Rif.html

Introduction to the Rif Package Version 1.1

The Rif package is a powerful collection of commands for the simplification and analysis of systems of polynomially nonlinear ODEs and PDEs.
The package includes a command to simplify systems of ODEs and PDEs by converting these systems to a canonical form (reduced involutive form), graphical display of results for ease of use, a command to determine the initial data required for the existence of formal power series solutions of a system, and a command to generate formal power series solutions of a system.
Rif is used by both dsolve and pdsolve to assist in solution of ODE/PDE systems (see dsolve,system , pdsolve,system ).
In addition, the PDEtools[casesplit] command extends the functionality of Rif by allowing differential elimination to proceed in the presence of nearly all non-polynomial objects known to Maple (over one hundred of these, including trigonometric, exponential, and generalized hypergeometric functions, fractional exponents, etc.).
Though the results obtained from the rifsimp command are similar to those obtained from the diffalg package, the commands use different approaches, one of which may work better for specific problems than the other.
The Rif package generalizes the Standard Form package for linear ODE/PDE systems to polynomially nonlinear ODE/PDE. The linear system capabilities of the Standard Form package for the simplification of ODE/PDE systems are also present as part of Rif.
The improvements over the most recently released version of Standard Form (1995) include

1) Full handling of polynomially nonlinear systems
2) Automatic case splitting
3) Flexible nonlinear ranking
4) Handling of inequation constraints (expr<>0)
5) Speed and memory efficiency

The improvements over the release 1.0 of Rif (Maple 6) include

1) New function maxdimsystems to find the most general
solutions for case splitting problems
2) Improvements to case visualization and initial data
computation
3) Greater flexibility in rifsimp with the addition of
new options for control of case splitting, declaration
of arbitrary functions and/or constants, detection
of empty cases, and much more
4) More efficient handling of nonlinear systems via new
nonlinear equation methods
5) Significant overall speed and memory enhancements
6) Automatic adjustment of results to remove inconsistent
cases, and their effect on the returned consistent
cases.

The functions available

rifsimp Simplifies systems of polynomially nonlinear ODEs and PDEs
to canonical form. Splits nonlinear equations into cases,
using Groebner basis techniques to handle algebraic
consequences of the system. Accounts for all differential
consequences of the system.
maxdimsystems Also simplifies systems of polynomially nonlinear ODEs
and PDEs, but performs case splitting automatically,
returning the most general cases (those with the highest
number of parameters in the initial data).
rifread Loads a partially completed rifsimp calculation for
viewing and/or manual manipulation. rifsimp must be
told to save partial calculations using the storage
options.
checkrank Provides information on ranking to allow determination
of an appropriate ranking to use with rifsimp .
caseplot Takes the case split output of rifsimp , and provides
a graphical display of the solution case tree.
initialdata Obtains the initial data required by an ODE/PDE system to
fully specify formal power series solutions of the system.
Typically the output of rifsimp is used as input for this
procedure, but any ODE/PDE system in the correct form can
be used.
rtaylor Calculates the Taylor series of solutions of an ODE/PDE
system to any order. Just as for initialdata , any
ODE/PDE system in the correct form can be used.

For theory used to produce the rifsimp and maxdimsystems algorithm, and related theory, please see the following:

Becker, T., Weispfenning, V. Groebner Bases: A Computational Approach to Commutative Algebra . New York: Springer-Verlag, 1993.
G.W. Bluman and S. Kumei, "Symmetries and Differential Equations", Springer-Verlag , vol. 81.
Boulier, F., Lazard, D., Ollivier, F., and Petitot, M. "Representation for the Radical of a Finitely Generated Differential Ideal". Proc. ISSAC 1995 . ACM Press, 158-166.
Carra-Ferro, G. "Groebner Bases and Differential Algebra". Lecture Notes in Comp. Sci. 356 (1987): 128-140.
Goldschmidt, H. "Integrability Criteria for Systems of Partial Differential Equations". J. Diff. Geom. 1 (1967): 269-307.
Mansfield, E. 1991. Differential Groebner Bases. Ph.D. diss., University of Sydney.
Ollivier, F. "Standard Bases of Differential Ideals". Lecture Notes in Comp. Sci. 508 (1991): 304-321.
G.J. Reid and A.D. Wittkopf, "Determination of Maximal Symmetry Groups of Classes of Differential Equations", Proc. ISSAC 2000 . ACM Press, 272-280
Reid, G.J., Wittkopf, A.D., and Boulton, A. "Reduction of Systems of Nonlinear Partial Differential Equations to Simplified Involutive Forms". Eur. J. Appl. Math. 7 (1996): 604-635.
Rust, C.J. 1998. "Rankings of Derivatives for Elimination Algorithms, and Formal Solvability of Analytic PDE". Ph.D. diss., University of Chicago.
Rust, C.J., Reid, G.J., and Wittkopf, A.D. "Existence and Uniqueness Theorems for Formal Power Series Solutions of Analytic Differential Systems". Proc. ISSAC 1999 . ACM Press, 105-112.

For a review of other algorithms and software (but more closely tied to symmetry analysis), please see the following.

Hereman, W. "Review of Symbolic Software for the Computation of Lie Symmetries of Differential Equations". Euromath Bull. 1 (1994): 45-79.

For a detailed guide to the use the Standard Form package, the predecessor of rifsimp , please see:

Reid, G.J. and Wittkopf, A.D. "The Long Guide to the Standard Form Package". 1993. Programs and documentation available on the web at http://wayback.cecm.sfu.ca/~wittkopf.

See Also

Rif , caseplot , PDEtools[casesplit] , checkrank , diffalg , dsolve,system , initialdata , maxdimsystems , pdsolve,system , rifsimp , rtaylor