GeM software package for computation of symmetries and conservation laws of differential equations

We present a recently developed Maple-based “GeM” software package for automated symmetry and conservation law analysis of systems of partial and ordinary differential equations (DE). The package contains a collection of powerful easy-to-use routines for mathematicians and applied researchers. A standard program that employs “GeM” routines for symmetry, adjoint symmetry or conservation law analysis of any given DE system occupies several lines of Maple code, and produces output in the canonical form. Classification of symmetries and conservation laws with respect to constitutive functions and parameters present in the given DE system is implemented. The “GeM” package is being successfully used in ongoing research. Run examples include classical and new results.

Program summary

Title of program: GeMCatalogue identifier: ADYK_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADYK_v1_0Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandLicensing provisions: noneComputers: PC-compatible running Maple on MS Windows or Linux; SUN systems running Maple for Unix on OS SolarisOperating systems under which the program has been tested: Windows 2000, Windows XP, Linux, SolarisProgramming language used: Maple 9.5Memory required to execute with typical data: below 100 MegabytesNo. of lines in distributed program, including test data, etc.: 4939No. of bytes in distributed program, including test data, etc.: 166 906Distribution format: tar.gzNature of physical problem: Any physical model containing linear or nonlinear partial or ordinary differential equations.Method of solution: Symbolic computation of Lie, higher and approximate symmetries by Lie's algorithm. Symbolic computation of conservation laws and adjoint symmetries by using multipliers and Euler operator properties. High performance is achieved by using an efficient representation of the system under consideration and resulting symmetry/conservation law determining equations: all dependent variables and derivatives are represented as symbols rather than functions or expressions.Restrictions on the complexity of the problem: The GeM module routines are normally able to handle ODE/PDE systems of high orders (up to order seven and possibly higher), depending on the nature of the problem. Classification of symmetries/conservation laws with respect to one or more arbitrary constitutive functions of one or two arguments is normally accomplished successfully.Typical running time: 1-20 seconds for problems that do not involve classification; 5-1000 seconds for problems that involve classification, depending on complexity.     

A reduce package for determining lie symmetries of ordinary and partial differential equations

Title of program: LIE0,LIE1,LIE2,LIE3,LIE4 Catalogue number: AAZB Program obtainable form: CPC Program library, Queen's University in Belfast, N. Ireland (see application form in this issue) Computer: Siemens 7.760 Operating system: BS 2000 Programming language used: LISP High speed storage required: depends on the problem, minimum about 400 000 bytes No. of bits in a word: 32 Number of cards in combined program and test deck: 200     

Automatically determining symmetries of partial differential equations

A REDUCE package for determining the group of Lie symmetries of an arbitrary system of partial differential equations is described. It may be used both interactively and in a batch mode. In many cases the system finds the full group completely automatically. In some other cases there are a few linear differential equations of the determining system left the solution of which cannot be found automatically at present. If it is provided by the user, the infinitesimal generators of the symmetry group are returned.     

Structural identifiability analysis via symmetries of differential equations

Results and derivations are presented for the generation of a local Lie algebra that represents the ‘symmetries’ of a set of coupled differential equations. The subalgebra preserving the observation defined on the model structure is found, thus giving all transformations of the system that preserve its structure. It is shown that this is equivalent to the similarity transformation approach (Evans, Chapman, Chappell, & Godfrey, 2002) for structural identifiability analysis and as such is a method of generating such transformations for this approach. This provides another method for performing structural identifiability analysis on nonlinear state-space models that has the possibility of extension to PDE type models. The analysis is easily automated and performed in Mathematica, and this is demonstrated by application of the technique to a number of practical examples from the literature.     

Automatic determination of dynamical symmetries of ordinary differential equations

We describe the application of a computer program CRACKSTAR for the exact analytic solution of overdetermined systems of differential equations which result in determining point-, contact- and dynamical symmetries of ordinary differential equations. Examples are discussed.     

几个非线性偏微分方程的非古典对称及相似解

研究了一些非线性偏微分方程的非古典势对称和非古典对称,得到了某些方程的新的势对称和新的对称,同时也得到了其伴随系统的新的对称,并求出了一些相似解．这些解对进一步研究这些非线性偏微分方程所描述的物理现象具有广泛的应用价值．     

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

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$n$ 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.     

[SADE] a Maple package for the symmetry analysis of differential equations

We present the package SADE (Symmetry Analysis of Differential Equations) for the determination of symmetries and related properties of systems of differential equations. The main methods implemented are: Lie, nonclassical, Lie–Bäcklund and potential symmetries, invariant solutions, first-integrals, Nöther theorem for both discrete and continuous systems, solution of ordinary differential equations, order and dimension reductions using Lie symmetries, classification of differential equations, Casimir invariants, and the quasi-polynomial formalism for ODE's (previously implemented by the authors in the package QPSI) for the determination of quasi-polynomial first-integrals, Lie symmetries and invariant surfaces. Examples of use of the package are given.

Program summary

Program title: SADECatalogue identifier: AEHL_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEHL_v1_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: Standard CPC license, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 27 704No. of bytes in distributed program, including test data, etc.: 346 954Distribution format: tar.gzProgramming language: MAPLE 13 and MAPLE 14Computer: PCs and workstationsOperating system: UNIX/LINUX systems and WINDOWSClassification: 4.3Nature of problem: Determination of analytical properties of systems of differential equations, including symmetry transformations, analytical solutions and conservation laws.Solution method: The package implements in MAPLE some algorithms (discussed in the text) for the study of systems of differential equations.Restrictions: Depends strongly on the system and on the algorithm required. Typical restrictions are related to the solution of a large over-determined system of linear or non-linear differential equations.Running time: Depends strongly on the order, the complexity of the differential system and the object computed. Ranges from seconds to hours.     

