The truth about differential equations 9

[GregLocock]

https://web.williams.edu/Mathematics/lg5/Rota.pdf

Engineering degrees should be far more focused on numerical methods than solving the few solvable equations.

Cheers

Greg Locock


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[MotorCity]

I have never once in my career had to use calculus.
Not because I used Mathcad or some software to do it for me, but because the need literally never presented itself.
If I had to solve a differential equation today, I think I'd have to study for a week and re-learn it.

 

[TheTick]

I have never had to use calculus. But an understanding of calculus has been helpful in visualizing certain things, especially organic geometry.

 

[dik]

I used to love non-linear, non-homogenious differential equations... over 50 years back, in the ACI publications were a series of articles on coupled shear walls for lateral design by Rico Rossman (sp?). It took me weeks to go through the material.

I'm kinda 'old school', archaic maybe, the advantage of having a formula as opposed to using a linear method is that it gives you an idea of what the variables are and how they act.

My $.02... other than helping a student with Grade 12 Calculus a couple of years back, I haven't used it often, but it still pops up.

So strange to see the singularity approaching while the entire planet is rapidly turning into a hellscape. -John Coates

-Dik

 

[SnTMan]

I was alway envious of people who were really, really fluent in calculus and math in general. Needless to say, I wasn't one of them.

I had a lecture in an upper level engineering course where the subject (don't recall) needed solution of a differential equation. The professor, working on the white board said "We are going to solve this by the graduate student method. We're going to just write down the answer." Both illustrative of how certain problems are well known and funny too I thought.

This same profesor in a lecture would ask "Why do we study mechanical engieering? (pause for effect) Electricity and chemistry." I loved his classes.

The problem with sloppy work is that the supply FAR EXCEEDS the demand

 

[cranky108]

Actually knowing that integrals work has been helpful, because analog circuits can do them, and computers can't. It is helpful to know how to simulate integrals in computers.

 

[dik]

I should have noted that I was working in advanced calculus in Grade 9, long before it was studied in school. My classmates were into Algebra back then...

So strange to see the singularity approaching while the entire planet is rapidly turning into a hellscape. -John Coates

-Dik

 

[mtu1972]

I used calculus once in my 40 year career. Working in Minnesota’s Iron Range and we were designing silo’s with conical hoppers to store taconite pellets at the port of Duluth. Our boss said “I wish there was a way to minimize the surface areas of the silos for the volume required.

I said It could be done with calculus. This was before computers, etc., so another engineer and I did the math and solved the problem.

My recollection is that our general arrangement group had already developed the sizes (diameters, height, and hopper angles), so our efforts weren’t implemented.

gjc

 

[Sparweb]

I've done enough math for work that by now I don't expect to ever need to use diff's. Which makes it a bit annoying since I struggled so much with them in University and the instructor was of no use because he didn't know what they were for, either. I asked and was sorely disappointed. I've found a number of uses for calculus in problem solving for many kinds of structures, but I didn't strictly *need* to use it. Just a shortcut, and only because I was using Mathcad which can solve the integral directly. Otherwise I wouldn't have bothered.
Several years ago, a complex problem came to me that I could have solved with a system of differential equations, but the computation and setup would be equivalent to doing it numerically. The iterative solution had the advantage of delivering mountains of step-by-step data that would describe the process in detail, not just the final result at the end.

 

[drawoh]

I do not fondly remember taking differential equations in college. At work, I have used double integration method to solve beams. When I am asked about vibration deflection and acceleration, I differentiate the basic motion equation.

At home, I was playing around with beam analysis, and I set up a spreadsheet to do basic beam theory and interation to find an accurate solution. Double integration actually is not correct. I also played with the basic vibration free body equation.

mx'' + cx' + kx...

...where c is the damping factor assuming a fluid damper. The force for fluid friction really ought to be Cx'². I assume the resulting differential equation is not solvable by algebra, although the numerical solution is easy, even with a spreadsheet.

--
JHG

 

[btrueblood]

Agree that classic diff. eq. class was a let down. As was the 400-level math class in extended diff eq.'s (required for grad students) covering Legendre Polynomials and Bessel functions. I can recall asking the droning professer in the latter class what use Bessel functions had, and he could not provide a cogent physical answer, so I skated through the course considering them useless. Until I ran into them again in my first year on the job...turns out Bessel functions are useful in solving acoustical resonances of cylindrical chambers...like rocket engines.

But yeah, solving diff. equations numerically (FEA, controls theory, CFD) is incredibly useful.

