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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 (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.

withdiffalg:

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:=differential_ringranking=z,y,derivations=x,notation=diff

R:=ODE_ring

(1)

eq1:=yx+xⅆⅆxyx+ⅆⅆxyx2+ⅆⅆxzx

eq1:=yx+xⅆⅆxyx+ⅆⅆxyx2+ⅆⅆxzx

(2)

eq2:=zx+xⅆⅆxzx+ⅆⅆxyxⅆⅆxzx

eq2:=zx+xⅆⅆxzx+ⅆⅆxyxⅆⅆxzx

(3)

P:=Rosenfeld_Groebnereq1,eq2,R

P:=characterizable,characterizable,characterizable

(4)

equationsP1,inequationsP1

yx+xⅆⅆxyx+ⅆⅆxyx2+ⅆⅆxzx,ⅆⅆxyx3+2ⅆⅆxyx2xⅆⅆxyxyx+ⅆⅆxyxx2+zxyxx,x2+4xⅆⅆxyx+3ⅆⅆxyx2yx

(5)

equationsP2,inequationsP2

6ⅆⅆxyxyx+2ⅆⅆxyxx29zx+7yxx+2x3,27zx218zxyxx4zxx34yx3yx2x2,3yx+x2,2x39yxx+27zx

(6)

equationsP3,inequationsP3

27zx+x3,3yx+x2,

(7)

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

G:=Rosenfeld_Groebnereq1,eq2,x2+4xⅆⅆxyxyx+3ⅆⅆxyx2,R

G:=characterizable

(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.

p1:=v1t+v1[]v1x+v2[]v1y+px:

p2:=v2t+v1[]v2x+v2[]v2y+py:

p3:=v1x+v2y:

R:=differential_ringderivations=x,y,t,ranking=v1,v2,p

R:=PDE_ring

(9)

Rosenfeld_Groebnerp1,p2,p3,R

characterizable,characterizable,characterizable

(10)

S:=differential_ringderivations=t,x,y,ranking=lexp,v1,v2

S:=PDE_ring

(11)

J:=Rosenfeld_Groebnerp1,p2,p3,S

J:=characterizable

(12)

is_orthonomicJ

true

(13)

Rosenfeld_Groebnerp1,p2,p3,R,J

characterizable

(14)

See Also

diffalg(deprecated), diffalg(deprecated)/differential_algebra, diffalg(deprecated)/Rosenfeld_Groebner_options, diffalg(deprecated)[differential_ring], diffalg(deprecated)[equations], diffalg(deprecated)[inequations], diffalg(deprecated)[is_orthonomic], diffalg(deprecated)[rewrite_rules],


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