Overview - Maple Help

Overview of the Rif Command Set Version 1.1

Description

 • The Rif command set is a powerful collection of commands for the simplification and analysis of systems of polynomially nonlinear ODEs and PDEs.
 • The Rif command set is part of the DEtools package. As such, the commands are invoked using the DEtools package. Each command in the DEtools package can be accessed by using either the long form or the short form of the command name in the command calling sequence.
 • The set 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 commands are 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 the Rif command set 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 DifferentialAlgebra package, the commands use different approaches, one of which may work better for specific problems than the other.
 • The Rif command set 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 command 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 following is a list of available commands.

 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. 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). Loads a partially completed rifsimp calculation for viewing and/or manual manipulation. rifsimp must be told to save partial calculations using the storage options. Provides information on ranking to allow determination of an appropriate ranking to use with rifsimp. Takes the case split output of rifsimp, and provides a graphical display of the solution case tree. 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. 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.

 • To display the help page for a particular Rif command in the DEtools package, see Getting Help with a Command in a Package.

Examples

For the following system

 > $\mathrm{sys}≔\left\{-\mathrm{\eta }\left(x,y\right)\left(6{a}^{2}{y}^{2}-2abx\right)-2\left(2{a}^{2}{y}^{3}-2abxy+b\right)\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),x\right)+\left(2{a}^{2}{y}^{3}-2abxy+b\right)\mathrm{diff}\left(\mathrm{\eta }\left(x,y\right),y\right)+2\mathrm{\xi }\left(x,y\right)aby+\mathrm{diff}\left(\mathrm{\eta }\left(x,y\right),x,x\right),-3\left(2{a}^{2}{y}^{3}-2abxy+b\right)\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),y\right)+2\mathrm{diff}\left(\mathrm{\eta }\left(x,y\right),x,y\right)-\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),x,x\right),\mathrm{diff}\left(\mathrm{\eta }\left(x,y\right),y,y\right)-2\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),x,y\right),\mathrm{diff}\left(\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),y\right),y\right)\right\}$
 ${\mathrm{sys}}{≔}\left\{\frac{{{\partial }}^{{2}}}{{\partial }{{y}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\eta }}{}\left({x}{,}{y}\right){-}{2}{}\frac{{{\partial }}^{{2}}}{{\partial }{x}{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\xi }}{}\left({x}{,}{y}\right){,}{-}{3}{}\left({2}{}{{a}}^{{2}}{}{{y}}^{{3}}{-}{2}{}{a}{}{b}{}{x}{}{y}{+}{b}\right){}\left(\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\xi }}{}\left({x}{,}{y}\right)\right){+}{2}{}\frac{{{\partial }}^{{2}}}{{\partial }{x}{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\eta }}{}\left({x}{,}{y}\right){-}\frac{{{\partial }}^{{2}}}{{\partial }{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\xi }}{}\left({x}{,}{y}\right){,}{-}{\mathrm{\eta }}{}\left({x}{,}{y}\right){}\left({6}{}{{a}}^{{2}}{}{{y}}^{{2}}{-}{2}{}{a}{}{b}{}{x}\right){-}{2}{}\left({2}{}{{a}}^{{2}}{}{{y}}^{{3}}{-}{2}{}{a}{}{b}{}{x}{}{y}{+}{b}\right){}\left(\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\xi }}{}\left({x}{,}{y}\right)\right){+}\left({2}{}{{a}}^{{2}}{}{{y}}^{{3}}{-}{2}{}{a}{}{b}{}{x}{}{y}{+}{b}\right){}\left(\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\eta }}{}\left({x}{,}{y}\right)\right){+}{2}{}{\mathrm{\xi }}{}\left({x}{,}{y}\right){}{a}{}{b}{}{y}{+}\frac{{{\partial }}^{{2}}}{{\partial }{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\eta }}{}\left({x}{,}{y}\right){,}\frac{{{\partial }}^{{2}}}{{\partial }{{y}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\xi }}{}\left({x}{,}{y}\right)\right\}$ (1)

