Re: [Maxima-discuss] Kovacic Algorithm

[Nijso B. <ni...@ho...>]

Dear all,

I have created a git repository for the Kovacic algorithm. You can find
it here:
https://github.com/bigfooted/maxima-odesolve

The usage is simple, just call kovacicODE:

load("kovacic.mac")'
declare(a,constant);
kovacicODE('diff(y,x,2)=a*y,y,x);


If kovacicODE finds a solution, it returns [y=solution] and if it
doesn't find a solution, it returns false.

The Kamke database has 445 ODE's, it has 363 ODEs with rational
coefficients, and 181 of those can be solved by Kovacic' algorithm.
Kovacic.mac currently returns 180 solutions.


Some solutions take a long time to compute, e.g. Kamke 2.152. This is
due to an intermediate integrate taking 10 minutes or so.
Some returned solutions are quite complicated and could be simplified,
but this is a difficult problem.
I am not a maxima expert. If you see strange things in the code, please
let me know. Have fun!

Best,
Nyso


 
On Sun, 2014-03-23 at 22:11 +0100, Nijso Beishuizen wrote:
> Dear list, please see below. If someone is interested in  kovacicODE.mac
> (around 34kb), just let me know. 
> 
> Best,
> Nyso
> 
> On Sun, 2014-03-23 at 22:02 +0100, Nijso Beishuizen wrote:
> > Dear Neeraj,
> > 
> > 
> > kovacicODE solves second order linear ODE's with Liouvillian solutions
> > In maxima, just do something like this:
> > 
> > load('kovacic.mac');
> > ode:'diff(y,x,2)+a*'diff(y,x)+b*x=c;
> > kovacicODE(ode,y,x);
> > 
> > Note that there is a lot of output before the final solution is given
> > 
> > kovacicODE was tested using the Kamke database, which is included in
> > maxima in share/contrib/diffequations. 
> > It solves around 50% or so of the second order linear equations.
> > 
> > 
> > The Kovacic algorithm is known to take a long time to finish for certain
> > ODE's, on my pc sometimes 5 minutes before it finally concludes that no
> > Liouvillian solutions exist.  
> > 
> > 
> > It is recommended to define constants appearing in the ODE as constants.
> > It might be necessary to restrict the constants further, i.e. say that
> > a>0 or even x>0.
> > It is sometimes necessary to load absimp to get rid of abs(a) when a>0.
> > 
> >  
> > 
> > After kovacicODE has found a solution, it tries to simplify the solution by
> > merging constants into the integration constants %k1 and %k2.
> > I also try some simplifications and check the number of operators in the
> > new result. If the result has less operators, the simplification is
> > accepted.
> > In some cases, you will still end up with a complex solution, or with a
> > solution containing exponential integrals or gamma_incomplete solutions,
> > even though the solution can be simplified further.  
> > 
> > 
> > 
> > 
> > On Wed, 2014-03-19 at 12:14 +0530, Neeraj Sangwan wrote:
> > > I am interested in development version. Could you please send it to me.
> > > 
> > > Thanking you
> > > Neeraj
> > > 
> > > On Wed, Mar 19, 2014 at 2:32 AM, nijso beishuizen <ni...@ho...> wrote:
> > > > Hello Neeraj,
> > > >
> > > > I have written an implementation of the Kovacic algorithm. It is based on
> > > > the thesis of Carolyn Smith which you can download from the university of
> > > > waterloo as a pdf:
> > > >
> > > > https://cs.uwaterloo.ca/research/tr/1984/CS-84-35.pdf
> > > >
> > > > I have a development version, if you are interested I can send it to you.
> > > >
> > > > Best,
> > > > Nijso
> > > >
> > > >
> > > >> Date: Sat, 8 Mar 2014 11:23:17 +0530
> > > >> Subject: Kovacic Algorithm
> > > >> From: ms...@gm...
> > > >> To: ni...@ho...
> > > >
> > > >>
> > > >> Respected Sir
> > > >>
> > > >> I was searching form some implemntation of kovacic algorithm and I
> > > >> fouund your question regarding the same somewhere. Did you find some
> > > >> implementation? If yes could you please send some inks or files
> > > >> regarding this?
> > > >>
> > > >> Thanking you
> > > >> Neeraj
> > > >> 5th year Integrated MS student
> > > >> IISER Mohali
> > > 
> > 
> > 
> > 
> 
> 
> 
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