Symmetry Classification Using Noncommutative Invariant Differential Operators

Ian Lisle, Gregory Reid

    Research output: Contribution to journalArticle

    9 Citations (Scopus)

    Abstract

    Given a class F of differential equations, the symmetry classification problem is to determine for each member f ∈ F the structure of its Lie symmetry group Gf or, equivalently, of its Lie symmetry algebra. The components of the symmetry vector fields of the Lie algebra are solutions of an associated overdetermined "defining system" of differential equations. The usual computer classification method which applies a sequence of total derivative operators and eliminations to this associated system often fails on problems of interest due to the excessive size of expressions generated in intermediate computations. We provide an alternative classification method which exploits the knowledge of an equivalence group G preserving the class. A noncommutative differential elimination procedure due to Lemaire, Reid, and Zhang, where each step of the procedure is invariant under G, can be applied and an existence and uniqueness theorem for the output used to classify the structure of symmetry groups for each f ∈ F. The method is applied to a class of nonlinear diffusion convection equations vx = u, vt = B(u) ux - K(u) which is invariant under a large but easily determined equivalence group G. In this example the complexity of the calculations is much reduced by the use of G-invariant differential operators.
    Original languageEnglish
    Pages (from-to)353-386
    Number of pages34
    JournalFoundations of Computational Mathematics
    Volume6
    Issue number3
    DOIs
    Publication statusPublished - 2006

    Fingerprint

    Invariant Differential Operators
    Mathematical operators
    Lie Symmetry
    Symmetry Group
    Symmetry
    Algebra
    Elimination
    Differential equations
    Total derivative
    Equivalence
    Invariant
    Existence and Uniqueness Theorem
    Nonlinear Diffusion
    System of Differential Equations
    Classification Problems
    Convection
    Vector Field
    Lie Algebra
    Classify
    Differential equation

    Cite this

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    title = "Symmetry Classification Using Noncommutative Invariant Differential Operators",
    abstract = "Given a class F of differential equations, the symmetry classification problem is to determine for each member f ∈ F the structure of its Lie symmetry group Gf or, equivalently, of its Lie symmetry algebra. The components of the symmetry vector fields of the Lie algebra are solutions of an associated overdetermined {"}defining system{"} of differential equations. The usual computer classification method which applies a sequence of total derivative operators and eliminations to this associated system often fails on problems of interest due to the excessive size of expressions generated in intermediate computations. We provide an alternative classification method which exploits the knowledge of an equivalence group G preserving the class. A noncommutative differential elimination procedure due to Lemaire, Reid, and Zhang, where each step of the procedure is invariant under G, can be applied and an existence and uniqueness theorem for the output used to classify the structure of symmetry groups for each f ∈ F. The method is applied to a class of nonlinear diffusion convection equations vx = u, vt = B(u) ux - K(u) which is invariant under a large but easily determined equivalence group G. In this example the complexity of the calculations is much reduced by the use of G-invariant differential operators.",
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    Symmetry Classification Using Noncommutative Invariant Differential Operators. / Lisle, Ian; Reid, Gregory.

    In: Foundations of Computational Mathematics, Vol. 6, No. 3, 2006, p. 353-386.

    Research output: Contribution to journalArticle

    TY - JOUR

    T1 - Symmetry Classification Using Noncommutative Invariant Differential Operators

    AU - Lisle, Ian

    AU - Reid, Gregory

    PY - 2006

    Y1 - 2006

    N2 - Given a class F of differential equations, the symmetry classification problem is to determine for each member f ∈ F the structure of its Lie symmetry group Gf or, equivalently, of its Lie symmetry algebra. The components of the symmetry vector fields of the Lie algebra are solutions of an associated overdetermined "defining system" of differential equations. The usual computer classification method which applies a sequence of total derivative operators and eliminations to this associated system often fails on problems of interest due to the excessive size of expressions generated in intermediate computations. We provide an alternative classification method which exploits the knowledge of an equivalence group G preserving the class. A noncommutative differential elimination procedure due to Lemaire, Reid, and Zhang, where each step of the procedure is invariant under G, can be applied and an existence and uniqueness theorem for the output used to classify the structure of symmetry groups for each f ∈ F. The method is applied to a class of nonlinear diffusion convection equations vx = u, vt = B(u) ux - K(u) which is invariant under a large but easily determined equivalence group G. In this example the complexity of the calculations is much reduced by the use of G-invariant differential operators.

    AB - Given a class F of differential equations, the symmetry classification problem is to determine for each member f ∈ F the structure of its Lie symmetry group Gf or, equivalently, of its Lie symmetry algebra. The components of the symmetry vector fields of the Lie algebra are solutions of an associated overdetermined "defining system" of differential equations. The usual computer classification method which applies a sequence of total derivative operators and eliminations to this associated system often fails on problems of interest due to the excessive size of expressions generated in intermediate computations. We provide an alternative classification method which exploits the knowledge of an equivalence group G preserving the class. A noncommutative differential elimination procedure due to Lemaire, Reid, and Zhang, where each step of the procedure is invariant under G, can be applied and an existence and uniqueness theorem for the output used to classify the structure of symmetry groups for each f ∈ F. The method is applied to a class of nonlinear diffusion convection equations vx = u, vt = B(u) ux - K(u) which is invariant under a large but easily determined equivalence group G. In this example the complexity of the calculations is much reduced by the use of G-invariant differential operators.

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