Nonlinear differential equations are more common in the real world than linear ones. Our textbooks tend to have a lot more linear ones simply because they are easier to handle and teach, not because of their prevalence.
Even when you get to the nitty gritty of real world circuits, you'll encounter nonlinearities and have to deal with them. The linear approach is fine for undergrad curricula, but having worked with some of the frontiers of circuit/semiconductor work professionally - you can't build great products by relying on linear diff eqs alone.
Furthermore, linear diff eq's are somewhat of a solved problem. A researcher is unlikely to discover a phenomenon today that is governed by a linear equation.
> Even for something as infamous as the Navier-Stokes equation, you can extract important parameters like the Reynolds number, Froude number, and so on -- which I hope I don't need to explain why they're extremely useful, even for a numerical design approach -- so even if you can't solve the equation, you can still get a lot of mileage out of the analysis itself.
Here I think we are merging a little: I took only one fluid mechanics/dynamics course (albeit 20 years ago), and we were taught the Reynolds number, etc. The thing is, I would say there were only two engineering courses I took in my whole undergrad where the analytical approach is not particularly helpful, and one really has to go full out "engineer" - either by looking up lots of (empirical) graphs and tables, or use numerical methods. Fluid mechanics was one of those courses.
Yes, I agree that doing some level of analytical work on the diff equations is very useful, but in the majority of cases, the main purpose of doing some theoretical analysis on the differential equation is to get it into a form where one can apply numerical techniques to it.
I'm not a fluid mechanics person, but I'd wager that most of the useful results we have from the Navier Stokes equation came out of some numerical technique (perhaps after a bit of analytical work).
In any case: Yes, I agree - it is useful to do some analytical work on it. But the bulk of the gains in the real world come from applying numerical technique on them. And more to the point: Most of what is taught in a typical diff eq course is of little use in doing that analytical work.
Edit: As another anecdote, for my grad research, I went the analytical route, as that was what my advisor preferred. Thing is, it was fine in his day when the field was young. Most of the people doing research use numerical methods (often from first principles, so almost no analytical approach applied). They basically "won". When you look at the useful results in the discipline in the last 10-15 years, the numerical ones outnumber the analytical ones easily by a factor of 10.
> But the bulk of the gains in the real world come from applying numerical technique on them.
> Most of the people doing research use numerical methods (often from first principles, so almost no analytical approach applied). They basically "won". When you look at the useful results in the discipline in the last 10-15 years, the numerical ones outnumber the analytical ones easily by a factor of 10.
I think this is probably true now in most fields (including fluid dynamics) but it doesn't mean that analytical techniques aren't useful. All it means is that they aren't used. Most people don't even attempt analytical techniques these days.
I've found time and time again people use very complicated (and difficult) numerical approaches where an analytical approach would get the same information faster, and often more information. It might take more thinking, yes, and not be as "sexy", but ultimately analytical approaches have their place.
I gave an example from my PhD in this other comment about how I analytically solved a series of nonlinear ODEs (approximately) to get a lot of insight into a problem that apparently people doing decades of numerical calculations couldn't figure out: https://news.ycombinator.com/item?id=24683038
I'm planning to publish another paper on the subject that builds on this work, and one point I'm going to make in the paper is that I don't see how someone doing experiments or simulations could figure out the main result of that paper. It's not impossible, but would be far more difficult
It looks like we agree on the core point, but have different lines of work and experience that colour our impressions of the relative frequency and importance of things. I think it's common to go through school and then never apply some fraction of what you learned in the rest of your career, but it will be a different fraction for each person, depending on their career path. So it's hard to go back and say "I spent so much time studying X, which was not useful in the real world".
As a particularly embarrassing example, I actually gave a presentation to first-year students at my alma mater and told them all that "one thing that never turned out to be useful was integration by parts. It's totally useless to learn these days because Maple and Mathematica are way better at solving integrals than I'll ever be". And then, literally the next week, I had to apply integration by parts to write code for integrating certain arbitrary functions that are passed in as an argument.
