Hereman, W. (1996). Symbolic Software for Lie Symmetry Analysis. CRC Handbook of Lie Group Analysis of Dierential Equations, Volume 3: New Trends in Theoretical Developments and Computational Methods. Chapter 13 (N.H. Ibragimov, ed.) Boca Raton, Florida: CRC Press, pp. 367-413.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....ansatze be investigated. One commonly used technique is to apply integrability conditions systematically by computing a characteristic system of the differential ideal [4] or differential Grobner basis (or a Pseudo differential Grobner basis) 5, 6, 7] Other related programs are listed in [3]. The algorithms to be described in this paper aim to extend such algorithms by enabling the integration of classes of equations either in parallel to the computation of integrability conditions or after a basis for the differential ideal differential Grobner basis has been computed. It appears to ....

....be 1. Further we assume the equations 0 = Delta to be at most polynomially non linear in the dependent variables u and their derivatives. A few minor generalizations will be added later. Even with this specification it appears that we are more general than most other programs that are listed in [3] which aim at solving over determined systems. Two strong programs to be mentioned are Dimsym of James Sherring, Geoff Prince and Michael Jerie and Lie of Alan Head. Integration techniques used in these programs are partially described in [2] 9] and [8] but to the best of the authors knowledge ....

W. Hereman. Symbolic software for lie symmetry analysis. In N.H. Ibragimov, editor, CRC Handbook of Lie Group Analysis of Differential Equations, Vol. 3: New trends in Theoretical Developments and Computational Methods, chapter 13, pages 367--413. CRC Press, Boca Raton, Florida, 1996.

....ansatze be investigated. One commonly used technique is to apply integrability conditions systematically by computing a characteristic system of the differential ideal [4] or a differential Grobner basis (or a Pseudo differential Grobner basis) 5, 6, 7] Other related programs are listed in [3]. The algorithms to be described in this paper aim to extend such algorithms by enabling the integration of classes of equations either in parallel to the computation of integrability conditions or after a basis for the differential ideal differential Grobner basis has been computed. It appears to ....

....be 1. Further we assume the equations 0 = Delta to be at most polynomially non linear in the dependent variables u and their derivatives. A few minor generalizations will be added later. Even with this specification it appears that we are more general than most other programs that are listed in [3] which aim at solving over determined systems. Two strong programs to be mentioned are Dimsym of James Sherring, Geoff Prince and Michael Jerie and Lie of Alan Head. The integration techniques used in these programs are partially described in [2] 8] and [9] but to the best of the author s ....

W. Hereman. Symbolic software for lie symmetry analysis. In N.H. Ibragimov, editor, CRC Handbook of Lie Group Analysis of Differential Equations, Vol. 3: New trends in Theoretical Developments and Computational Methods, chapter 13, pages 367--413. CRC Press, Boca Raton, Florida, 1996.

.... a differential ideal (pseudo differential Grobner basis) Algorithms for applying integrability conditions to a system of differential equations in a systematic way in order to generate simplified differential equations are implemented in a number of programs ( 3, 4, 7, 8, 10, 11, 12] and more in [6]) Such calculations result in the radical or a (pseudo) differential Grobner Basis of the differential ideal generated by the original system. A common problem of these algorithms, and consequently their implementations, is an explosive expression swell. Optimizations like Buchbergers 2 nd ....

Hereman, W., Symbolic Software for Lie Symmetry Analysis. Chapter 13 in CRC Handbook of Lie Group Analysis of Differential Equations, vol. 3, ed. NH Ibragimov. Boca Raton, Florida: CRC Press, 1996, pp. 367--413

....analysis. This approach has made the strongest impact on computer algebra applications to differential equations. There exist so many implementations that it is very difficult to keep an overview. We must refer here to a recent survey by Hereman [20] an updated and extended version will appear in [21]) Most packages just set up the determining system for symmetry generating vector fields. A few of the packages also try some heuristics to solve it [19, 44] usually they can cope, however, only with rather simple examples. Singularity analysis. We must distinguish two approaches: In the ....

