diffalg(deprecated)/Rosenfeld_Groebner - Maple Help

diffalg

 Rosenfeld_Groebner
 compute a characteristic decomposition of the radical differential ideal generated by a finite set of differential polynomials

 Calling Sequence Rosenfeld_Groebner (S, H, R, J)

Parameters

 S - list or set of differential polynomials of R H - (optional) list or a set of differential polynomials of R R - differential polynomial ring J - (optional) radical differential ideal

Description

 • Important: The diffalg package has been deprecated. Use the superseding package DifferentialAlgebra instead.
 • For an informal presentation, see the diffalg overview.
 • Rosenfeld_Groebner computes a characteristic decomposition of the radical differential ideal P = {S}:(H)^infinity.
 • If the parameter H is omitted, Rosenfeld_Groebner computes a characteristic decomposition of the radical differential ideal P={S} generated by the differential polynomials of S.
 • R is a differential polynomial ring constructed with the differential_ring command.
 • The output of Rosenfeld_Groebner depends on the ranking defined on R.
 • Rosenfeld_Groebner returns a list of characterizable differential ideals.
 The empty list denotes the unit ideal (meaning that there is no solution).
 Each characterizable differential ideal is stored in a table. Only the name of the table ($\mathrm{characterizable}$) is printed on the screen. To access their defining characteristic sets you can use the commands rewrite_rules, equations, and inequations.
 • If the fourth parameter J is present, it is assumed to be another representation of $P$ with respect to another ranking. It is used to spare some splittings. It can be used to speed up the computation, for example, if there is a natural ranking to compute the representation of $P$.
 • The command with(diffalg,Rosenfeld_Groebner) allows the use of the abbreviated form of this command.

Examples

Important: The diffalg package has been deprecated. Use the superseding package DifferentialAlgebra instead.

 > $\mathrm{with}\left(\mathrm{diffalg}\right):$

The first example illustrates how the Rosenfeld_Groebner command splits a system of differential equations into a system representing the general solution and  systems representing the singular solutions.

 > $R≔\mathrm{differential_ring}\left(\mathrm{ranking}=\left[\left[z,y\right]\right],\mathrm{derivations}=\left[x\right],\mathrm{notation}=\mathrm{diff}\right)$
 ${R}{≔}{\mathrm{ODE_ring}}$ (1)
 > $\mathrm{eq1}≔-y\left(x\right)+x\mathrm{diff}\left(y\left(x\right),x\right)+{\mathrm{diff}\left(y\left(x\right),x\right)}^{2}+\mathrm{diff}\left(z\left(x\right),x\right)$
 ${\mathrm{eq1}}{≔}{-}{y}{}\left({x}\right){+}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)}^{{2}}{+}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{z}{}\left({x}\right)$ (2)
 > $\mathrm{eq2}≔-z\left(x\right)+x\mathrm{diff}\left(z\left(x\right),x\right)+\mathrm{diff}\left(y\left(x\right),x\right)\mathrm{diff}\left(z\left(x\right),x\right)$
 ${\mathrm{eq2}}{≔}{-}{z}{}\left({x}\right){+}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{z}{}\left({x}\right)\right){+}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{z}{}\left({x}\right)\right)$ (3)
 > $P≔\mathrm{Rosenfeld_Groebner}\left(\left[\mathrm{eq1},\mathrm{eq2}\right],R\right)$
 ${P}{≔}\left[{\mathrm{characterizable}}{,}{\mathrm{characterizable}}{,}{\mathrm{characterizable}}\right]$ (4)
 > $\mathrm{equations}\left(P\left[1\right]\right),\mathrm{inequations}\left(P\left[1\right]\right)$
 $\left[{-}{y}{}\left({x}\right){+}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)}^{{2}}{+}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{z}{}\left({x}\right){,}{\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)}^{{3}}{+}{2}{}{\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)}^{{2}}{}{x}{-}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){}{y}{}\left({x}\right){+}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){}{{x}}^{{2}}{+}{z}{}\left({x}\right){-}{y}{}\left({x}\right){}{x}\right]{,}\left[{{x}}^{{2}}{+}{4}{}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{3}{}{\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)}^{{2}}{-}{y}{}\left({x}\right)\right]$ (5)
 > $\mathrm{equations}\left(P\left[2\right]\right),\mathrm{inequations}\left(P\left[2\right]\right)$
 $\left[{6}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){}{y}{}\left({x}\right){+}{2}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){}{{x}}^{{2}}{-}{9}{}{z}{}\left({x}\right){+}{7}{}{y}{}\left({x}\right){}{x}{+}{2}{}{{x}}^{{3}}{,}{27}{}{{z}{}\left({x}\right)}^{{2}}{-}{18}{}{z}{}\left({x}\right){}{y}{}\left({x}\right){}{x}{-}{4}{}{z}{}\left({x}\right){}{{x}}^{{3}}{-}{4}{}{{y}{}\left({x}\right)}^{{3}}{-}{{y}{}\left({x}\right)}^{{2}}{}{{x}}^{{2}}\right]{,}\left[{3}{}{y}{}\left({x}\right){+}{{x}}^{{2}}{,}{-}{2}{}{{x}}^{{3}}{-}{9}{}{y}{}\left({x}\right){}{x}{+}{27}{}{z}{}\left({x}\right)\right]$ (6)
 > $\mathrm{equations}\left(P\left[3\right]\right),\mathrm{inequations}\left(P\left[3\right]\right)$
 $\left[{27}{}{z}{}\left({x}\right){+}{{x}}^{{3}}{,}{3}{}{y}{}\left({x}\right){+}{{x}}^{{2}}\right]{,}\left[\right]$ (7)

