Computations involving differential operators and their actions on functions

The algorithms derived by Grossmann and Larson (1989) are further developed for rewriting expressions involving differential operators. The differential operators involved arise in the local analysis of nonlinear dynamical systems. These algorithms are extended in two different directions: the algorithms are generalized so that they apply to differential operators on groups and the data structures and algorithms are developed to compute symbolically the action of differential operators on functions. Both of these generalizations are needed for applications.

[1]  H. Knapp,et al.  NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS BY GROEBNER'S METHOD OF LIE-SERIES. , 1968 .

[2]  Jean Della Dora,et al.  Formal solutions of linear difference equations: method of Pincherle-Ramis , 1986, SYMSAC '86.

[3]  Robert Hermann,et al.  Sophus Lie's 1880 transformation group paper , 1975 .

[4]  Bruce W. Char,et al.  Using Lie transformation groups to find closed form solutions to first order ordinary differential equations , 1981, SYMSAC '81.

[5]  M. J. Prelle,et al.  Elementary first integrals of differential equations , 1981, SYMSAC '81.

[6]  H. Hermes.  Nilpotent approximations of control systems and distributions , 1986 .

[7]  E. Stein,et al.  Hypoelliptic differential operators and nilpotent groups , 1976 .

[8]  C. Scovel.  Symplectic Numerical Integration of Hamiltonian Systems , 1991 .

[9]  Arthur J. Krener,et al.  Bilinear and Nonlinear Realizations of Input-Output Maps , 1975 .

[10]  Fritz Schwarz,et al.  Symmetries of Differential Equations: From Sophus Lie to Computer Algebra , 1988 .

[11]  Kenneth R. Meyer.  Lie Transform Tutorial — II , 1991 .

[12]  A. Krener,et al.  Nonlinear controllability and observability , 1977 .

[13]  C. Rockland,et al.  Intrinsic nilpotent approximation , 1987 .

[14]  H. Knapp,et al.  LIESE: A PROGRAM FOR ORDINARY DIFFERENTIAL EQUATIONS USING LIE-SERIES. , 1968 .

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

[16]  Robert L. Grossman.  Evaluation of expressions involving higher order derivations , 1990 .

[17]  G. Rota,et al.  Finite operator calculus , 1975 .

[18]  Robert L. Grossman,et al.  Hopf-algebraic structure of combinatorial objects and differential operators , 1990 .

[19]  Warren D. Nichols.  The Kostant structure theorems for Kk-Hopf algebras , 1985 .

[20]  H. Busemann.  Advances in mathematics , 1961 .

[21]  Paul S. Wang,et al.  FINGER: A Symbolic System for Automatic Generation of Numerical Programs in Finite Element Analysis , 1986, J. Symb. Comput..

[22]  Robert G. Grossman,et al.  Using trees to compute approximate solutions to ordinary differential equations exactly , 1991 .

[23]  Gerald J. Sussman,et al.  Intelligence in scientific computing , 1989, CACM.

[24]  J. Butcher.  The numerical analysis of ordinary differential equations: Runge-Kutta and general linear methods , 1987 .

[25]  B. Weisfeiler,et al.  Differential Formal Groups of J. F. Ritt , 1982 .

[26]  Stanly Steinberg,et al.  Using MACSYMA to Write FORTRAN Subroutines , 1986, J. Symb. Comput..

[27]  Joachim Apel,et al.  An Extension of Buchberger's Algorithm and Calculations in Enveloping Fields of Lie Algebras , 1988, J. Symb. Comput..

[28]  Pierre Leroux,et al.  A Combinatorial Approach to Nonlinear Functional Expansions: An Introduction with an Example , 1991, Theor. Comput. Sci..

[29]  Shlomo Sternberg,et al.  An algebraic model of transitive differential geometry , 1964 .

[30]  R. D. Vogelaere,et al.  Methods of Integration which Preserve the Contact Transformation Property of the Hamilton Equations , 1956 .

[31]  H. Hermes,et al.  Nilpotent bases for distributions and control systems , 1984 .

[32]  M. Grayson,et al.  Models for Free Nilpotent Lie Algebras , 1990 .

[33]  E. Stein,et al.  Estimates for the complex and analysis on the heisenberg group , 1974 .

[34]  H. Hermes.  Control systems which generate decomposable Lie algebras , 1982 .

[35]  J. Cole,et al.  Similarity methods for differential equations , 1974 .

[36]  Victor Guillemin,et al.  Infinite dimensional primitive Lie algebras , 1970 .

[37]  Marshall Hall,et al.  A basis for free Lie rings and higher commutators in free groups , 1950 .

[38]  Gérard Viennot,et al.  Combinatorial resolution of systems of differential equations, I. Ordinary differential equations , 1986 .

[39]  Gian-Carlo Rota,et al.  Coalgebras and Bialgebras in Combinatorics , 1979 .

[40]  Robert L. Grossman,et al.  Symbolic Computation of Derivations Using Labeled Trees , 1992, J. Symb. Comput..

[41]  Robert Hermann,et al.  Sophus Lie's 1884 differential invariant paper , 1975 .

[42]  Robert L. Grossman,et al.  Solving nonlinear equations from higher order derivations in linear stages , 1990 .

[43]  Richard G. Larson,et al.  Hopf-algebraic structure of families of trees , 1989 .

[44]  John C. Butcher.  An Order Bound for Runge–Kutta Methods , 1975 .

[45]  André Deprit,et al.  Canonical transformations depending on a small parameter , 1969 .

[46]  Robert L. Grossman,et al.  Labeled trees and the efficient computation of derivations , 1989, ISSAC '89.

[47]  D. Armbruster,et al.  "Perturbation Methods, Bifurcation Theory and Computer Algebra" , 1987 .