Solving First Order Differential Equations with Julia

https://ritog.github.io/posts/1st-order-DE-julia/1st_order_DE_julia.html

by __rito__ • 1 year ago

125 32 1 year ago

ritog.github.io

anigbrowl• 1 year ago

X^2 - 3x - 18 =0

[...]

This is how the equation is created: x(x+3) = 18

wat

Maybe don't do very fast recaps if you're not going to proofread them. Incidentally I assume the formulae in this article were done with MathJax or its Julia equivalent, they render great but can't be copied from the text.

Overall a good article (and a great ad for Julia) but stumbling blocks like the one above ensure some readers won't make it any farther.

grandempire• 1 year ago

Is the issue here quadratic factorization is not obvious or that the sign is wrong?

Note that x(x+3) = 18 and x(x-3) = 18 are valid models of the same problem. One finds the longer side in terms of the shorter, or vice versa

hansvm• 1 year ago

Looks like proofreading (sign being wrong) to me.

zombot• 1 year ago

Both variants (+3 & -3) have the same shape, and so do their derivatives, but they do land in different places on the X axis. Is that irrelevant?

__rito__10• 1 year ago

"Note that x(x+3) = 18 and x(x-3) = 18 are valid models of the same problem. One finds the longer side in terms of the shorter, or vice versa"

That is the reason I made the mistake initially. They both were correct in my mind.

But that's no excuse for the mistake, and I made a correction.

beepbooptheory• 1 year ago

Well if they made it that far, they also read that this article is for "people who are already familiar with Differential Equations from Mathematics." I think if you're already familiar, a single wrong sign is not something thats going to be particularly hindering. It's more just something to jump on if you are, for some reason, itching to criticize it.

__rito__10• 1 year ago

It was my fault, and was created by a moment of absent-mindedness.

They are both correct individually, but definitely wrong when read together. I corrected it.

Thanks for reading the article and pointing out the obvious mistake. I will proofread more carefully and use a spell checker from the next time. There were some typos, too, that I corrected since then. Like 'langauge', 'equaion', etc.

I wrote the article in Jupyter Lab with a Julia kernel. I just wrote LaTeX in Markdown cells. I rendered the article with quarto and quarto is also how the blog is created. I am ignorant about how it renders LaTeX in webpages. Pandoc and MathJax seems to be involved. I am not sure.

"Overall a good article (and a great ad for Julia) but stumbling blocks like the one above ensure some readers won't make it any farther."

Thank you for your compliment. I absolutely don't want people not making it any farther due to a small mistake I made. I will be more mindful about it from the next time.

forgotpwd16• 1 year ago

The first example is also the very first example in the docs[1], but rather f!(du,u,p,t) that modifies du, it passes f(u,p,t) that returns du. Searching the help for ODEProblem() function, it's explained[2] that the first way is more memory-efficient, but if mutation is not allowed, the second way is more suitable. It will be of interest to write a follow-up post elaborating on this part.

[1] https://docs.sciml.ai/DiffEqDocs/stable/examples/classical_p... [2]: https://docs.sciml.ai/DiffEqDocs/stable/types/ode_types/#Sci...

selimthegrim• 1 year ago

I think I had volunteered to do something similar to this - let me check

__rito__10• 1 year ago

Author here.

Yes, f(u,p,t) returning du is the suggested way to do things, but this pattern of updating du through in-place updating is more memory efficient when a system of equations (say 10-100) is involved.

The suggested way, when we don't need to think about memory efficiency, is easier to write and reason about.

I have plans to write more on this topic, and make it to a series. After I finish that, if there is enough interest, I will try and do a benchmark of mutating versus non-mutating styles with varying numbers of equations in a system of DEs.

Thanks for reading the article and providing your opinion.

fussypart3• 1 year ago

can anyone recommend a good textbook on differential equations?

__rito__10• 1 year ago

Could you tell me a bit about your background, your level in mathematics, and the type of book you prefer? Do you enjoy highly rigorous, proof-heavy texts, or do you prefer books where the author explains concepts in a more conversational and detailed way—still rigorous, of course, but more wordy?

fussypart3• 1 year ago

I have a CS degree but haven't done math in a few years. Last course I completed was an MITx - intro to probability. While I enjoyed its rigor and detail, I am looking for something practical without proofs.

seanhunter2• 1 year ago

I wish more people did this (get the required context) before steaming in with recommendations tbh.

That said, I will steam in with some starting points that may be helpful.

