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[Licence| Download | New Version Template] adpr_v2_0.tar.gz(31 Kbytes)
Manuscript Title: PSsolver: A Maple implementation to solve first order ordinary differential equations with Liouvillian solutions
Authors: J. Avellar, L.G.S. Duarte, L.A.C.P. da Mota
Program title: PSsolver
Catalogue identifier: ADPR_v2_0
Distribution format: tar.gz
Journal reference: Comput. Phys. Commun. 183(2012)2313
Programming language: Maple 14 (also tested using Maple 15 and 16).
Computer: Intel Pentium Processor P6000, 1.86 GHz.
Operating system: Windows 7.
RAM: 4 GB DDR3 Memory
Keywords: First order ordinary differential equations (FOODEs), Symbolic computation, Prelle-Singer procedure, Liouvillian and elementary functions, General purpose.
PACS: 02.30.Hq.
Classification: 4.3.

Does the new version supersede the previous version?: Yes

Nature of problem:
Symbolic solution of first order differential equations via the Prelle-Singer method.

Solution method:
The method of solution is based on the standard Prelle-Singer method, with extensions for the cases when the FOODE contains elementary functions. Additionally, an extension of our own which solves FOODEs with Liouvillian solutions is included.

Reasons for new version:
The program was not running anymore due to changes in the latest versions of Maple. Additionally, we corrected/changed some bugs/details that were hampering the smoother functioning of the routines.

Summary of revisions:
  • As time went by, many commands in Maple were deprecated. So, in order to make the program able to run with the newer versions, we have checked and changed some of those. For instance, the command sum had changed, and some program lines were substituted so that the package works properly.
  • In the old version we must supply the degree of the Darboux polynomials we want to determine. In the present version the user can set the degree by typing Deg=number in the command call (e.g., PSsolve(ode,Deg=3); telling the command PSsolve that it must use Darboux polynomials of degree up to three). If the user does not specify the degree, the routines use, as default, the degree 1.

Restrictions:
If the integrating factor for the FOODE under consideration has factors of high degree in the dependent and independent variables and in the elementary functions appearing in the FOODE, the package may spend a long time finding the solution. Also, when dealing with FOODEs containing elementary functions, it is essential that the algebraic dependency between them is recognized. If that does not happen, our program can miss some solutions.

Unusual features:
Our implementation of the Prelle-Singer approach not only solves FOODEs, but can also be used as a research tool that allows the user to follow all the steps of the procedure. For example the Darboux polynomials (eigenpolynomials) of the D-operator associated with a FOODE can be calculated. In addition, our package is successful in solving FOODEs that were not solved by some of the most commonly available solvers. Finally, our package implements a theoretical extension (for details see [1,2]) to the original Prelle-Singer approach that enhances its scope, allowing it to tackle some FOODEs whose solutions involve non-elementary Liouvillian functions.

Running time:
This depends strongly on the FOODE, but usually under 2 seconds when running our 'arena' test file: The non linear FOODEs presented in the book by Kamke [3]. These times were obtained using a Intel Pentium Processor P6000, 1.86 GHz, with 4 GB RAM.

References:
[1] M. Singer, Liouvillian First Integrals of Differential Equations Trans. Amer. Math. Soc., 333 673-688 (1992).
[2] L.G.S. Duarte, S.E.S. Duarte, L.A.C.P. da Mota and J.E.F. Skea, Tackling FOODEs with Liouvillian Solutions within the PS-procedure, submitted to J. Phys. A: Mathematical, Nuclear and General.
[3] E. Kamke, Differentialgleichungen: Lösungsmethoden und Lösungen. Chelsea Publishing Co, New York (1959).