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DEtools

  

Closure

  

compute the closure of a linear differential operator

 

Calling Sequence

Parameters

Description

Examples

References

Compatibility

Calling Sequence

Closure(L, Dx, x, p, func, tord)

Parameters

L

-

polynomial in Dx with coefficients that are polynomials in x

Dx

-

variable, denoting the differential operator w.r.t. x

x

-

variable

p

-

(optional) irreducible polynomial in x

func

-

(optional) Maple command or user-defined procedure

tord

-

(optional) equation of the form termorder=TO

Description

• 

Let L be a linear differential operator, given as a polynomial in Dx with univariate polynomial coefficients in x over a field k of characteristic zero. The command Closure(L,Dx,x) constructs a basis of the closure of L, whose elements R satisfy P·L=f·R for an operator P and polynomial f in k[x] not dividing P on the left.

• 

If an optional fourth argument p is provided, Closure(L,Dx,x,p) constructs a local closure of L at the irreducible polynomial p. The output is a list of generators whose elements R satisfy P·L=p·R.

• 

A function may be specified using the optional argument func. It is applied to the coefficients of the collected result. Often factor or expand will be used.

• 

A Groebner basis computation with respect to a particular term ordering can be applied to the closure with the optional argument 'termorder'=TO where TO is of type MonomialOrder.

Examples

withDEtools:

For the given differential operator L

LDx4x24Dx3x+6x42x3Dx2+2Dxx2+x5+x42x

LDx4x24Dx3x+x42x3+6Dx2+2Dxx2+x5+x42x

(1)

compute the closure of L:

CClosureL,Dx,x

Cx2Dx44xDx3+x42x3+6Dx2+2x2Dx+x5+x42x,xDx6x2x+2Dx42x4x+5Dx3+x4+x312x6Dx2+2x25x+4Dx+4x5x+3,xDx7+2Dx6x2x+2Dx53x4x+5Dx4+x4+x336x30Dx3+14x3+10x224Dx2+2x30x+19Dx+x4+x3+60x+22

(2)

In the following example, we apply the Groebner basis computation with term ordering plexDx,x to the computed differential closure.

AOre_algebra:-diff_algebraDx,x,polynom=x:

TOGroebner:-MonomialOrderA,'plex'Dx,x:

ClosureL,Dx,x,'termorder'=TO

x2Dx4x4Dx22x3Dx2+x54xDx3+x4+2x2Dx+6Dx22x,Dx6x4Dx2xDx42x3Dx2+x512xDx3x2Dx2+2x414Dx3+12x2Dx+x36Dx2+10xDx+18x+10

(3)

Compute the local closure of L at p = x^2+1.  Only one of the polynomials in C satisfies P·L=p·R.

ClosureL,Dx,x,x2+1

x2Dx44xDx3+x42x3+6Dx2+2x2Dx+x5+x42x

(4)

References

  

Tsai, H. "Weyl closure of a linear differential operator." Journal of Symbolic Computation Vol. 29 No. 4-5 (2000): 747-775.

  

Chyzak, F.; Dumas, P.; Le, H.Q.; Martins, J.; Mishna, M.; Salvy, B. "Taming apparent singularities via Ore closure." In preparation.

Compatibility

• 

The DEtools[Closure] command was introduced in Maple 15.

• 

For more information on Maple 15 changes, see Updates in Maple 15.

See Also

DEtools/Desingularize

Groebner

Ore_algebra