W. Hereman. Symbolic software for Lie symmetry analysis. In N.H. Ibragimov, editor, CRC Handbook of Lie Group Analysis of Differential Equations, Volume 3: New Trends in Theoretical Development and Computational Methods, chapter 13. CRC Press, Boca Raton, Florida, to appear (1995).

.... mde : detools: modode(F,EULER1, z] t,h,4) 1] ModF : proc(t,y) begin [subs(mde,h=0.1,z=y[1] end proc: modsol : numeric: odesolve(0. 0. 95,ModF, 1] RKF78,Alldata) mod4 : detools: odeplot(modsol, 1] Title= ModEuler ,TitlePosition=[9,3]] plot2d(Title= Modified Equation for Euler Method ,TitlePosition=Below, FontSize=15,ForeGround= 0,0,0] BackGround= 1,1,1] Scaling=UnConstrained, euler,exact,mod4[1] Graphics Modified Equation for Euler method Euler Exact ModEuler The first line defines our differential equation ....

....Thus it is not surprising that already very early in the history of computer algebra the first packages for the symmetry analysis of differential equations emerged. Today it is almost impossible to maintain an overview on the many existing packages; comparatively recent surveys can be found in [8, 9]. MuPAD has been the only one among the major general purpose computer algebra system without a symmetry package. Therefore the first larger component of the detools library is such a package. In this first version it contains basically only methods for setting up determining systems; later ....

W. Hereman. Symbolic software for Lie symmetry analysis. In N.H. Ibragimov, editor, CRC Handbook of Lie Group Analysis of Differential Equations, Volume 3: New Trends in Theoretical Development and Computational Methods, chapter 13. CRC Press, Boca Raton, Florida, 1995.

....bibliography is of course far from being exhaustive. As a further source of references one should mention the survey by Singer [115] It gives much more details, especially on the more algebraic approaches, and contains a large bibliography. The same holds for the more focused surveys by Hereman [57, 58] covering symmetry theory and related fields and the one by MacCallum [74] on the integration of ordinary differential equations. In addition there have been three conferences devoted exclusively to differential equations and computer algebra. Their proceedings [114, 123, 124] contain a number of ....

....than the original equation. Often the repeated application of rather simple heuristics suffices to completely solve at least the determining system. There exist so many implementations of symmetry methods that it is rather difficult to keep an overview; we refer again to the surveys by Hereman [57, 58] with their huge bibliography. In almost any computer algebra system one can find at least a package for setting up the determining system which sometimes also tries to solve it. Some packages are even able to use the symmetries to compute automatically closed form solutions for some classes of ....

W. Hereman. Symbolic software for Lie symmetry analysis. In N.H. Ibragimov, editor, CRC Handbook of Lie Group Analysis of Differential Equations, Volume 3: New Trends in Theoretical Development and Computational Methods, chapter 13. CRC Press, Boca Raton, Florida, 1995.

....bibliography is of course far from being exhaustive. As a further source of references one should mention the survey [104] by Singer. It gives much more details, especially on the more algebraic approaches, and contains a large bibliography. The same holds for the more focused surveys by Hereman [50, 51] covering symmetry theory and related fields and the one by MacCallum [65] on the integration of ordinary differential equations. In addition there have been three conferences devoted exclusively to differential equations and computer algebra. Their proceedings [103, 112, 113] contain a number of ....

....in many cases of practical interest it turns out that is much simpler to solve these linear partial differential equations than the original equation. There exist so many implementations of symmetry methods that it is rather difficult to keep an overview; we refer again to the surveys by Hereman [50, 51]. In almost any computer algebra system one can find a package for setting up the determining system. A few of the packages try furthermore some heuristics to solve it automatically. Again it is rather surprising how often this suffices to obtain the complete symmetry algebra. The symmetry package ....

W. Hereman. Symbolic software for Lie symmetry analysis. In N.H. Ibragimov, editor, CRC Handbook of Lie Group Analysis of Differential Equations, Volume 3: NewTrendsin TheoreticalDevelopmentandComputational Methods, chapter 13. CRC Press, Boca Raton, Florida, 1995.

