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

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

New command maxdimsystems to find the most general solutions for case splitting problems

Improvements to case visualization and initial data computation

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

More efficient handling of nonlinear systems via new nonlinear equation methods

Significant overall speed and memory enhancements

Automatic adjustment of results to remove inconsistent cases, and their effect on the returned consistent cases.

The following is a list of available commands.

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

$\mathrm{sys}≔\left\{-\mathrm{\eta }\left(x\,y\right)\left(6a^2{y}^2-2abx\right)-2\left(2a^2{y}^3-2abxy+b\right)\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,x\right)+\left(2a^2{y}^3-2abxy+b\right)\mathrm{diff}\left(\mathrm{\eta }\left(x\,y\right)\,y\right)+2\mathrm{\xi }\left(x\,y\right)aby+\mathrm{diff}\left(\mathrm{\eta }\left(x\,y\right)\,x\,x\right)\,-3\left(2a^2{y}^3-2abxy+b\right)\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,y\right)+2\mathrm{diff}\left(\mathrm{\eta }\left(x\,y\right)\,x\,y\right)-\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,x\,x\right)\,\mathrm{diff}\left(\mathrm{\eta }\left(x\,y\right)\,y\,y\right)-2\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,x\,y\right)\,\mathrm{diff}\left(\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,y\right)\,y\right)\right\}$

$\mathrm{sys}≔\left\{\frac{{\partial }^2}{\partial y^2}\mathrm{\eta }\left(x\,y\right)-2\frac{{\partial }^2}{\partial x\partial y}\mathrm{\xi }\left(x\,y\right)\,-3\left(2a^2{y}^3-2abxy+b\right)\left(\frac{\partial }{\partial y}\mathrm{\xi }\left(x\,y\right)\right)+2\frac{{\partial }^2}{\partial x\partial y}\mathrm{\eta }\left(x\,y\right)-\frac{{\partial }^2}{\partial x^2}\mathrm{\xi }\left(x\,y\right)\,-\mathrm{\eta }\left(x\,y\right)\left(6a^2{y}^2-2abx\right)-2\left(2a^2{y}^3-2abxy+b\right)\left(\frac{\partial }{\partial x}\mathrm{\xi }\left(x\,y\right)\right)+\left(2a^2{y}^3-2abxy+b\right)\left(\frac{\partial }{\partial y}\mathrm{\eta }\left(x\,y\right)\right)+2\mathrm{\xi }\left(x\,y\right)aby+\frac{{\partial }^2}{\partial x^2}\mathrm{\eta }\left(x\,y\right)\,\frac{{\partial }^2}{\partial y^2}\mathrm{\xi }\left(x\,y\right)\right\}$