we can obtain all cases of the reduced form via

 > $\mathrm{with}\left(\mathrm{DEtools}\right):$
 > $\mathrm{cases}≔\mathrm{rifsimp}\left(\mathrm{sys},\mathrm{casesplit}\right)$
 ${\mathrm{cases}}{≔}{table}{}\left(\left[{1}{=}{table}{}\left(\left[{\mathrm{Pivots}}{=}\left[{a}{\ne }{0}{,}{b}{\ne }{0}\right]{,}{\mathrm{Solved}}{=}\left[{\mathrm{\eta }}{}\left({x}{,}{y}\right){=}{0}{,}{\mathrm{\xi }}{}\left({x}{,}{y}\right){=}{0}\right]{,}{\mathrm{Case}}{=}\left[\left[{a}{\ne }{0}{,}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\xi }}{}\left({x}{,}{y}\right)\right]{,}\left[{b}{\ne }{0}{,}{\mathrm{\eta }}{}\left({x}{,}{y}\right)\right]\right]\right]\right){,}{2}{=}{table}{}\left(\left[{\mathrm{Pivots}}{=}\left[{a}{\ne }{0}\right]{,}{\mathrm{Solved}}{=}\left[\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\eta }}{}\left({x}{,}{y}\right){=}{0}{,}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\xi }}{}\left({x}{,}{y}\right){=}{-}\frac{{\mathrm{\eta }}{}\left({x}{,}{y}\right)}{{y}}{,}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\eta }}{}\left({x}{,}{y}\right){=}\frac{{\mathrm{\eta }}{}\left({x}{,}{y}\right)}{{y}}{,}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\xi }}{}\left({x}{,}{y}\right){=}{0}{,}{b}{=}{0}\right]{,}{\mathrm{Case}}{=}\left[\left[{a}{\ne }{0}{,}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\xi }}{}\left({x}{,}{y}\right)\right]{,}\left[{b}{=}{0}{,}{\mathrm{\eta }}{}\left({x}{,}{y}\right)\right]\right]\right]\right){,}{3}{=}{table}{}\left(\left[{\mathrm{Solved}}{=}\left[\frac{{{\partial }}^{{3}}}{{\partial }{x}{\partial }{{y}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\eta }}{}\left({x}{,}{y}\right){=}{0}{,}\frac{{{\partial }}^{{3}}}{{\partial }{{y}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\eta }}{}\left({x}{,}{y}\right){=}{0}{,}\frac{{{\partial }}^{{2}}}{{\partial }{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\eta }}{}\left({x}{,}{y}\right){=}{b}{}\left({2}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\xi }}{}\left({x}{,}{y}\right){-}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\eta }}{}\left({x}{,}{y}\right)\right){,}\frac{{{\partial }}^{{2}}}{{\partial }{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\xi }}{}\left({x}{,}{y}\right){=}{-}{3}{}{b}{}\left(\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\xi }}{}\left({x}{,}{y}\right)\right){+}{2}{}\frac{{{\partial }}^{{2}}}{{\partial }{x}{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\eta }}{}\left({x}{,}{y}\right){,}\frac{{{\partial }}^{{2}}}{{\partial }{x}{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\xi }}{}\left({x}{,}{y}\right){=}\frac{\frac{{{\partial }}^{{2}}}{{\partial }{{y}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\eta }}{}\left({x}{,}{y}\right)}{{2}}{,}\frac{{{\partial }}^{{2}}}{{\partial }{{y}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\xi }}{}\left({x}{,}{y}\right){=}{0}{,}{a}{=}{0}\right]{,}{\mathrm{Case}}{=}\left[\left[{a}{=}{0}{,}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\xi }}{}\left({x}{,}{y}\right)\right]\right]\right]\right){,}{\mathrm{casecount}}{=}{3}\right]\right)$ (2)

and there are

 > $\mathrm{cases}\left[\mathrm{casecount}\right]$
 ${3}$ (3)

cases. We can get an idea of the structure of the case splitting via:

 > $\mathrm{caseplot}\left(\mathrm{cases}\right)$

and for a case we can obtain the initial data

 > $\mathrm{id}≔\mathrm{initialdata}\left(\mathrm{cases}\left[\mathrm{cases}\left[\mathrm{casecount}\right]\right]\right)$
 ${\mathrm{id}}{≔}{table}{}\left(\left[{\mathrm{Infinite}}{=}\left[\right]{,}{\mathrm{Finite}}{=}\left[{\mathrm{\eta }}{}\left({{x}}_{{0}}{,}{{y}}_{{0}}\right){=}{\mathrm{_C1}}{,}{{\mathrm{D}}}_{{1}}{}\left({\mathrm{\eta }}\right){}\left({{x}}_{{0}}{,}{{y}}_{{0}}\right){=}{\mathrm{_C2}}{,}{{\mathrm{D}}}_{{2}}{}\left({\mathrm{\eta }}\right){}\left({{x}}_{{0}}{,}{{y}}_{{0}}\right){=}{\mathrm{_C3}}{,}{{\mathrm{D}}}_{{1}{,}{2}}{}\left({\mathrm{\eta }}\right){}\left({{x}}_{{0}}{,}{{y}}_{{0}}\right){=}{\mathrm{_C4}}{,}{{\mathrm{D}}}_{{2}{,}{2}}{}\left({\mathrm{\eta }}\right){}\left({{x}}_{{0}}{,}{{y}}_{{0}}\right){=}{\mathrm{_C5}}{,}{\mathrm{\xi }}{}\left({{x}}_{{0}}{,}{{y}}_{{0}}\right){=}{\mathrm{_C6}}{,}{{\mathrm{D}}}_{{1}}{}\left({\mathrm{\xi }}\right){}\left({{x}}_{{0}}{,}{{y}}_{{0}}\right){=}{\mathrm{_C7}}{,}{{\mathrm{D}}}_{{2}}{}\left({\mathrm{\xi }}\right){}\left({{x}}_{{0}}{,}{{y}}_{{0}}\right){=}{\mathrm{_C8}}{,}{b}{=}{\mathrm{_C9}}\right]\right]\right)$ (4)