....every author designs his own structures. For simple tasks this approach may suffice, but for more complex problems it faces limitations. Imagine that you want to apply the Lie symmetry theory to some system of partial differential equations. You will probably use one of the many symmetry packages [10] to set up the determining equations. In order to solve these it is often useful to complete them to a passive or involutive system. Again you can choose from a number of packages, e.g. 14] Finally, you may have found a symmetry and want to perform a corresponding change of variables. Procedures ....

W. Hereman, Symbolic software for Lie symmetry analysis, CRC Handbook of Lie Group Analysis of Differential Equations, Volume 3: New Trends in Theoretical Development and Computational Methods (N.H. Ibragimov, ed.), CRC Press, Boca Raton, Florida, 1995.

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Hereman, W. (1996). Symbolic Software for Lie Symmetry Analysis. CRC Handbook of Lie Group Analysis of Dierential Equations, Volume 3: New Trends in Theoretical Developments and Computational Methods. Chapter 13 (N.H. Ibragimov, ed.) Boca Raton, Florida: CRC Press, pp. 367-413.

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Hereman, W. (1996). Symbolic Software for Lie Symmetry Analysis. CRC Handbook of Lie Group Analysis of Di#erential Equations, Volume 3: New Trends in Theoretical Developments and Computational Methods. Chapter 13, 367--413. (N.H. Ibragimov, ed.) Boca Raton, Florida: CRC Press.

....for the computation of recursion operators, based on the knowledge of a few conserved densities and symmetries. The computation of Lie point symmetries and generalized symmetries via prolongation techniques is purposely omitted. That topic and related software were covered extensively in [21, 22]. Space limitations also prevent the inclusion of the wellknown Painlev e test, which is a widely applied and successful integrability detector for nonlinear ODEs and PDEs. We refer to [23] for survey papers, books, and software related to Painlev e analysis. This chapter is organized as follows. ....

....GUIDE Higher order symmetries can be computed with prolongation methods, and numerous software packages are available that can aid in the tedious computations inherent to such methods. A 50 page survey of software for Lie symmetry computations, including generalized symmetries, can be found in [21], and a short update in [22] We will not repeat these software reviews here. A survey of packages for conservation laws was rst given in [15] However, to keep this chapter self contained, we present a summary of that survey. Based on dilation invariance, Ito s programs in REDUCE (see [25, 26, ....

[Article contains additional citation context not shown here]

W. Hereman, Symbolic software for Lie symmetry analysis, in: CRC Handbook of Lie Group Analysis of Dierential Equations, Volume 3: New Trends in Theoretical Developments and Computational Methods, Ed.: N. H. Ibragimov, CRC Press, Boca Raton, Florida, 1996, pp. 367-413.

....the calculation of the continuous symmetry group of a modest system of di erential equations is prone to errors, if done with pencil and paper. Computer algebra systems (CAS) such as Mathematica, MACSYMA, Maple, REDUCE, AXIOM and MuPAD are extremely useful for such computations. Symbolic packages [40, 41, 43], written in the language of these CAS, can nd the determining equations of the Lie symmetry group. The most sophisticated packages then reduce these into an equivalent but more suitable system, subsequently solve that system in closed form, and go on to calculate the in nitesimal generators that ....

....In Section 6 we draw some conclusions. Lack of space forces us to give only a few key references for the Lie symmetry packages. A comprehensive survey of the literature devoted to theoretical as well as computational aspects of Lie symmetries, with over 300 references, can be found elsewhere [43]. 1 Methods and algorithms 1.1 Computing the determining equations The classical Lie symmetry group of a system of di erential equations is a local group of point transformations, meaning di eomorphisms on the space of independent and dependent variables, which map solutions of the system ....

[Article contains additional citation context not shown here]

W. Hereman, Symbolic Software for Lie Symmetry Analysis, In CRC Handbook of Lie Group Analysis of Dierential Equations. Volume 3: New Trends in Theoretical Developments and Computational Methods, (Edited by N.H. Ibragimov), Chapter 13, CRC Press, Boca Raton, Florida (1995) (in press).

