FracSym: Automated symbolic computation of Lie symmetries of fractional differential equations

Abstract In this paper, we present an algorithm for the systematic calculation of Lie point symmetries for fractional order differential equations (FDEs) using the method as described by Buckwar & Luchko (1998) and Gazizov, Kasatkin & Lukashchuk (2007, 2009, 2011). The method has been generalised here to allow for the determination of symmetries for FDEs with n independent variables and for systems of partial FDEs. The algorithm has been implemented in the new MAPLE package FracSym (Jefferson and Carminati 2013) which uses routines from the MAPLE symmetry packages DESOLVII (Vu, Jefferson and Carminati, 2012) and ASP (Jefferson and Carminati, 2013). We introduce FracSym by investigating the symmetries of a number of FDEs; specific forms of any arbitrary functions, which may extend the symmetry algebras, are also determined. For each of the FDEs discussed, selected invariant solutions are then presented. Program summary Program title: FracSym Catalogue identifier: AERA_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AERA_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.: 16802 No. of bytes in distributed program, including test data, etc.: 165364 Distribution format: tar.gz Programming language: MAPLE internal language. Computer: PCs and workstations. Operating system: Linux, Windows XP and Windows 7. RAM: Will depend on the order and/or complexity of the differential equation or system given (typically MBs). Classification: 4.3. Nature of problem: Determination of the Lie point symmetries of fractional differential equations (FDEs). Solution method: This package utilises and extends the routines used in the MAPLE symmetry packages DESOLVII (Vu, Jefferson and Carminati [1]) and ASP (Jefferson and Carminati [2]) in order to calculate the determining equations for Lie point symmetries of FDEs. The routines in FracSym automate the method of finding symmetries for FDEs as proposed by Buckwar & Luchko [3] and Gazizov, Kasatkin & Lukashchuk in [4,5] and are the first routines to automate the symmetry method for FDEs in MAPLE. Some extensions to the basic theory have been used in FracSym which allow symmetries to be found for FDEs with n independent variables and for systems of partial FDEs (previously, symmetry methods as applied to FDEs have only been considered for scalar FDEs with two independent variables and systems of ordinary FDEs). Additional routines (some internal and some available to the user) have been included which allow for the simplification and expansion of infinite sums, identification and expression in MAPLE of fractional derivatives (of Riemann–Liouville type) and calculation of the extended symmetry operators for FDEs. Restrictions: Sufficient memory may be required for large and/or complex differential systems. Running time: Depends on the order and complexity of the differential equations given. Usually seconds. References: [1] K.T. Vu, G.F. Jefferson, J. Carminati, Finding generalised symmetries of differential equations using the MAPLE package DESOLVII, Comput. Phys. Commun. 183 (2012) 1044. [2] G.F. Jefferson, J. Carminati, ASP: Automated Symbolic Computation of Approximate Symmetries of Differential Equations, Comput. Phys. Commun. 184 (2013) 1045. [3] E. Buckwar, Y. Luchko, Invariance of a partial differential equation of fractional order under the lie group of scaling transformations, J. Math. Anal. Appl. 227 (1998) 81. [4] R.K. Gazizov, A.A. Kasatkin and S.Y. Lukashchuk, Continuous transformation groups of fractional differential equations, Vestn. USATU 9 (2007) 125. [5] R.K. Gazizov, A.A. Kasatkin and S.Y. Lukashchuk, Symmetry properties of fractional diffusion equations, Phys. Scr. T136 (2009) 014016.