Notes on Triangular Sets and Triangulation-Decomposition Algorithms II: Differential Systems

This is the second in a series of two tutorial articles devoted to triangulation-decomposition algorithms. The value of these notes resides in the uniform presentation of triangulation-decomposition of polynomial and differential radical ideals with detailed proofs of all the presented results.We emphasize the study of the mathematical objects manipulated by the algorithms and show their properties independently of those. We also detail a selection of algorithms, one for each task. The present article deals with differential systems. It uses results presented in the first article on polynomial systems but can be read independently.

[1]  Gregory J. Reid,et al.  Rankings of partial derivatives , 1997, ISSAC.

[2]  Wang Dongming.  AN ELIMINATION METHOD FOR DIFFERENTIAL POLYNOMIAL SYSTEMS I , 1996 .

[3]  Evelyne Hubert,et al.  Notes on Triangular Sets and Triangulation-Decomposition Algorithms I: Polynomial Systems , 2001, SNSC.

[4]  J. Chazy,et al.  Sur les équations différentielles du troisième ordre et d'ordre supérieur dont l'intégrale générale a ses points critiques fixes , 1911 .

[5]  Gabriel Thomas.  Contributions théoriques et algorithmiques à l'étude des équations différencielles-algébriques : Approche par le calcul formel , 1997 .

[6]  E. Cartan,et al.  Les systèmes différentiels extérieurs et leurs applications géométriques , 1945 .

[7]  Peter A. Clarkson,et al.  The classical, direct, and nonclassical methods for symmetry reductions of nonlinear partial differential equations , 1997 .

[8]  Joachim Apel.  Passive Complete Orthonomic Systems of PDEs and Involutive Bases of Polynomial Modules , 2001, SNSC.

[9]  H P Wynn,et al.  Differential algebra methods for the study of the structural identifiability of rational function state-space models in the biosciences. , 2001, Mathematical biosciences.

[10]  D. Eisenbud.  Commutative Algebra: with a View Toward Algebraic Geometry , 1995 .

[11]  Gregory J. Reid,et al.  Reduction of systems of nonlinear partial differential equations to simplified involutive forms , 1996, European Journal of Applied Mathematics.

[12]  Elizabeth L. Mansfield,et al.  Diffgrob2: A symbolic algebra package for analysing systems of PDE using MAPLE , 1993 .

[13]  I. Kaplansky.  An introduction to differential algebra , 1957 .

[14]  William Y. Sit.  THE RITT–KOLCHIN THEORY FOR DIFFERENTIAL POLYNOMIALS , 2002 .

[15]  A. Seidenberg.  An elimination theory for differential algebra , 1959 .

[16]  Gregory J. Reid,et al.  Algorithms for reducing a system of PDEs to standard form, determining the dimension of its solution space and calculating its Taylor series solution , 1991, European Journal of Applied Mathematics.

[17]  François Lemaire,et al.  Computing canonical representatives of regular differential ideals , 2000, ISSAC.

[18]  Dongming Wang,et al.  Reasoning about Surfaces Using Differential Zero and Ideal Decomposition , 2000, Automated Deduction in Geometry.

[19]  Alexandre Sedoglavic.  A probabilistic algorithm to test local algebraic observability in polynomial time , 2001, ISSAC '01.

[20]  P. Clarkson,et al.  Nonclassical Reductions of a 3+1-Cubic Nonlinear Schrodinger System , 1998 .

[21]  P. Olver.  Applications of Lie Groups to Differential Equations , 1986 .

[22]  J. F. Ritt.  On the Singular Solutions of Algebraic Differential Equations , 1936 .

[23]  Michael Kalkbrener,et al.  A Generalized Euclidean Algorithm for Computing Triangular Representations of Algebraic Varieties , 1993, J. Symb. Comput..

[24]  K. B. O’Keefe,et al.  The differential ideal $[uv]$ , 1966 .

[25]  E. S. Cheb-Terrab,et al.  Computer Algebra Solving of Second Order ODEs Using Symmetry Methods , 1997 .

[26]  Sally Morrison.  The Differential Ideal [P]: Minfty , 1999, J. Symb. Comput..

