Lie symmetries of differential equations

[Furkan Semih Dündar]

Dearl All,

I recently bought the book that I studied in my PhD years. It is "Differential Equations: their solution using symmetries" by Hans Stephani. https://www.amazon.com/Differential-Equations-Their-Solution-Symmetries/dp/0521355311/ref=sr_1_1?dchild=1&keywords=hans+stephani+differential+equation&qid=1591870417&sr=8-1

In the book, methods to solve nonlinear ODE's or PDE's are given if the differential equation has an underlying Lie group symmetry. I remember using the program "Reduce" in the course. It may be useful.

My aim is to incorporate the methods given in the book to Sage Math. So if you are interested maybe we can form a small group to contribute to Sage in this respect. I believe the ability to solve complex differential equations is quite important for a symbolic math software. Please let me know if you are interested.

Best regards,

--

F. Semih Dündar

[Mikhail Malakhaltsev]

Dear Furcan, 

The topic is very interesting, there are several implementations of "symmetry analysis of differential equations" in software, for example, see Gerd Baumann "Symmetry Analysis of Differential Equations with Mathematica", or the project jets.math.slu.cz (I did not know about the book by Hans Stephani, thank you). Of course, it would be interesting to do similar things in SAGE, and  I could also contribute a bit to this project :)

Best wishes,

Mikhail 

[David Joyner]

Is the command
sympy.solvers.ode.infinitesimals
(in sympy, therefore in Sage) related to what you want?

https://docs.sympy.org/latest/modules/solvers/ode.html#infinitesimals


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[Furkan Semih Dündar]

Dear David,

Yes it is related.

Dear Mikhail,

Let us check what is developed so far as a starting point. This looks like a good initial step.

I have been travelling since last evening. Sorry for the relatively late reply.

Best wishes,

Furkan.

13 Haz 2020 Cmt 16:21 tarihinde David Joyner <wdjo...@gmail.com> şunu yazdı:

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[Mikhail Malakhaltsev]

Dear David and Furkan,

For sure, it is related. About the theory of symmetries of ODE  one can also look in V. Arnold's text books on ODE,  in the books of list of references of John Starrett's paper "Solving equations by symmetry groups", there are various texts about the symmetries of PDE, for example, the book www.amazon.com/Symmetries-Conservation-Differential-Mathematical-Translations/dp/082180958X. Therefore,  theoretically the area of symmetries of DE is well-developed. Another thing is to design a package for SAGE...  

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[Tapir]

Hi!

I've started to write a package for my own sake some month ago, but didn't get very far by now. I'm still in the state of playing around, just to learn SAGE, trying to compute Janet bases as the will provide a valuable tool at its own (as simple Lie symmetry stuff is already existing). Anyone to join later on?

https://github.com/tapir442/delierium/tree/main

BR

Martin