ASP: Automated symbolic computation of approximate symmetries of differential equations

Abstract A recent paper (Pakdemirli et al. (2004) [12] ) compared three methods of determining approximate symmetries of differential equations. Two of these methods are well known and involve either a perturbation of the classical Lie symmetry generator of the differential system (Baikov, Gazizov and Ibragimov (1988) [7] , Ibragimov (1996) [6] ) or a perturbation of the dependent variable/s and subsequent determination of the classical Lie point symmetries of the resulting coupled system (Fushchych and Shtelen (1989)  [11] ), both up to a specified order in the perturbation parameter. The third method, proposed by Pakdemirli, Yurusoy and Dolapci (2004)  [12] , simplifies the calculations required by Fushchych and Shtelen’s method through the assignment of arbitrary functions to the non-linear components prior to computing symmetries. All three methods have been implemented in the new MAPLE package ASP (Automated Symmetry Package) which is an add-on to the MAPLE symmetry package DESOLVII (Vu, Jefferson and Carminati (2012)  [25] ). To our knowledge, this is the first computer package to automate all three methods of determining approximate symmetries for differential systems. Extensions to the theory have also been suggested for the third method and which generalise the first method to systems of differential equations. Finally, a number of approximate symmetries and corresponding solutions are compared with results in the literature. Program summary Program title: ASP Catalogue identifier: AEOG_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEOG_v1_0.html Program obtainable from: CPC Program Library, Queen’s University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 15819 No. of bytes in distributed program, including test data, etc.: 185052 Distribution format: tar.gz Programming language: MAPLE internal language. Computer: PCs and workstations. Operating system: Linux, Windows XP and Windows 7. RAM: Depends on the type of problem and the complexity of the system (small ≈ MB, large ≈ GB) Word size: Supports 32 and 64 bit platforms Classification: 4.3, 5. Nature of problem: Calculates approximate symmetries for differential equations using any of the three methods as proposed by Baikov, Gazizov and Ibragimov [1,2], Fushchych and Shtelen [3] or Pakdemirli, Yurusoy and Dolapci [4]. Package includes an altered version of the DESOLVII package (Vu, Jefferson and Carminati [5]). Solution method: See Nature of problem above. Restrictions: Sufficient memory may be required for large systems. Running time: Depends on the type of problem and the complexity of the system (small ≈ seconds, large ≈ hours). References: [1] V. A. Baikov, R.K. Gazizov, N.H. Ibragimov, Approximate symmetries of equations with a small parameter, Mat. Sb. 136 (1988), 435-450 (English Transl. in Math. USSR Sb. 64 (1989), 427–441). [2] N.H. Ibragimov (Ed.), CRC Handbook of Lie Group Analysis of Differential Equations, Vol. 3, CRC Press, Boca Raton, FL, 1996. [3] W. I. Fushchych, W.H. Shtelen, On approximate symmetry and approximate solution of the non-linear wave equation with a small parameter, J. Phys. A: Math. Gen. 22 (1989), 887–890. [4] M. Pakdemirli, M. Yurusoy I. T. Dolapci, Comparison of Approximate Symmetry Methods for Differential Equations, Acta Applicandae Mathematicae 80 (2004), 243–271. [5] K. T. Vu, G. F. Jefferson, J. Carminati, Finding generalised symmetries of differential equations using the MAPLE package DESOLVII, Computer Physics Communications 183 (2012), 1044–1054.