 

[GregLocock]

drawoh-as soon as you introduce proper drag, v², the whole mess is no longer analytically solvable for acceleration, so far as I know.

Cheers

Greg Locock


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[NorthCivil]

The math i use in my day to day practice, I could do half asleep. occasionally some challenging maths arise, that take a bit of brainpower, but nothing near what i learned in university.

But potentially it was all that intense maths i did in university that make these daily maths i do, that most people couldnt do, seem "easy"

 

[CWB1]

Until I started in govt diff eq popped up at least 1-2x annually in everything from product-definition/business-case development to actual engineering. We modeled and studied most everything to optimize efficiency.

Rota has some very valid points but I would argue that "Calc II" (integral) as taught stateside is the most difficult due to the volume of material, ordinary diff eq was easy by comparison. I'd also argue that word problems are essential to show the usefulness of the material and their subjects should vary to force students to engage the brain a bit. The bigger issue with textbooks today IMO is that they're written by academics with little-no real-world experience, if word or other problems stink its bc the author stinks at writing them. I started collecting obscure/niche textbooks while still working in the trades bc I found they were great reference material for machine/tool setups in the shop. In college I started picking up engineering/other texts and one thing I have found is that even among math/calc books, older texts were written in more plain/simple language and thus more understandable. Half of the books today make me wonder if they weren't ghost-written by an algorithm at IBM.

 

[dik]

It's all part of the learning process, including the ability to solve problems. There is a lot of stuff that you studied during school that you never use. It's the process of leaning by studying that's achieved, and also problem solving is part of the process.

So strange to see the singularity approaching while the entire planet is rapidly turning into a hellscape. -John Coates

-Dik

 

[xnuke]

As a controls engineer, I've used linear and nonlinear ODEs a couple of times to develop computer models for simulation purposes so I could develop/improve upon control/protection schemes. One was a first-principles model of an acid waste neutralization system under closed-loop control, one was a system identification model of the overspeed behavior of a generator undergoing a full load rejection under closed-loop control. I enjoyed both of those projects very much due to the use of the higher-level math. I seem to recall using diff eq's on a few other occasions, but can't remember the details. Haven't used them very often, though.

When I was a professor, I taught students to perform system modeling using first principles with ODEs and block diagram editors, and also to develop diff eq models using system identification methods fed with experimental data obtained from existing systems. Did this for a buck converter in my power electronics class, a water flow control system in my controls class, and a few other lab projects. Once the model had been developed and validated, then the students would design a control algorithm for the system model, implement the algorithm in a controller (e.g., analog circuit made from op-amps for the buck circuit, tuned PID or discrete-time filter in PLC/microcontroller), and test it on the real system. All these projects started with obtaining the diff eq's through analysis or experimentation. The students really enjoyed the theory-to-practice side of things, as did I.

xnuke
"Live and act within the limit of your knowledge and keep expanding it to the limit of your life." Ayn Rand, Atlas Shrugged.
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[davefitz]

xnuke is right- the problem is first simplified, an ODE is associated with the process, the initial conditions and boundary conditions are defined, and the solution is normally solved numerically. Bright colorful charts of the solution are then presented to management/customer, and there is a tendency to think these charts represent the "truth" , and decisions are then made.

There are several faults to this approach that the decison maker needs to be made aware of, including:
a) The initial simplification of the process , as well as the initial conditions and boundary conditions are assumptions and often do not capture the intricacies of reality. For example , 3D CFD solution of fluid flow assumes continuity of processes and ignores the possibility of fractal or chaos effects.

b) The knowledge or accuracy of assumed physical properties is sometimes poor, yet the assumed accuracy of the used properties is never evaluated. For example, for thermal radiation heat transfer of participating media, the complex index of refraction of the entrained particles is usually poorly known, likewise isotropic properties of metals is nearly always assumed yet non-isotropic properties dominate weld interfaces and crystalline differences throughout thick objects that were heated and cooled quickly.

c) The computer program used by most users is normally a commercial program, and the details of that program are deliberately made unknowable to the end user. The numerical solution of differential equations is affected by the chosen integration method, the stability of the solution is a function of the discrete spacial and time steps utilized as well as the details of the programming method. For example , numerical round off errors can propagate, and instabilities can go unnoticed. The fact that the program ended without error messages and the colorful charts are impressive to all can hide the fact that the numerical solution is wrong.

d) In the worst case, if the analyst has a financial interest in the final solution, it can affect their chosen assumptions, so as to affect the indicated "solution". The current 30 yr effort to predict "climate change" stands out as a notable instance of this problem.





"...when logic, and proportion, have fallen, sloppy dead..." Grace Slick