Introducing Weird Differential Equations: Delay, Fractional, Integro, Stochastic!
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- čas přidán 16. 06. 2024
- A brief standalone video that introduces weird types of differential equations, where 'weird' means differential equations that aren't conventionally taught unless you get to super specialized areas. Here, they include delay differential equations, integro-differential equations, stochastic differential equations, and fractional differential equations.
Questions/requests? Let me know in the comments!
Pre-requisites: know what a differential equation is (so basically 1st year undergrad calculus should be more than enough).
Lecture Notes: drive.google.com/file/d/1zufF...
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Wow that was quick, haha. Again very clear and concise, I love it thanks Khan. Worth being a patreon supporter!
My ODE teacher had a special section of our coursework. That semester's special topic was DDE because we were in the early part of the pandemic. I really enjoyed that course and he was an amazing instructor.
I want my handwriting to look that clean
I would be ecstatic if you could make further videos on these weird DEs. I'm an EE/Signal Processing undergrad just about to start my upper-level classes at a University. I also have a huge interest in mathematical finance, so naturally I have a huge interest in delay differential equations and stochastic differential equations.
And I thought it was bad when I first glossed through a control systems textbook.
Imagine the punch to the gut when I downloaded an *introductory* book on stochastic DEs and saw the shear breadth of mathematical prerequisites to even begin understanding them. Set theory, partial differential equations (haven't gotten to them yet), rigorous analysis, advanced linear algebra, advanced statistics... a practical undergrad degree's worth of material unto itself. Stuff I'll never be able to touch on doing engineering.
I really appreciate the videos you make, since they let me get a taste of the things I really want to learn about but would otherwise never have the opportunity to do so.
clear and concise, deserving MORE subs!
What an awesome intro! Do you plan making a series for solving then? Like introducing Riemann-Liouville's and Caputo's integral along with Mitag-Lefler's functions to solve FODEs (Fractional ODEs) or Itō's integral for SODEs (Stochastic ODEs), etc? Would be cool, since there's almost no content available on CZcams on these "weird" topics haha.
Little late but I've got a video on solving a SODE on my channel. It's not technical at all so sorry if that's what you're looking for.
@@MathematicalToolbox gonna watch It later
@@felipegabriel9220 thank you sir! 😄
Interesting. Never heard of delayed DE and fractional DE and that says something! Follow up videos would be very exciting, maybe a few examples and where these weird DE are frequently encountered
They are quite commonly used in practice, especially delayed differential equations (DDEs). Think of a phenomena where delay exist, hence one can formulate a DDE for it. I will provide some examples. There exist delayed SIR models to model epidemics where the delay parameter represents the incubation time. DDEs are frequently used in optoelectronic feedback models and oscillators and so on. DDEs are very interesting since the state space is infinite dimensional (a function space) instead of a fininite dimensional state space like one has for ODEs. This make a spectral analysis more difficult, but on the other hand more interesting. To handle with this infinite dimensional state space, a couple of mathematicians (Odo Diekmann for example) applied the sun-star calculus framework to DDEs to obtain a variation of constants formula which is a key part in the central manifold theorem. This theorem relates DDEs to ODEs in a certain sense.
@@bramlentjes huh wow, sounds highly interesting. Maybe my scope is limitied as I never had a lecture covering population dynamics (for example the SIR model) in more detail. I will certainly look into these types of DE:) Thank you very much
@@bramlentjes Can you suggest me some books on Delay differential equations?
@@sheeba7779
A good theoretical reference is the book from Odo Diekmann, the pioneer of DDEs: www.springer.com/de/book/9780387944166
Another good reference is the book of Jack Hale: www.springer.com/gp/book/9781461298946
@@bramlentjes thank you
Great video. Keep it up!
a great video. Thank you!
I'm about to take this course next semester, so uh, I hope this makes sense to me by then
More like this please.
Hi! What application do you use to make fantastic videos like this? How to make the words appear as in your video?
Mind blown. The more you learn the more you realize you know very little
Can you suggest any book on continuous delay differential equation?
First.
Was just in the mood for weird differential equations!
Can you please solve an example using method of steps to solve DDE ?
Hello. Can you recommend me good reference/textbooks on delay differential equations?
I found Hal Smith's "An Introduction to Delay Differential Equations with Applications to the Life Sciences" to be a good, gentle introduction.
@@drumstixkml Thanks!
I did my thesis on Delay Differential Equations (DDEs). A good theoretical reference is the book from Odo Diekmann, the pioneer of DDEs: www.springer.com/de/book/9780387944166
Another good reference is the book of Jack Hale: www.springer.com/gp/book/9781461298946
Sir what software are you using for black board?
could you make a sequel? I bet there are many more
friend great video :-)
What does an antiderivative of a fraction differential look like?? Some sort of "fractional integral" ??
i am 12 and i love physics and math
You are doing God's work
I'm curious to see the kinds of phenomena that these DE's can model.
There's a bunch! DDEs, for example, can be used to model respiration and other physiologic feedback systems. CO2 in the blood is used to drive breathing, but there is a delay between sensing the CO2 and conveying that to the lungs to drive up breathing. There's other examples, but this is just one I learned back in undergrad!
Variable order/random order DE's come up in the quantitative description of permeability in certain porous materials. You can keep complicating them infinitely.
stochastic DE usually gets introduced in Brownian motion
I've got a video on my channel going over the noisy pendulum (a stochastic ODE) if you're interested. It discusses how to solve it rather than talking about application and analysis of the physical problem though.
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