$\mathrm{with}\left(\mathrm{DEtools}\right)\:$

$\mathrm{cases}≔\mathrm{rifsimp}\left(\mathrm{sys}\,\mathrm{casesplit}\right)$

$\mathrm{cases}≔table\left(\left[1=table\left(\left[\mathrm{Solved}=\left[\mathrm{\eta }\left(x\,y\right)=0\,\mathrm{\xi }\left(x\,y\right)=0\right]\,\mathrm{Case}=\left[\left[a\ne 0\,\frac{\partial }{\partial y}\mathrm{\xi }\left(x\,y\right)\right]\,\left[b\ne 0\,\mathrm{\eta }\left(x\,y\right)\right]\right]\,\mathrm{Pivots}=\left[a\ne 0\,b\ne 0\right]\right]\right)\,2=table\left(\left[\mathrm{Solved}=\left[\frac{\partial }{\partial x}\mathrm{\eta }\left(x\,y\right)=0\,\frac{\partial }{\partial x}\mathrm{\xi }\left(x\,y\right)=-\frac{\mathrm{\eta }\left(x\,y\right)}{y}\,\frac{\partial }{\partial y}\mathrm{\eta }\left(x\,y\right)=\frac{\mathrm{\eta }\left(x\,y\right)}{y}\,\frac{\partial }{\partial y}\mathrm{\xi }\left(x\,y\right)=0\,b=0\right]\,\mathrm{Case}=\left[\left[a\ne 0\,\frac{\partial }{\partial y}\mathrm{\xi }\left(x\,y\right)\right]\,\left[b=0\,\mathrm{\eta }\left(x\,y\right)\right]\right]\,\mathrm{Pivots}=\left[a\ne 0\right]\right]\right)\,3=table\left(\left[\mathrm{Solved}=\left[\frac{{\partial }^3}{\partial x\partial y^2}\mathrm{\eta }\left(x\,y\right)=0\,\frac{{\partial }^3}{\partial y^3}\mathrm{\eta }\left(x\,y\right)=0\,\frac{{\partial }^2}{\partial x^2}\mathrm{\eta }\left(x\,y\right)=b\left(2\frac{\partial }{\partial x}\mathrm{\xi }\left(x\,y\right)-\frac{\partial }{\partial y}\mathrm{\eta }\left(x\,y\right)\right)\,\frac{{\partial }^2}{\partial x^2}\mathrm{\xi }\left(x\,y\right)=-3b\left(\frac{\partial }{\partial y}\mathrm{\xi }\left(x\,y\right)\right)+2\frac{{\partial }^2}{\partial x\partial y}\mathrm{\eta }\left(x\,y\right)\,\frac{{\partial }^2}{\partial x\partial y}\mathrm{\xi }\left(x\,y\right)=\frac{\frac{{\partial }^2}{\partial y^2}\mathrm{\eta }\left(x\,y\right)}{2}\,\frac{{\partial }^2}{\partial y^2}\mathrm{\xi }\left(x\,y\right)=0\,a=0\right]\,\mathrm{Case}=\left[\left[a=0\,\frac{\partial }{\partial y}\mathrm{\xi }\left(x\,y\right)\right]\right]\right]\right)\,\mathrm{casecount}=3\right]\right)$

$\mathrm{cases}\left[\mathrm{casecount}\right]$

$3$

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

$\mathrm{id}≔\mathrm{initialdata}\left(\mathrm{cases}\left[\mathrm{cases}\left[\mathrm{casecount}\right]\right]\right)$

$\mathrm{id}≔table\left(\left[\mathrm{Finite}=\left[\mathrm{\eta }\left(x_0\,y_0\right)=\mathrm{\_C1}\,{\mathrm{D}}_1\left(\mathrm{\eta }\right)\left(x_0\,y_0\right)=\mathrm{\_C2}\,{\mathrm{D}}_2\left(\mathrm{\eta }\right)\left(x_0\,y_0\right)=\mathrm{\_C3}\,{\mathrm{D}}_{1,2}\left(\mathrm{\eta }\right)\left(x_0\,y_0\right)=\mathrm{\_C4}\,{\mathrm{D}}_{2,2}\left(\mathrm{\eta }\right)\left(x_0\,y_0\right)=\mathrm{\_C5}\,\mathrm{\xi }\left(x_0\,y_0\right)=\mathrm{\_C6}\,{\mathrm{D}}_1\left(\mathrm{\xi }\right)\left(x_0\,y_0\right)=\mathrm{\_C7}\,{\mathrm{D}}_2\left(\mathrm{\xi }\right)\left(x_0\,y_0\right)=\mathrm{\_C8}\,b=\mathrm{\_C9}\right]\,\mathrm{Infinite}=\left[\right]\right]\right)$

$\mathrm{rtaylor}\left(\mathrm{cases}\left[\mathrm{cases}\left[\mathrm{casecount}\right]\right]\left[\mathrm{Solved}\right]\,\mathrm{id}\right)$

$\left[\mathrm{\eta }\left(x\,y\right)=\mathrm{\_C1}+\mathrm{\_C2}\left(x-x_0\right)+\mathrm{\_C3}\left(y-y_0\right)+\frac{\mathrm{\_C9}\left(2\mathrm{\_C7}-\mathrm{\_C3}\right){\left(x-x_0\right)}^2}{2}+\mathrm{\_C4}\left(x-x_0\right)\left(y-y_0\right)+\frac{\mathrm{\_C5}{\left(y-y_0\right)}^2}{2}\,\mathrm{\xi }\left(x\,y\right)=\mathrm{\_C6}+\mathrm{\_C7}\left(x-x_0\right)+\mathrm{\_C8}\left(y-y_0\right)+\frac{\left(-3\mathrm{\_C8}\mathrm{\_C9}+2\mathrm{\_C4}\right){\left(x-x_0\right)}^2}{2}+\frac{\mathrm{\_C5}\left(x-x_0\right)\left(y-y_0\right)}{2}\,a=0\,\mathrm{\_C9}=\mathrm{\_C9}\right]$

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.