John Carminati

A comparative study of some computer algebra packages which determine the lie point symmetries of differential equations

The study seeks to determine which of five computer algebra packages is best at finding the Lie point symmetries of systems of partial differential equations with minimal user intervention. The chosen packages are LIEPDE and DIMSYM for REDUCE, LIE and BIGLIE for MUMATH, DESOLV for MAPLE, and MATHLIE for MATHEMATICA. A series of systems of partial differential equations are used in the study. The paper concludes that while all of the computer packages are useful, DESOLV appears to be the most successful system at determining the complete set of Lie point symmetries of systems of partial differential equations.

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.

Similarity solutions of partial differential equations using DESOLV

We present and describe new reduction routines included in DESOLV which, in many cases, may allow the complete automation of the determination of similarity solutions of partial differential equations.

Simultaneous estimation of optical flow and object state: A modified approach to optical flow calculation

Differential optical flow methods are widely used within the computer vision community. They are classified as being either local, as in the Lucas-Kanade method, or global, as in the Horn-Schunck technique. As the physical dynamics of an object is inherently coupled into the behavior of its image in the video stream, in this paper, we use such dynamic parameter information in calculating optical flow when tracking a moving object using a video stream. Indeed, we use a modified error function in the minimization that contains physical parameter information. Further, the refined estimates of optical flow is used for better estimation of the physical parameters of the object in the simultaneous estimation of optical flow and object state(SEOS).

Shear-free perfect fluids with solenoidal magnetic curvature and a γ-law equation of state

We show that shear-free perfect fluids obeying an equation of state p = (γ − 1)μ are non-rotating or non-expanding under the assumption that the spatial divergence of the magnetic part of the Weyl tensor is zero.

Purely magnetic locally rotationally symmetric spacetimes

We consider all purely magnetic, locally rotationally symmetric (LRS) spacetimes. It is shown that such spacetimes belong to either LRS class I or III by the Ellis classification. For each class the most general solution is found exhibiting a disposable function and three parameters. A Segre classification of purely magnetic LRS spacetimes is given together with the compatibility requirements of two general energy–momentum tensors. Finally, implicit solutions are obtained, in each class, when the energy–momentum tensor is a perfect fluid.

A new algorithm for the Segre classification of the trace-free Ricci tensor

A new algorithm, based on the introduction of new spinor quantities, for the Segre classification of the trace-free Ricci tensor is presented. It is capable of automatically distinguishing between the two Segre types [1,1(11)] and [(1,1)11] where all other known algorithms fail to do so.

On the problem of algebraic completeness for the invariants of the Riemann Tensor: I

We study the set CZ of invariants [Zakhary and Carminati, J. Math. Phys. 42, 1474 (2001)] for the class of space–times whose Ricci tensors possess a null eigenvector. We show that all cases are maximally backsolvable, in terms of sets of invariants from CZ, but that some cases are not completely backsolvable and these all possess an alignment between an eigenvector of the Ricci tensor with a repeated principal null vector of the Weyl tensor. We provide algebraically complete sets for each canonically different space–time and hence conclude with these results and those of a previous article [Carminati, Zakhary, and McLenaghan, J. Math. Phys. 43, 492 (2002)] that the CZ set is determining or maximal.

On an alignment condition of the weyl tensor

We generalize an alignment condition of the Weyl tensor given by Barnes and Rowlingson. The alignment condition is then applied to Petrov type D perfect fluid spacetimes. In particular, purely magnetic, Petrov type D, shear-free perfect fluids are shown to be locally rotationally symmetric.

Shear-free perfect fluids with a solenoidal electric curvature

We prove that the vorticity or the expansion vanishes for any shear-free perfect fluid solution of the Einstein field equations where the pressure satisfies a barotropic equation of state and the spatial divergence of the electric part of the Weyl tensor is zero.