Cyclic animation using partial differential equations

This work presents an efficient and fast method for achieving cyclic animation using partial differential equations (PDEs). The boundary-value nature associated with elliptic PDEs offers a fast analytic solution technique for setting up a framework for this type of animation. The surface of a given character is thus created from a set of pre-determined curves, which are used as boundary conditions so that a number of PDEs can be solved. Two different approaches to cyclic animation are presented here. The first of these approaches consists of attaching the set of curves to a skeletal system, which is responsible for holding the animation for cyclic motions through a set mathematical expressions. The second approach exploits the spine associated with the analytic solution of the PDE as a driving mechanism to achieve cyclic animation. The spine is also manipulated mathematically. In the interest of illustrating both approaches, the first one has been implemented within a framework related to cyclic motions inherent to human-like characters. Spine-based animation is illustrated by modelling the undulatory movement observed in fish when swimming. The proposed method is fast and accurate. Additionally, the animation can be either used in the PDE-based surface representation of the model or transferred to the original mesh model by means of a point to point map. Thus, the user is offered with the choice of using either of these two animation representations of the same object, the selection depends on the computing resources such as storage and memory capacity associated with each particular application.     

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.     

Solutions of delay differential equations by using differential transform method

Differential transform method (DTM) is extended for delay differential equations. By using DTM, we manage to obtain the numerical, analytical, and exact solutions of both linear and nonlinear equations. In comparison with the existing techniques, the DTM is a reliable method that needs less work and does not require strong assumptions and linearization.     

Checking system boundedness using ordinary differential equations

Boundedness is one of the most important properties of discrete Petri nets. Determining the boundedness of a Petri net is usually done through building coverability graph or coverability tree. However, the computation is infeasible for complex applications because the size of the coverability graph may increase faster than any primitive recursive functions. This paper proposes a new technique to check the boundedness without causing this problem. Let a concurrent system be represented by a (discrete) Petri net. By relaxing the (discrete) Petri net to a continuous Petri net, we can model the concurrent system by a family of ordinary differential equations. It has been shown that the boundedness of the discrete Petri net is equivalent to the boundedness of the solutions of the corresponding ordinary differential equations. Hence, we can check the boundedness of a (discrete) Petri net by analyzing the solutions of a family of ordinary differential equations. A case study demonstrates the benefits of our technique.     

Numerical solution of differential equations using Haar wavelets

Haar wavelet techniques for the solution of ODE and PDE is discussed. Based on the Chen–Hsiao method [C.F. Chen, C.H. Hsiao, Haar wavelet method for solving lumped and distributed-parameter systems, IEE Proc.—Control Theory Appl. 144 (1997) 87–94; C.F. Chen, C.H. Hsiao, Wavelet approach to optimising dynamic systems, IEE Proc. Control Theory Appl. 146 (1997) 213–219] a new approach—the segmentation method—is developed. Five test problems are solved. The results are compared with the result obtained by the Chen–Hsiao method and with the method of piecewise constant approximation [C.H. Hsiao, W.J. Wang, Haar wavelet approach to nonlinear stiff systems, Math. Comput. Simulat. 57 (2001) 347–353; S. Goedecker, O. Ivanov, Solution of multiscale partial differential equations using wavelets, Comput. Phys. 12 (1998) 548–555].     

基于微分方程的程序性能分析

基于连续Petri网模型,用一组常微分方程来描述程序,通过研究微分方程的解来研究程序的性能。每个微分方程描述程序状态的变化,每个状态可由介于0和1之间的数来度量,显示程序到达状态的程度。该方法的好处在于在做程序分析时,可避开状态爆炸问题。     

Interconnections and symmetries of linear differential systems

In this paper we study the interplay between control problems and symmetries in the context of linear systems. In particular, we establish sufficient conditions under which it is possible to control a symmetric system in order to make it achieve control objectives, without breaking its symmetry.     

Time-series forecasting using a system of ordinary differential equations

This paper presents a hybrid evolutionary method for identifying a system of ordinary differential equations (ODEs) to predict the small-time scale traffic measurements data. We used the tree-structure based evolutionary algorithm to evolve the architecture and a particle swarm optimization (PSO) algorithm to fine tune the parameters of the additive tree models for the system of ordinary differential equations. We also illustrate some experimental comparisons with genetic programming, gene expression programming and a feedforward neural network optimized using PSO algorithm. Experimental results reveal that the proposed method is feasible and efficient for forecasting the small-scale traffic measurements data.     

Numerical solution of nonlinear differential equations using computer algebra

This paper describes a numerical method for finding periodic solutions to nonlinear ordinary differential equations. The solution is approximated by a trigonometric series. The series is substituted into the differential equation using the FORMAC computer algebra system for the resulting lengthy algebraic manipulations. This lead to a set of nonlinear algebraic equations for the series coefficients. Modern search methods are used to solve for the coefficients. The method is illustrated by application to Duffing’ equation.     

Numerical integration of partial differential equations using cubic splines

This paper presents techniques for the numerical solution of partial differential equations using cubic spline collocation.

The main spline relations are presented and incorporated into solution procedures for partial differential equations. The computational algorithm in every case is a tridiagonal matrix system amenable to efficient inversion methods. Truncation errors and stability are briefly discussed. Finally, some examples of their application to parabolic and hyperbolic systems with mixed boundary conditions are presented.

The results obtained are encouraging and justify further research in this field.