Formal Analysis of PDEs

Organizer

Description

The session is to be devoted to the algorithmic and applied aspects of analysis of algebraic PDEs, based on their transformation into the special forms which are called in the literature as formal integrable, passive, involutive or standard. Such forms are much like to a Groebner basis and allow one to extract an important information on the equation system without its explicit integration. This information includes, in particular, compatibility analysis, finding dimension of the solution space, formulation of the initial conditions providing uniqueness and holomorphy of solution. Recently a number of different algorithmic techniques were designed and some of them have been implemented in computer algebra systems Axiom, Maple and Reduce. Over the session one expects presentation of some algorithmic approaches and their discussion in view of differential algebra, interconnection of the above special forms of PDEs with differential Groebner bases and characteristic sets, application to real problems of theoretical and mathematical physics, applied mathematics. One of the most important applications is Lie symmetry analysis of nonlinear differential equations which is the most universal tool of their integration.

Talks

• The Application of a System REDUCE for the Construction of the Solutions of Gas Dynamics Equations by the Method of differential Constraints.
  S.V. Meleshko, V.P. Shapeev.
  Abstract: The REDUCE was used for finding the compatibility conditions of overdetermined system of differential equations. Such systems are obtained in the applications of the method of differential constraints to the two-dimensional gas dynamics equations. In the given case, the REDUCE system was used for symbolic manipulations according to the algorithm of compatibility analysis: various substitutions, solving the systems of linear algebraic equations, various transformations of equations. Finally, a classification of the solutions satisfying additionally to one, two or three differential constraints of the first order was performed for two-dimensional stationary equations of gas dynamics. The solutions of gas dynamics equations are constructed by an integration of obtained overdetermined equations. Such solutions are interesting for the applications.

• Some improvements of a lemma of Rosenfeld.
  François Boulier
  Abstract: We give some improvements of a lemma of Rosenfeld which permit us to optimize some algorithms in differential algebra: we prove the lemma with weaker hypotheses and we demonstrate an analogue of Buchberger's second criterion, which avoids non necessary reductions for detecting coherent sets of differential polynomials. We try also to clarify the relations between the theorems in differential algebra and some more widely known results in the Gröbner bases theory.

• Minimal Involutive Bases
  Vladimir P. Gerdt, Yuri A. Blinkov
  Abstract: In this paper we study of the uniqueness properties of involutive polynomial bases which are redundant Gröobner bases of the special form. The most general involutive algorithmic techniques is based on the concept of involutive monomial division which allows one to separate all the variables into multiplicative and non-multiplicative subsets. The separation gives thereby the self-consistent computational procedure for constructing an involutive basis by performing non-multiplicative prolongations and multiplicative reductions. Every specific involutive division generates a particular form of involutive computational procedure. In addition to three involutive divisions used by Thomas, Janet and Pommaret for analysis of partial differential equations we introduce two new ones. These two divisions much as Thomas division do not depend on the order of variables. We prove noetherity and continuity of the new divisions. Given noetherian and continuous division, we present an algorithm for constructing of the minimal involutive basis for a polynomial ideal. This minimal basis is uniquely defined for any admissible monomial ordering.

• Involutive systems and the numerical analysis of constrained Hamiltonian Systems
  Werner M. Seiler
  Abstract: We will give a brief introduction into the theory of involutive systems concentrating on the completion of a given system of PDEs to an involutive one. Then we will show how such problems naturally occur in the numerical analysis of differential algebraic equations and in the the theory of constrained Hamiltonian systems.

• Triangular Matrices, Differential Resultants and Systems of Linear Homogeneous PDE's Giuseppa Carrà Ferro
  Abstract

• A Generalization of Rosenfeld's Lemma.
  Sally D. Morrison
  Abstract: In this paper we explore connections between Mansfield's work on differential Gröbner bases and Boulier's work on differential radical ideals. In particular, we obtain a generalization of the result in differential algebra known as Rosenfeld's Lemma. This generalization replaces Rosenfeld's `auto-reduced' hypothesis (very effectively used by Boulier in his computation of radical differential ideals) with Mansfield's weaker `almost complete' hypothesis, and thus permits some of Boulier's techniques to be adapted to the more general case.

• The rif and radical rif algorithms.
  Gregory J. Reid
  Abstract: The rif algorithm uses a finite number of differentiations and algebraic operations to simplify analytic nonlinear systems of partial differential equations to what we call reduced involutive form. This form includes the integrability conditions of the system and satisfies a constant rank condition.

  The rif algorithm is effective provided it can effectively perform elimination on the algebraic systems it generates and transform such systems to constant rank form. The radical rif algorithm is a realization of the rif algorithm for polynomially nonlinear pde systems which uses Buchberger's algorithm for elimination and algorithms for constructing the radical of a polynomial ideal to realize the constant rank condition. Various applications including the symmetry analysis of differential equations are discussed.

  Reference:
  G. J. Reid, A. D. Wittkopf and A. Boulton (1994).
  Reduction of systems of nonlinear partial differential equations to simplified involutive forms.
  Available at http://www.iam.ubc.ca/tr/1994/iam94-14.

Potential Speakers

• Francois Boullier, University of Waterloo, Canada
• David Hartley, GMD, St-Augustin, Germany
• Elizabeth Mansfeld, University of Kent at Canterbury, U.K.
• Gregary Reid, University of British Columbia, Vancouver, Canada
• Fritz Schwarz, GMD, St-Augustin, Germany
• Werner Seiler, Universität Karlsruhe, Germany