> Nonlinear differential equations are more common in the real world than linear ones
It really depends how you look at it. I'd say linear equations are far more common in practical problems, but perhaps common to the point that you hardly notice them and only need to pull out your thinking cap when you hit a nonlinear one. But in a world where I'd never learned how to solve linear equations (other than numerically), I'd probably be really frustrated by how often I encounter them.
> Even when you get to the nitty gritty of real world circuits, you'll encounter nonlinearities and have to deal with them.
Yes. Usually at that point, the main "concept" of the circuit and most modules have already been designed, and you need to pull out SPICE to help design some workarounds when you notice that your capacitor's capacitance is also a function of its voltage, say. You can still think of it as being linearish, with modifications. For most problems, that is (not for everything).
> by looking up lots of (empirical) graphs and tables... Fluid mechanics was one of those courses.
If you're talking about, say, the Moody chart, those nondimensional numbers are what allows you to compress a function of fluid density, viscosity, flow rate, pipe diameter, and pipe roughness (5 inputs) down to a readable chart with just two inputs (Reynolds number and roughness). And someone trying to come up with the empirical measurements to generate such a function would have to take proportionately more measurements, not knowing in advance that most of them will collapse to the same line in Reynolds-number-space.
> A researcher is unlikely to discover a phenomenon today that is governed by a linear equation
I'm really not trying to be argumentative here, but that's simply not true! First of all, engineers are often more interested in modelling and controlling phenomena than discovering them, and frequently they'll find themselves modelling something that nobody's modelled before. But moreso, I know because even I've done it, and I'm not that special. (It was a novel application of the dynamic Timoshenko beam equation). It's often just a matter of noticing the linear equation that is there, or else pretending that it's linear and then noticing that the result works pretty well.
And again, a lot of interesting non-linear equations reduce to linear ones in the small-signal limit, but remain important in that limit -- for example, most of the domain of acoustics and vibrations deals with small stresses, strains, and displacements, such that all but the most exotic materials behave linear-elastically. But this is still a very active field of research and modelling, and you can bet that design is not just a matter of throwing everything into ANSYS and waiting for a result. Even gravitational waves are studied based on a linearized approximation to the famously nonlinear equations of general relativity.
Even when you get to the nitty gritty of real world circuits, you'll encounter nonlinearities and have to deal with them. The linear approach is fine for undergrad curricula, but having worked with some of the frontiers of circuit/semiconductor work professionally - you can't build great products by relying on linear diff eqs alone.
Furthermore, linear diff eq's are somewhat of a solved problem. A researcher is unlikely to discover a phenomenon today that is governed by a linear equation.
> Even for something as infamous as the Navier-Stokes equation, you can extract important parameters like the Reynolds number, Froude number, and so on -- which I hope I don't need to explain why they're extremely useful, even for a numerical design approach -- so even if you can't solve the equation, you can still get a lot of mileage out of the analysis itself.
Here I think we are merging a little: I took only one fluid mechanics/dynamics course (albeit 20 years ago), and we were taught the Reynolds number, etc. The thing is, I would say there were only two engineering courses I took in my whole undergrad where the analytical approach is not particularly helpful, and one really has to go full out "engineer" - either by looking up lots of (empirical) graphs and tables, or use numerical methods. Fluid mechanics was one of those courses.
Yes, I agree that doing some level of analytical work on the diff equations is very useful, but in the majority of cases, the main purpose of doing some theoretical analysis on the differential equation is to get it into a form where one can apply numerical techniques to it.
I'm not a fluid mechanics person, but I'd wager that most of the useful results we have from the Navier Stokes equation came out of some numerical technique (perhaps after a bit of analytical work).
In any case: Yes, I agree - it is useful to do some analytical work on it. But the bulk of the gains in the real world come from applying numerical technique on them. And more to the point: Most of what is taught in a typical diff eq course is of little use in doing that analytical work.
Edit: As another anecdote, for my grad research, I went the analytical route, as that was what my advisor preferred. Thing is, it was fine in his day when the field was young. Most of the people doing research use numerical methods (often from first principles, so almost no analytical approach applied). They basically "won". When you look at the useful results in the discipline in the last 10-15 years, the numerical ones outnumber the analytical ones easily by a factor of 10.