and use it to obtain a multi-dimensional Taylor series for that case

 > $\mathrm{rtaylor}\left(\mathrm{cases}\left[\mathrm{cases}\left[\mathrm{casecount}\right]\right]\left[\mathrm{Solved}\right],\mathrm{id}\right)$
 $\left[{\mathrm{\eta }}{}\left({x}{,}{y}\right){=}{\mathrm{_C1}}{+}{\mathrm{_C2}}{}\left({x}{-}{{x}}_{{0}}\right){+}{\mathrm{_C3}}{}\left({y}{-}{{y}}_{{0}}\right){+}\frac{{\mathrm{_C9}}{}\left({2}{}{\mathrm{_C7}}{-}{\mathrm{_C3}}\right){}{\left({x}{-}{{x}}_{{0}}\right)}^{{2}}}{{2}}{+}{\mathrm{_C4}}{}\left({x}{-}{{x}}_{{0}}\right){}\left({y}{-}{{y}}_{{0}}\right){+}\frac{{\mathrm{_C5}}{}{\left({y}{-}{{y}}_{{0}}\right)}^{{2}}}{{2}}{,}{\mathrm{\xi }}{}\left({x}{,}{y}\right){=}{\mathrm{_C6}}{+}{\mathrm{_C7}}{}\left({x}{-}{{x}}_{{0}}\right){+}{\mathrm{_C8}}{}\left({y}{-}{{y}}_{{0}}\right){+}\frac{\left({-}{3}{}{\mathrm{_C8}}{}{\mathrm{_C9}}{+}{2}{}{\mathrm{_C4}}\right){}{\left({x}{-}{{x}}_{{0}}\right)}^{{2}}}{{2}}{+}\frac{{\mathrm{_C5}}{}\left({x}{-}{{x}}_{{0}}\right){}\left({y}{-}{{y}}_{{0}}\right)}{{2}}{,}{a}{=}{0}{,}{\mathrm{_C9}}{=}{\mathrm{_C9}}\right]$ (5)

References

 For theory used to produce the rifsimp and maxdimsystems algorithm, and related theory, please see the following:
 Becker, T., and Weispfenning, V. Groebner Bases: A Computational Approach to Commutative Algebra. New York: Springer-Verlag, 1993.
 Bluman, G.W., and Kumei, S. Symmetries and Differential Equations, Vol. 81. Springer-Verlag.
 Boulier, F.; Lazard, D.; Ollivier, F.; and Petitot, M. "Representation for the Radical of a Finitely Generated Differential Ideal." Proc. ISSAC 1995, pp. 158-166. ACM Press.
 Carra-Ferro, G. "Groebner Bases and Differential Algebra." Lecture Notes in Comp. Sci., Vol. 356, (1987): 128-140.
 Goldschmidt, H. "Integrability Criteria for Systems of Partial Differential Equations." J. Diff. Geom., Vol. 1, (1967): 269-307.
 Mansfield, E. "Differential Groebner Bases." Ph.D. diss., University of Sydney, 1991.
 Ollivier, F. "Standard Bases of Differential Ideals." Lecture Notes in Comp. Sci., Vol. 508, (1991): 304-321.
 Reid, G.J., and Wittkopf, A.D. "Determination of Maximal Symmetry Groups of Classes of Differential Equations." Proc. ISSAC 2000, pp. 272-280. ACM Press.
 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., Vol. 7, (1996): 604-635.
 Rust, C.J. "Rankings of Derivatives for Elimination Algorithms, and Formal Solvability of Analytic PDE." Ph.D. diss., University of Chicago, 1998.
 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, pp. 105-112. ACM Press.
 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., Vol. 1, (1994): 45-79.
 For a detailed guide to the use the Standard Form package, the predecessor of rifsimp, refer to:
 Reid, G.J. and Wittkopf, A.D. The Long Guide to the Standard Form Package. 1993.