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16
Existence and Uniqueness Theorems for Formal Power Series Solutions of Analytic Differential Systems
, 1999
"... We present Existence and Uniqueness Theorems for formal power series solutions of analytic systems of pde in a certain form. This form can be obtained by a finite number of differentiations and eliminations of the original system, and allows its formal power series solutions to be computed in an alg ..."
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Cited by 12 (6 self)
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We present Existence and Uniqueness Theorems for formal power series solutions of analytic systems of pde in a certain form. This form can be obtained by a finite number of differentiations and eliminations of the original system, and allows its formal power series solutions to be computed in an algorithmic fashion. The resulting reduced involutive form (rif 0 form) produced by our rif 0 algorithm is a generalization of the classical form of Riquier and Janet, and that of Cauchy Kovalevskaya. We weaken the assumption of linearity in the highest derivatives in those approaches to allow for systems which are nonlinear in their highest derivatives. A new formal development of Riquier's theory is given, with proofs, modeled after those in Grobner Basis Theory. For the nonlinear theory, the concept of relative Riquier Bases is introduced. This allows for the easy extension of ideas from the linear to the nonlinear theory. The essential idea is that an arbitrary nonlinear system can ...
Fast Differential Elimination in C: The CDiffElim Environment
, 2000
"... We introduce the CDiffElim environment, written in C, and an algorithm developed in this environment for simplifying systems of overdetermined partial differential equations by using differentiation and elimination. This environment has strategies for addressing difficulties encountered in different ..."
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Cited by 9 (7 self)
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We introduce the CDiffElim environment, written in C, and an algorithm developed in this environment for simplifying systems of overdetermined partial differential equations by using differentiation and elimination. This environment has strategies for addressing difficulties encountered in differential elimination algorithms, such as exhaustion of computer memory due to intermediate expression swell, and failure to complete due to the massive number of calculations involved. These strategies include lowlevel memory management strategies and data representations that are tailored for efficient differential elimination algorithms. These strategies, which are coded in a lowlevel C implementation, seem much more difficult to implement in highlevel general purpose computer algebra systems. A differential elimination algorithm written in this environment is applied to the determination of symmetry properties of classes of n+1dimensional coupled nonlinear partial differential equations of form iut+r2u+ i a(t)jxj2 + b(t) \Delta x + c(t) + djuj 4n j u = 0; where u is an mcomponent vectorvalued function. The resulting systems of differential equations for the symmetries have been made available on the web, to be used as benchmark systems for other researchers. The new differential elimination algorithm in C, runs on the test suite an average of 400 times faster than our RifSimp algorithm in Maple.
Determination of Maximal Symmetry Groups of Classes of Differential Equations
 in: Proceedings of the 2000 International Symposium on Symbolic and Algebraic Computation
, 2000
"... A symmetry of a dierential equation is a transformation which leaves invariant its family of solutions. As the functional form of a member of a class of dierential equations changes, its symmetry group can also change. We give an algorithm for determining the structure and dimension of the symmetry ..."
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Cited by 5 (1 self)
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A symmetry of a dierential equation is a transformation which leaves invariant its family of solutions. As the functional form of a member of a class of dierential equations changes, its symmetry group can also change. We give an algorithm for determining the structure and dimension of the symmetry group(s) of maximal dimension for classes of partial dierential equations. It is based on the application of dierential elimination algorithms to the linearized equations for the unknown symmetries. Existence and Uniqueness theorems are applied to the output of these algorithms to give the dimension of the maximal symmetry group. Classes of dierential equations considered include ode of form uxx = f(x; u; ux ), ReactionDiusion Systems of form u t uxx = f(u; v); v t vxx = g(u; v), and Nonlinear Telegraph Systems of form v t = ux ; vx = C(u; x)ux +B(u;x). 1. INTRODUCTION The symmetries, or transformations leaving invariant a system of partial dierential equations, are generally not...
Symmetry classification using noncommutative invariant differential operators
 Found. Comput. Math
"... 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 G f, or equivalently of its Lie symmetry algebra. The components of the symmetry vector fields of the Lie algebra are solutions of an associate ..."
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Cited by 4 (0 self)
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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 G f, 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 Ginvariant differential operators.
ORDERINGS OF MONOMIAL IDEALS
, 2003
"... We study the set of monomial ideals in a polynomial ring as an ordered set, with the ordering given by reverse inclusion. We give a short proof of the fact that every antichain of monomial ideals is finite. Then we investigate ordinal invariants for the complexity of this ordered set. In particular ..."
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Cited by 4 (1 self)
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We study the set of monomial ideals in a polynomial ring as an ordered set, with the ordering given by reverse inclusion. We give a short proof of the fact that every antichain of monomial ideals is finite. Then we investigate ordinal invariants for the complexity of this ordered set. In particular, we give an interpretation of the height function in terms of the HilbertSamuel polynomial, and we compute upper and lower bounds on the maximal order type.
SymbolicNumeric Completion of Differential Systems by Homotopy Continuation
 Proc. ISSAC 2005. ACM
, 2005
"... Two ideas are combined to construct a hybrid symbolicnumeric differentialelimination method for identifying and including missing constraints arising in differential systems. First we exploit the fact that a system once differentiated becomes linear in its highest derivatives. Then we apply diagona ..."