1) An “anti-recommendation”: Probably don’t get Edwards and Penney unless you are forced to for some course. It’s _fine_ I guess - like you will learn from it, but it is staggeringly overpriced for what it is and there are enough typesetting errors and other little niggles that grate when a book is as expensive as that. The one good part about it is there are tons of problems but for many/most of them it just gives the answer not a full solution, so it’s not very helpful if you are stuck or your solution looks very different from theirs and you don’t know where to go from there.

2) If you want a free pdf or online resource, mathematics libretexts has “Differential equations for engineers” by Jiri Lebl, which is at least as good as Edwards and Penney and is free and you get the pdf if you want to download it https://math.libretexts.org/Bookshelves/Differential_Equatio...

3) Dover Books publish “Ordinary Differential Equations” by Tennenbaum, Morris and Pollard, which I don’t have personally but a lot of people recommend. It’s a Dover book which means it is cheap and some of the terminology and notation is probably a little bit old-fashioned but it’s going to be a lot cheaper than Edwards and Penney if you want a physical book and as I say a lot of people recommend it.

So to complement what the parent said, one approach if you’re not sure what type of book you prefer, is check out Lebl (because it’s online and free so easy to dip into) and then you can explore from there.

But don’t get Edwards and Penney. I got a cheap second-hand copy and I still think I probably overpaid.

I would add get a CAS. You could use wxmaxima (which is OSS) or get Mathematica or something if you can get a cheap student license. It’s going to help a lot to develop intuition by allowing you to plot direction fields etc much more easily as well as doing some of the heavy lifting of verifying solutions etc (although you really need to do that a bunch yourself so you get good at it).

alok-g• 1 year ago

Here's an interesting book to consider:

https://www.amazon.com/Advanced-Engineering-Mathematics-Math...

https://www.amazon.in/Advanced-Engineering-Mathematics-Mathu...

https://khannapublishers.in/index.php?route=product/product&...

(This sells in India for a mere US $4. Costlier elsewhere.)

Pros:

* This has the most exhaustive catalog of differential equations and methods to solving them that I have ever seen. An example would be the discussion on non-homogenous differential equations that I have never seen anywhere else at all.

* While the subject matter is not easy, the book is complete in itself. A reader would not be led into a "dependency hell" or worse a cyclic dependencies of other materials to read.

Cons:

* Very low quality typesetting, print and paper quality. I wish a improved edition could be in the pipeline, however, the book is rather out of print.

* The book is not focused on differential equations. The other material covered is however a bonus.

Note: The authors were professors at my engineering college. Am however not associated with them in any way and I have no vested interest in recommending the book.

cashsterling• 1 year ago

I'd consider Advanced Engineering Mathematics by Zill, 6th Edition which is available used for as little as 25 USD. There is also a print solution manual for this book which is great for self study.

nxpnsv3• 1 year ago

I didn't find a link to the package in the article. It seems to be a quite capable package, and the given two examples that are trivial to solve analytically don't really do it justice.

__rito__10• 1 year ago

I have added a link to the package now. The first mention of the package in the body now contains a link to the official page for the package.

and the given two examples that are trivial to solve analytically don't really do it justice.

Yes, I know. I plan to write more in the future.

nxpnsv3• 1 year ago

Cool, the link is helpful, and the package is pretty cool. I'd just add that the examples are simple and be done with it, you can revisit with more later. Keep writing :)

leephillips2• 1 year ago

It’s probably the best library for DEs in any language. I’m pretty sure that DifferentialEquations.jl, by itself, is the reason many scientists and engineers are using Julia.

https://docs.sciml.ai/DiffEqDocs/stable/

__rito__10• 1 year ago

I use Diffrax and my own stuff with Numba/CuPy JIT-ing. Can you compare the Python ecosystem for DEs with Julia according to your experience?

nxpnsv3• 1 year ago

I didn't try it yet, but the docs looks great.

sundarurfriend• 1 year ago

doggo dot jl mention!

Thanks for the article. It gave me a little bit of familiarity with the DiffEq side of Julia, but even more valuable to me was the comparison between ordinary equations and diffeqs:

In normal equations, we solve the equation to find out values of variables previously unknown to us. In case of Differential Equations, we solve to find out functions previously unknown to us.

That's a simple, easy to remember nugget of information that gives me some context to this whole field.

__rito__10• 1 year ago

I really wish that somebody told me this when I was in High School. In HS, DEs were just solving integrals with a simple step away around as far as the student is concerned. I learned about them much more deeply when in college, and later independently.

As you know this now- if you come across a young person studying DEs, feel free to share this line with him/her.

Thanks for reading the article and sharing your opinion.

Solving First Order Differential Equations with Julia

Related Stories