taught was extremely abstract: here is the equation, this is integration, this is separation, this is your SLP, now…
I took, we called it Diff-e-q, my freshman year in college and the professor died a couple weeks into the course and the new guy was not pleased about having to teach it. We did the applied aspects with the springs and the little ants running away from a candle heating the corner of a plate.
What I didn't like about it was just all the memorization. I had no desire to memorize a bunch of formulas that I knew full well in the real world I'd look up in a table or type into a computer. What I wanted to learn how to do was solve problems using math, not memorize patterns of formulas to apply to problems.
So I didn't memorize them and instead went to work and earned money to pay for college. Still passed the class but it was one of my lowest grades. It's a hard class even without the memorization.
Re: Ten lessons I wish I had learned before teaching differential equations (1997) [pdf]
#38He was the author of Indiscrete Thoughts[1], a great book on Mathematics.
[1] Review here: http://www.maa.org/publications/maa-reviews/indiscrete-thoug...
Re: Ten lessons I wish I had learned before teaching differential equations (1997) [pdf]
#39Could someone give some examples of applications of numerical methods for solving differential equations that are relevant to a HN crowd? Also, where might I find some introductory material that teaches it well, according to the the suggestions in the OP?
https://www.udacity.com/course/differential-equations-in-act...
I'm sure there are others, but this is a pretty good introduction. It focuses quite a bit on numerical solutions (using python programs that are automatically graded.)
Re: Ten lessons I wish I had learned before teaching differential equations (1997) [pdf]
#40Could someone give some examples of applications of numerical methods for solving differential equations that are relevant to a HN crowd? Also, where might I find some introductory material that teaches it well, according to the the suggestions in the OP?
I'm not sure what the HN crowd finds useful as a group, but I personally use a couple of different techniques in my work (geophysics). One of them is finite difference and finite element methods. I have a book called 'Introduction to Numerical Geodynamic Modeling' by Teras Gerya that teaches finite difference modeling of plate tectonic phenomena, particularly of 2nd order differential equations (Poisson equation and variants) that are useful in modeling heat flow, diffusion and so forth through space and time. It's a great book but written for a specialized audience. I've used finite element models a lot, and the occasional boundary element method, but never written any.
I also use Green's functions, which are equations that describe the response of a medium to an impulse (think the propagation of sound waves from a source, though I do different stuff), by using convolution.
But I think jofer is the only other geophysicist on HN so we're probably not representative. Nonetheless, a lot of HNers have a physics, classical engineering or chemistry background and use similar tools... just not to find out what happened tens of millions of years ago.