To obtain the characterizable differential ideal representing the  general solution alone, we can proceed as follows.

 > $G≔\mathrm{Rosenfeld_Groebner}\left(\left[\mathrm{eq1},\mathrm{eq2}\right],\left[{x}^{2}+4x\mathrm{diff}\left(y\left(x\right),x\right)-y\left(x\right)+3{\mathrm{diff}\left(y\left(x\right),x\right)}^{2}\right],R\right)$
 ${G}{≔}\left[{\mathrm{characterizable}}\right]$ (8)

It is sometimes the case that the radical differential ideal $P$ generated by S is prime. This can be proved by exhibiting a ranking for which the characteristic decomposition of P consists of only one orthonomic characterizable differential ideal.

Before computing a representation of P with respect to the ranking of R, it may be useful to proceed as follows. Search for a ranking for which the characteristic decomposition is as described above. Assign J this computed characteristic decomposition. Then call Rosenfeld_Groebner with J as fourth parameter.

With such a fourth parameter, whatever the ranking of R is, the computed representation of P consists of only one characterizable differential ideal.

If J consists of a single non-orthonomic component or has more than one characterizable component, Rosenfeld_Groebner uses the information to avoid unnecessary splittings.

The example below illustrates this behavior for Euler's equations for an incompressible fluid in two dimensions.

 > $\mathrm{p1}≔\mathrm{v1}\left[t\right]+\mathrm{v1}\left[\right]\mathrm{v1}\left[x\right]+\mathrm{v2}\left[\right]\mathrm{v1}\left[y\right]+p\left[x\right]:$
 > $\mathrm{p2}≔\mathrm{v2}\left[t\right]+\mathrm{v1}\left[\right]\mathrm{v2}\left[x\right]+\mathrm{v2}\left[\right]\mathrm{v2}\left[y\right]+p\left[y\right]:$
 > $\mathrm{p3}≔\mathrm{v1}\left[x\right]+\mathrm{v2}\left[y\right]:$
 > $R≔\mathrm{differential_ring}\left(\mathrm{derivations}=\left[x,y,t\right],\mathrm{ranking}=\left[\left[\mathrm{v1},\mathrm{v2},p\right]\right]\right)$
 ${R}{≔}{\mathrm{PDE_ring}}$ (9)
 > $\mathrm{Rosenfeld_Groebner}\left(\left[\mathrm{p1},\mathrm{p2},\mathrm{p3}\right],R\right)$
 $\left[{\mathrm{characterizable}}{,}{\mathrm{characterizable}}{,}{\mathrm{characterizable}}\right]$ (10)
 > $S≔\mathrm{differential_ring}\left(\mathrm{derivations}=\left[t,x,y\right],\mathrm{ranking}=\left[\mathrm{lex}\left[p,\mathrm{v1},\mathrm{v2}\right]\right]\right)$
 ${S}{≔}{\mathrm{PDE_ring}}$ (11)
 > $J≔\mathrm{Rosenfeld_Groebner}\left(\left[\mathrm{p1},\mathrm{p2},\mathrm{p3}\right],S\right)$
 ${J}{≔}\left[{\mathrm{characterizable}}\right]$ (12)
 > $\mathrm{is_orthonomic}\left(J\right)$
 $\left[{\mathrm{true}}\right]$ (13)
 > $\mathrm{Rosenfeld_Groebner}\left(\left[\mathrm{p1},\mathrm{p2},\mathrm{p3}\right],R,J\right)$
 $\left[{\mathrm{characterizable}}\right]$ (14)