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Cited by 4 (1 self)
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Two ideas are combined to construct a hybrid symbolicnumeric differentialelimination method for identifying and including missing constraints arising in differential systems. First we exploit the fact that a system once differentiated becomes linear in its highest derivatives. Then we apply diagonal homotopies to incrementally process new constraints, one at a time. The method is illustrated on several examples, combining symbolic differential elimination (using rifsimp) with numerical homotopy continuation (using phc).
Symbolicnumeric Computation of Implicit Riquier Bases for PDE
 Proc. ISSAC ’07, ACM
, 2007
"... Riquier Bases for systems of analytic pde are, loosely speaking, a differential analogue of Gröbner Bases for polynomial equations. They are determined in the exact case by applying a sequence of prolongations (differentiations) and eliminations to an input system of pde. We present a symbolicnumer ..."
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Cited by 2 (0 self)
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Riquier Bases for systems of analytic pde are, loosely speaking, a differential analogue of Gröbner Bases for polynomial equations. They are determined in the exact case by applying a sequence of prolongations (differentiations) and eliminations to an input system of pde. We present a symbolicnumeric method to determine Riquier Bases in implicit form for systems which are dominated by pure derivatives in one of the independent variables and have the same number of pde and unknowns. The method is successful provided the prolongations with respect to the dominant independent variable have a block structure which is uncovered by Linear Programming and certain Jacobians are nonsingular when evaluated at points on the zero sets defined by the functions of the pde. For polynomially nonlinear pde, homotopy continuation methods from Numerical Algebraic Geometry can be used to compute approximations of the points. We give a differential algebraic interpretation of Pryce’s method for ode, which generalizes to the pde case. A major aspect of the method’s efficiency is that only prolongations with respect to a single (dominant) independent variable are made, possibly after a random change of coordinates. Potentially expensive and numerically unstable eliminations are not made. Examples are given to illustrate theoretical features of the method, including a curtain of Pendula and the control of a crane.
Universal characteristic decomposition of radical differential ideals
, 2006
"... We call a differential ideal universally characterizable, if it is characterizable w.r.t. any ranking on partial derivatives. We propose a factorizationfree algorithm that represents a radical differential ideal as a finite intersection of universally characterizable ideals. The algorithm also cons ..."
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Cited by 2 (1 self)
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We call a differential ideal universally characterizable, if it is characterizable w.r.t. any ranking on partial derivatives. We propose a factorizationfree algorithm that represents a radical differential ideal as a finite intersection of universally characterizable ideals. The algorithm also constructs a universal characteristic set for each universally characterizable component, i.e., a finite set of differential polynomials that contains a characterizing set of the ideal w.r.t. any ranking. As a part of the proposed algorithm, the following problem of satisfiability by a ranking is efficiently solved: given a finite set of differential polynomials with a derivative selected in each polynomial, determine whether there exists a ranking w.r.t. which the selected derivatives are leading derivatives and, if so, construct such a ranking. Key words: differential algebra, radical differential ideals, factorizationfree algorithms, characteristic decomposition, universal characteristic sets, differential rankings 1.
Implicit Riquier Bases for PDAE and their SemiDiscretizations
, 2008
"... Complicated nonlinear systems of pde with constraints (called pdae) arise frequently in applications. Missing constraints arising by prolongation (differentiation) of the pdae need to be determined to consistently initialize and stabilize their numerical solution. In this article we review a fast pr ..."
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Cited by 1 (0 self)
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Complicated nonlinear systems of pde with constraints (called pdae) arise frequently in applications. Missing constraints arising by prolongation (differentiation) of the pdae need to be determined to consistently initialize and stabilize their numerical solution. In this article we review a fast prolongation method, a development of (explicit) symbolic Riquier Bases, suitable for such numerical applications. Our symbolicnumeric method to determine Riquier Bases in implicit form, without the unstable eliminations of the exact approaches, applies to square systems which are dominated by pure derivatives in one of the independent variables. The method is successful provided the prolongations with respect to a single dominant independent variable have a block structure which is uncovered by Linear Programming and certain Jacobians are nonsingular when evaluated at points on the zero sets defined by the functions of the pdae. For polynomially nonlinear pdae, homotopy continuation methods from Numerical Algebraic Geometry can be used to compute approximations of the points. Our method generalizes Pryce’s method for dae to pdae. Given a dominant independent time variable, for an initial value problem for a system of pdae we show that its semidiscretization
www.elsevier.com/locate/jsc The Gröbner fan and Gröbner walk for modules
, 2005
"... This paper extends the theory of the Gröbner fan and Gröbner walk for ideals in polynomial rings to the case of submodules of free modules over a polynomial ring. The Gröbner fan for a submodule creates a correspondence between a pair consisting of a cone in the fan and a point in the support of the ..."
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This paper extends the theory of the Gröbner fan and Gröbner walk for ideals in polynomial rings to the case of submodules of free modules over a polynomial ring. The Gröbner fan for a submodule creates a correspondence between a pair consisting of a cone in the fan and a point in the support of the cone and a pair consisting of a leading monomial submodule (or equivalently, a reduced marked Gröbner basis) and a grading of the free module over the ring that is compatible with the choice of leading monomials. The Gröbner walk is an algorithm based on the Gröbner fan that converts a given Gröbner basis to a Gröbner basis with respect to a different monomial order; the point being that the Gröbner walk can be more efficient than the standard algorithms for Gröbner basis computations with difficult monomial orders. Algorithms for generating the Gröbner fan and for the Gröbner walk are given.