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[Licence| Download | New Version Template] aeql_v2_0.tar.gz(34 Kbytes)
Manuscript Title: Improving a family of Darboux methods for rational second order ordinary differential equations
Authors: J. Avellar, L.G.S. Duarte, L.A.C.P. da Mota
Program title: FiOrDi
Catalogue identifier: AEQL_v2_0
Distribution format: tar.gz
Journal reference: Comput. Phys. Commun. 195(2015)221
Programming language: Maple (release 17).
Computer: PC.
Operating system: Windows 7. Windows Vista.
RAM: 128 Mb
Keywords: First integrals, Second order ordinary differential equations, Darboux type approach, Computer algebra, Darboux polynomials.
PACS: 02.30.Hq.
Classification: 4.3, 5.

Does the new version supersede the previous version?: Yes

Nature of problem:
Determination of first order differential invariants for rational second order ordinary differential equations.

Solution method:
The method of solution is based on a Darboux type approach.

Reasons for new version:
We have been working on analyzing and solving systems of first and second order differential equations (1ODEs and 2ODEs, respectively) from a numerical point of view, using Lie methods and Darboux type approaches. For this latter class of methods, we have been developing (semi) algorithms to deal with classes of ODEs. In these algorithms, one fact has been always present: the most (computationally) costly step is the determination of the associated Darboux polynomials. Based on this realization, here we will be focused on speeding the process of finding Darboux polynomials for a class of ODEs of our interest. In particular, in this paper, we will talk about a class of rational 2ODEs.

Summary of revisions:
We have realized that one can extract information regarding the Darboux polynomials, correspondent to the D-operator related to the 2ODE (please see [1]) being studied in a very straightforward way. This, although a simple procedure, will prove essential to solve (or at least reduce) some 2ODEs. As mentioned in [1], the fact that our method uses the differential operator defines as in eq. (1) is very advantageous. So, let us first define the D-operator we have been using:

D ≡ Nδx + zNδy + Mδz

where z=dy(x)/dx, y=y(x), and M and N are polynomials in (x,y,z).

Considering this D-operator, one can see, by inspection on (1), that the cases that will be of interest to us are the ones listed below:

1. Darboux polynomials as factors of the numerator (in M) that are polynomials on (z) only;
2. Darboux polynomials as factors of the denominator (in N) that are polynomials without z;

So, the main revision we introduce here is to implement routines, in our new Maple code, to look for the needed Darboux polynomials for 2ODEs belonging to the classes mentioned above, just by inspecting the general expression for the 2ODE. With this, we make it available for the researcher, using our package, a more powerful weapon. These implementation is basically done via a modification to the command Invar, now one can use an extra argument MNDarboux in order to make use of the ideas explained above. Apart from that, some bugs were removed.

If, for the 2ODE under consideration, the Darboux Polynomials are of high degree ( > 3 ) in the dependent and independent variables, the package may spend an impractical amount of time to obtain the solution. That restriction is in part lifted by the modifications hereby introduced.

Unusual features:
Since our package is based on our theoretical developments [1], it can successfully reduce some rational 2ODEs that were not solved (or reduced) by some of the best-known methods available. This situation is even enhanced via the improvements that we have made here.
Let us present an example that shows the power of the change introduced. The command now is able to (automatically) find Darboux polynomials from M (depending on (z)) and N (depending on (x,y)):

d2 / dx2 (y) = -1/7 ((z7-3)2 (-9y8zx7+5y9x6+8y9zx+y10-6x5+z)) / (z6(-x6+y)(xy9-1))

Using what we have been learning, we can see that we have Darboux polynomials coming from both M and N. Bellow we will display them and their corresponding co-factors:

v1=z7-3 → g1 = -7z6(z7-3) (9x7y8z-5x6y9-8xy9z-y10+6x5-z)
v2 = x6-y → g2 = 7(xy9-1)(6x5-z)z6
v3 = xy9-1 → g3 = 7(x6-y)(9xz+y)y8z6

Using the method briefly described above one conclude that, for this ODE, we have the following results for the parameters and functions needed to find the differential invariant for the ODE:

P = 1/81 ((z7-3)2(9x7y8-8xy9-1))
Q = 1/81
R = 1/ ((z7-3)2(x6-y)(xy9-1))

and, using the theory described in [1], we finally find the first order invariant given by:

1/81 ((ln(-x6+y)z7 - ln(xy9-1)z7 - 3ln(-x6+y) + 3ln(xy9-1) + 1)/(z7-3))

It worth mention that the presence of the Darboux polynomials of such a high degree (as can be seen above) , with terms up to the power of 10, makes the regular process of determining it very "expensive" in time expenditure and memory. After applying the method here presented, which very quickly determined the needed Darboux polynomials, the algorithm we introduced in [1] finds the results (4) and (5). For this particular instance, the inbuilt Maple (very powerful) dsolve command fails to reduce this ode. Our procedure takes some minutes but reduces it.

Running time:
This depends strongly on the ODE, but usually under 4 seconds.

[1] L.G.S. Duarte and L.A.C.P. da Mota, Finding Elementary First Integrals for Rational Second Order Ordinary Differential Equations, Journal of Mathematical Physics, Volume 50, Issue 1, pp. 013514-013514-17 (2009)