[27]  Uri M. Ascher,et al.  Computer methods for ordinary differential equations and differential-algebraic equations , 1998 .

[28]  T. Glad,et al.  An Algebraic Approach to Linear and Nonlinear Control , 1993 .

[29]  Daniel Lazard,et al.  Solving Zero-Dimensional Algebraic Systems , 1992, J. Symb. Comput..

[30]  A. Rosenfeld.  Specializations in differential algebra , 1959 .

[31]  Hamid Maarouf,et al.  Unmixed-dimensional Decomposition of a Finitely Generated Perfect Differential Ideal , 2001, J. Symb. Comput..

[32]  E. Hairer,et al.  Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems , 2010 .

[33]  Marc Moreno Maza,et al.  Polynomial Gcd Computations over Towers of Algebraic Extensions , 1995, AAECC.

[34]  François Boulier,et al.  Representation for the radical of a finitely generated differential ideal , 1995, ISSAC '95.

[35]  Liu Ziming.  COHERENT, REGULAR AND SIMPLE SYSTEMS IN ZERO DECOMPOSITIONS OF PARTIAL DIFFERENTIAL SYSTEMS , 1999 .

[36]  Sally Morrison.  The Differential Ideal [ P ] : M ∞ , 1999 .

[37]  Gregory J. Reid,et al.  Existence and uniqueness theorems for formal power series solutions of analytic differential systems , 1999, ISSAC '99.

[38]  Heinz Kredel,et al.  Gröbner Bases: A Computational Approach to Commutative Algebra , 1993 .

[39]  Evelyne Hubert,et al.  Factorization-free Decomposition Algorithms in Differential Algebra , 2000, J. Symb. Comput..

[40]  Sette Diop,et al.  Differential-Algebraic Decision Methods and some Applications to System Theory , 1992, Theor. Comput. Sci..

[41]  Hyman Bass,et al.  Selected works of Ellis Kolchin with commentary , 1999 .

[42]  Bernard Malgrange.  Differential Algebra and Differential Geometry , 2002 .

[43]  Evelyne Hubert,et al.  Resolvent Representation for Regular Differential Ideals , 2003, Applicable Algebra in Engineering, Communication and Computing.

[44]  R. Lathe.  Phd by thesis , 1988, Nature.

[45]  Sofi Stenström.  Differential Gröbner bases , 2002 .

[46]  Josselin Visconti,et al.  Résolution numérique des équations algébro-différentielles, estimation de l'erreur globale et réduction formelle de l'indice , 1999 .

[47]  Wu Wen-tsun,et al.  Mechanical theorem proving of differential geometries and some of its applications in mechanics , 1991 .

[48]  François Lemaire.  Contribution à l'algorithmique en algèbre différentielle , 2002 .

[49]  Frann Cois Ollivier,et al.  Canonical Bases: Relations with Standard Bases, Finiteness Conditions and Application to Tame Automorphisms? , 1994 .

[50]  Evelyne Hubert,et al.  Essential Components of an Algebraic Differential Equation , 1999, J. Symb. Comput..

[51]  Giuseppa Carra'Ferro,et al.  Groebner Bases and Differential Algebra , 1987 .

[52]  Fritz Schwarz.  Symmetries of Second- and Third-Order Ordinary Differential Equations , 2001, SNSC.

[53]  Gregory J. Reid,et al.  Finding abstract Lie symmetry algebras of differential equations without integrating determining equations , 1991, European Journal of Applied Mathematics.

[54]  François Boulier,et al.  Étude et implantation de quelques algorithmes en algèbre différentielle. (Study and implementation of some algorithms in differential algebra) , 1994 .

[55]  L.A.C.P. da Mota,et al.  Computer algebra solving of first order ODEs using symmetry methods , 1997 .

[56]  Elizabeth L. Mansfield,et al.  Symmetry reductions and exact solutions of a class of nonlinear heat equations , 1993, solv-int/9306002.

[57]  J. Ritt,et al.  Differential Equations from the Algebraic Standpoint , 1933, Nature.

[58]  N. G. Parke,et al.  Ordinary Differential Equations. , 1958 .