compute a minimal characteristic decomposition
essential_components (p, R)
differential polynomial in R
differential polynomial ring
Important: The diffalg package has been deprecated. Use the superseding package DifferentialAlgebra instead.
The function essential_components returns a minimal characteristic decomposition of the radical differential ideal generated by the single differential polynomial p.
Each of the characterizable components returned has a characteristic set consisting of only one differential polynomial, say a1,...,ak.
This means that the set of solutions of the differential equation p=0 is minimally described as the union of the general solutions of a1=0, ... , ak=0.
The set of irreducible factors of a1,...,ak does not depend on the ranking chosen for R.
This function proceeds by eliminating the redundancy in the characteristic decomposition computed by Rosenfeld_Groebner applied to ([p], R).
The command with(diffalg,essential_components) allows the use of the abbreviated form of this command.
Ordinary differential polynomials of first order:
R ≔ differential_ring⁡derivations=t,ranking=y,notation=diff:
p ≔ ⅆⅆt⁢y⁡t3−4⁢t⁢y⁡t⁢ⅆⅆt⁢y⁡t+8⁢y⁡t2
This differential polynomial has two singular zeros: the cubic y⁡t=4⁢t327 and y⁡t=0. Nonetheless, the general zero can be expressed as y⁡t=_C⁢t−_C2. Therefore, y⁡t=0 is a particular case (_C=0) of the general solution. This is uncovered by essential_components without solving the differential equation. The function essential_components gives a minimal description of the zero set.
Let us consider the two similar differential polynomials p and q.
R ≔ differential_ring⁡derivations=t,ranking=y:
p ≔ yt2−4⁢y
q ≔ yt2−4⁢y3
Cp ≔ equations⁡Rosenfeld_Groebner⁡p,R
Cq ≔ equations⁡Rosenfeld_Groebner⁡q,R
Both p and q admit y⁡t=0 as a singular zero. Nonetheless:
Mp ≔ equations⁡essential_components⁡p,R
Mq ≔ equations⁡essential_components⁡q,R
y⁡t=0 is an essential singular zero of p but not of q. This has an analytic interpretation: y⁡t=0 is an envelope of the non singular zeros of p while it is a limit of the non singular zeros of q.
Incidentally: the general zero of q can be expressed as y⁡t=_C_C⁢t−12. Thus, y⁡t=0 is a particular case of the general zero of q.
Partial differential polynomials:
This illustrates the fact that the characteristic sets of the components of the minimal characteristic decomposition have only one element.
R ≔ differential_ring⁡derivations=x,y,ranking=u:
p ≔ −u+y⁢uy+x⁢ux−ux2−uy2
C ≔ equations⁡Rosenfeld_Groebner⁡p,R
M ≔ equations⁡essential_components⁡p,R
A differential polynomial in several variables:
R ≔ differential_ring⁡derivations=x,y,ranking=u,v:
p ≔ ux,y2⁢vy−ux,y⁢vy⁢uy−uy⁢ux,y+uy2
It would seem that there several types of zeros, the general zero of p and several singular zeros. Nonetheless,
MR ≔ essential_components⁡p,R
ER ≔ equations⁡MR
This show that the singular zeros exhibited by the Rosenfeld_Groebner decomposition are in fact particular zeros of the general zero of p.
We illustrate now the fact that the underlying prime minimal decomposition of the obtained characteristic minimal decomposition is independent of the ranking.
Q ≔ differential_ring⁡derivations=x,y,ranking=v,u:
MQ ≔ essential_components⁡p,Q
EQ ≔ equations⁡MQ
We check that the two differential polynomials appearing in this decompositions are the two factors of differential polynomials appearing in MR.
Higher order differential polynomials:
The following equation arose in Chazy's work to extend the Painleve analysis to third order differential equations. In the process, he uncovered certain differential equations whose non-singular solutions have no movable singularity whereas one of the singular solutions does.
R ≔ differential_ring⁡ranking=y,derivations=x:
chazy ≔ −yx,x+y3⁢yx2+y⁢yx2⁢4⁢yx+y4
The singular zeros are given by y⁡x=_C and y⁡x3=13⁢x4+_C. Only the second kind is essential.
The zeros of the following 4th order, homogeneous differential equation of degree 7 have the property that they can be used to approximate piecewisely any smooth function. This was shown by Rubel (1981).
R ≔ differential_ring⁡ranking=y,z,derivations=x:
rubel ≔ 3⁢yx4⁢yx$2⁢yx$42−4⁢yx4⁢yx$32⁢yx$4+6⁢yx3⁢yx$22⁢yx$3⁢yx$4+24⁢yx2⁢yx$24⁢yx$4−12⁢yx3⁢yx$2⁢yx$33−29⁢yx2⁢yx$23⁢yx$32+12⁢yx$27
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