....Higher order symmetries can be computed with prolongation methods and numerous software packages are available that can aid in the tedious computations inherent to such methods. An extensive review of software for Lie symmetry computations, including generalized symmetries, can be found in [26, 27]. With prolongation methods one generates and subsequently reduces and solves a determining system of linear homogeneous partial di erential equations for the coecients of the unknown higher order symmetry generator. In many cases, due to the length and complexity of that system, the general ....

....ansatz for the symmetry. Although restricted to polynomial higher order symmetries, we believe that the method presented in this paper is much more straightforward. Furthermore, it avoids the application of prolongation methods or Lie algebraic techniques. To avoid retreading the surveys [26, 27], we restrict our discussion to algorithms and symbolic packages that allow one to compute generalized symmetries of PDEs, as they were de ned in Section 2.1. We are not aware of software packages for DDEs to calculate directly the type of symmetries de ned in Section 3.1. Yamilov and co workers ....

W. Hereman, Symbolic software for Lie symmetry analysis, in CRC Handbook of Lie Group Analysis of Dierential Equations, Volume 3: New Trends in Theoretical Developments and Computational Methods, Chapter 13, N. H. Ibragimov ed., CRC Press, Boca Raton, Florida, 1996, pp. 367-413.

.... 1 = u 2 v 2 ; 58) 2 = 1 2 (u 2 v 2 ) 2 uv x u x v u 2 ; 59) 3 = 1 4 (u 2 v 2 ) 3 1 2 (u 2 x v 2 x ) u 3 v x v 3 u x 4 (u 4 v 4 ) 60) the remaining two can be found in [22] 5 Lie Point Symmetries 5. 1 Purpose The program SYMMGRP.MAX [3, 5, 6, 7] automatically computes the determining equations for the coecients in the vector eld that realizes the Lie algebra of point symmetries. With a feedback mechanism, these determining equations can then be solved explicitly. 11 The program SYMMGRP.MAX has been adapted in two ways. It now allows ....

....a symbolic package, will give the explicit forms of the i (x; u) and l (x; u) The procedure, which is thoroughly discussed in [19] consists of two major steps: deriving the determining equations, and solving them explicitly. Details on how this can be done by computer can be found in [5, 6, 7]. In these papers we also presented a survey of available codes, including a discussion of their strengths and weaknesses. 5.3 Examples Consider the Dym Kruskal equation [1] u t u 3 u xxx = 0: 69) The program SYMMGRP.MAX automatically computes the determining equations for the coecients ....

W. Hereman, Symbolic Software for Lie Symmetry Analysis, in: \New Trends in Theoretical Developments and Computational Methods", CRC Handbook of Lie Group Analysis of Dierential Equations, Vol. 3, N.H. Ibragimov, ed. (CRC Press, Boca Raton, Florida, 1995) Chapter 13, in press.

....not only automatically compute the system of determining equations of the Lie symmetry group, but also reduce these into an equivalent yet more suitable system, subsequently solve it in closed form, and go on to calculate the in nitesimal generators that span the Lie algebra of symmetries. In [1] detailed information is given about numerous Lie symmetry computer packages, together with a review of their strengths and weaknesses. The classical Lie symmetry group of a system of di erential equations is a local group of point transformations, meaning di eomorphisms on the space of ....

....generators, one can obtain the Lie group of point transformations upon integration of a system of rst order characteristic equations. A detailed review of innovative ways of classifying, subsequently reducing, and nally solving overdetermined systems of linear homogeneous PDEs is given in [1]. Lie Symmetry Software To design a reliable and powerful integration algorithm for a system of determining equations the system needs to be brought into a standard form. Standard form procedures can be viewed as generalizations to systems of linear PDEs of the Gaussian reduction method for ....

[Article contains additional citation context not shown here]

W. Hereman, Symbolic Software for Lie Symmetry Analysis, In: CRC Handbook of Lie Group Analysis of Dierential Equations. Volume 3: New Trends in Theoretical Developments and Computational Methods, (Edited by N.H. Ibragimov), CRC Press, Boca Raton, Florida (1996) Chapter 13, pp. 367-413.

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