|[Licence| Download | New Version Template] adzf_v2_0.tar.gz(21 Kbytes)|
|Manuscript Title: Determining Liouvillian first integrals for dynamical systems in the
plane and an integrability analysis|
|Authors: J. Avellar, M.S. Cardoso, L.G.S. Duarte, L.A.C.P. da Mota|
|Program title: Lsolver|
|Catalogue identifier: ADZF_v2_0|
Distribution format: tar.gz
|Journal reference: Comput. Phys. Commun. 185(2014)1496|
|Programming language: Maple.|
|Computer: Any computer running Maple.|
|Operating system: Windows XP, Windows 7.|
|RAM: 20 Mb|
|Keywords: Symbolic Computation, Dynamical systems, Integrating factors, Integrability.|
Does the new version supersede the previous version?: Yes
Nature of problem:
We deal with the analytical solving of 1ODEs via a Darboux
method. We also introduce an integrability analysis for this present case (1ODEs)
that was previously presented in the context of 2ODEs.
The method of solution is based on the standard Prelle-Singer
(Darboux) method , with extensions for the cases when the 1ODEs of our own,
which solves 1ODEs with Liouvillian solutions, is included. The first part of the
Method depends on the finding of the Darboux polynomials for the ODE. The
integrability analysis mentioned above is performed at this stage.
Reasons for new version:
- Check and enforce compatibility with the newer versions of Maple (in this
case, Maple release 17).
- Some features of the program were modified in order to improve its
- The Integrability analysis was implemented.
Summary of revisions:
Regarding the question about the program running with
the newer versions of Maple, namely, Maple 17, the answer is that the deprecation
of certain commands in the Maple system has not damaged the running capability
of Lsolver. It runs properly. Nevertheless, we have introduced some changes in the
way some of its features are implemented:
- During the running of the routines in Lsolver, one has to use a degree for
the Darboux polynomials and another for the co-factors. There is, of course,
a default for those values: namely, 1 for the Darboux polynomials and the
maximum value possible for the co-factors. This maximum value is determined from
the Ode itself. It is the maximum degree (on both x or y) either of the numerator
or the denominator of the rational function defining the Ode minus 1. In this new
version of Lsolver, when the user decides to use a different value for the degree
of the Darboux polynomials (and/or for the co-factors) the user has to provide an
extra argument to the Lsolve command. This argument is of the form:
Deg = [nf, np] where nf is the degree for the Darboux
polynomials and np the one for the co-factors. If np is bigger than
it can be, the program tells you so and states which value was actually used
(the maximum possible one). If the value for np provided is smaller than
the maximum value, the one provided will be the one used.
- If by any means the user knows the Darboux polynomials and the corresponding
co-factors, he/she can feed it to the command Lsolve via the following extra
argument: DP = [[Darboux], [co-factors]]. Where
[Darboux] is a list with the Darboux polynomials and [co-factors] is
a list with the corresponding co-factors.
So, the calling of the Darboux procedure will be as follows:
The changes above were requested by the users that contacted us and suggested
some changes along those lines. So, we decided to oblige.
Finally, perhaps the biggest change we have introduced is the capability to perform
an Integrability analysis concerning the free parameters (possibly) appearing
on the Ode. Basically, we here implement for first order ordinary differential
equations (1ODEs) the idea we have presented in [2, 3]. The idea, although very
simple to implement, is powerful in regards to its effectiveness. We have
introduced a new routine, called Lintegrability. In a few words, if the user
provides the ODE and an argument in the form of a set sequence with the free
parameters to this command:
of the free parameters});
it produces an output of the form:
[[the Darboux Polynomials], [the co-factors], [the free parameters]]
i.e., a list with three lists inside it. Each one of these lists has the same
number of operands.
The command operates by using the Darboux commands internally. So, in
order to prepare this command for this new task, some changes had to be made to
it. Therefore, some new lines where introduced in the Darboux routine. For the
user, the Darboux command works as if it has not been modified.
So, in the cases where the integration can be made only after one knows the values
(or the range of values) for the free parameters present on the ODE (and the
corresponding Darboux polynomials), the user has to collect the Darboux
polynomials discovered by the usage of the Lintegrability command and feed them
to the Lsolve command to perform the integration.
The finding of the Darboux polynomials and the correspondent co-factors is still
time-demanding task of the method. So, one could say that, for the general case, if
the degree for the Darboux polynomials is "4", for instance, it would make the case
hard to deal with.
The possibility of searching for integrability regions, as described above.
This depends strongly on the 1ODE being analyzed. As mentioned
above, a degree for the Darboux polynomials "4" or above would (in general) make
dealing with the case unpractical. For the degree "3", the routines may take more
than a minute to deal with the task. Generally, the program takes, at most, for the
other cases, a few seconds.
| ||M. Prelle, M. Singer, Trans. Amer. Math. Soc. 279 (1983) 215.|
| ||J. Avellar, L.G.S. Duarte, S.E.S. Duarte, L.A.C.P. da Mota, in: Theodore
Simos, George Maroulis (Eds.), Lecture Series on Computer and Computational
Sciences, vol. 4B, 2005, p. 1786.|
| ||J. Avellar, L.G.S. Duarte, S.E.S. Duarte, L.A.C.P. da Mota, Appl. Math. Comput.
184 (2007) 2.|