Finding higher symmetries of differential equations using the MAPLE package DESOLVII

Abstract

We present and describe, with illustrative examples, the MAPLE computer algebra package DESOLVII, which is a major upgrade of DESOLV. DESOLVII now includes new routines allowing the determination of higher symmetries (contact and Lie–Bäcklund) for systems of both ordinary and partial differential equations. Program summaryProgram title: DESOLVIICatalogue identifier: ADYZ_v2_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADYZ_v2_0.htmlProgram obtainable from: CPC Program Library, Queenʼs University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 10 858No. of bytes in distributed program, including test data, etc.: 112 515Distribution format: tar.gzProgramming language: MAPLE internal languageComputer: PCs and workstationsOperating system: Linux, Windows XP and Windows 7RAM: Depends on the type of problem and the complexity of the system (small≈MB, large≈GB)Classification: 4.3, 5Catalogue identifier of previous version: ADYZ_v1_0Journal reference of previous version: Comput. Phys. Comm. 176 (2007) 682Does the new version supersede the previous version?: YesNature of problem: There are a number of approaches one may use to find solutions to systems of differential equations. These include numerical, perturbative, and algebraic methods. Unfortunately, approximate or numerical solution methods may be inappropriate in many cases or even impossible due to the nature of the system and hence exact methods are important. In their own right, exact solutions are valuable not only as a yardstick for approximate/numerical solutions but also as a means of elucidating the physical meaning of fundamental quantities in systems.One particular method of finding special exact solutions is afforded by the work of Sophus Lie and the use of continuous transformation groups. The power of Lieʼs group theoretic method lies in its ability to unify a number of ad hoc integration methods through the use of symmetries, that is, continuous groups of transformations which leave the differential system “unchanged”. These symmetry groups may then be used to find special solutions. Solutions found in this manner are called similarity or invariant solutions.The method of finding symmetry transformations initially requires the generation of a large overdetermined system of linear, homogeneous, coupled PDEs. The integration of this system is usually reasonably straightforward requiring the (often elementary) integration of equations by splitting the system according to dependency on different orders and degrees of the dependent variable/s. Unfortunately, in the case of contact and Lie–Bäcklund symmetries, the integration of the determining system becomes increasingly more difficult as the order of the symmetry is increased. This is because the symmetry generating functions become dependent on higher orders of the derivatives of the dependent variables and this diminishes the overall resulting “separable” differential conditions derived from the main determining system. Furthermore, typical determining systems consist of tens to hundreds of equations and this, combined with standard mechanical solution methods, makes the process well suited to automation using computer algebra systems.The new MAPLE package DESOLVII, which is a major upgrade of DESOLV, now includes routines allowing the determination of higher symmetries (contact and Lie–Bäcklund) for systems of both ordinary and partial differential equations. In addition, significant improvements have been implemented to the algorithm for PDE solution. Finally, we have made some improvements in the overall automated process so as to improve user friendliness by reducing user intervention where possible.Solution method: See “Nature of problem” above.Reasons for new version: New and improved functionality.Summary of revisions:1.New functionality – can now compute generalised symmetries.2.Much improved efficiency (speed and memory use) of existing routines.Restrictions: Sufficient memory may be required for complex systems.Running time: Depends on the type of problem and the complexity of the system (small≈seconds, large≈hours).