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Ore_algebra[Ore_to_diff] - convert a differential operator to a differential equation

Ore_algebra[Ore_to_shift] - convert a shift operator to a recurrence equation

Ore_algebra[Ore_to_DESol] - convert a differential operator to a DESol structure

Ore_algebra[Ore_to_RESol] - convert a shift operator to an RESol structure

Calling Sequence

Ore_to_diff(G, f, A)

Ore_to_diff(G, f, A, 'D')

Ore_to_shift(G, u, A)

Ore_to_shift(G, u, A, 'indexed')

Ore_to_DESol(P, f, A)

Ore_to_RESol(P, u, A)

Parameters

G

-

list of operators of the Ore algebra A

P

-

operator of the Ore algebra A

f

-

expression denoting a mathematical function

A

-

Ore algebra table

Description

• 

The Ore_to_diff command converts a differential operator or a list of differential operators of the skew algebra A into a differential equation or a list of differential equations in the function f.  The output is expressed in terms of the diff function by default, or in terms of the D function when the optional parameter is set.

• 

The Ore_to_DESol command converts a single differential operator of the skew algebra A into a DESol structure in the function f.

• 

The Ore_to_shift command converts a shift operator or a list of shift operators of the skew algebra A into a recurrence equation or a list of recurrence equations in the sequence u.  The output is expressed in functional notation ( un,... ) by default, or in the indexed notation ( un,... ) when the optional argument is set.

• 

The Ore_to_RESol command converts a single recurrence operator of the skew algebra A into an RESol structure in the sequence u.

Examples

withOre_algebra:

Differential case.

A:=diff_algebraDx,x,comm,μ:

P:=x2Dx2+xDx+x2μ2:

Ore_to_diffP,f,A

x2ⅆ2ⅆx2fx+xⅆⅆxfx+μ2+x2fx

(1)

Ore_to_diffP,f,A,D

x2D2fx+xDfx+μ2+x2fx

(2)

Ore_to_DESolP,f,A

DESolx2ⅆ2ⅆx2fx+xⅆⅆxfx+μ2+x2fx,fx

(3)

normalapplyoprP,,A

0

(4)

Euler case.

A:=skew_algebraeuler=Tx,x,comm=μ:

P:=Tx2+x2μ2:

Ore_to_diffP,f,A

xⅆⅆxfx+xⅆ2ⅆx2fx+μ2+x2fx

(5)

Recurrence case.

A:=shift_algebraSn,n,comm,α:

P:=Sn2+αSn+1:

Ore_to_shiftP,u,A

un+2+αun+1+un

(6)

Ore_to_shiftP,u,A,indexed

αun+1+un+un+2

(7)

Ore_to_RESolP,u,A

RESolun+2+αun+1+un=0,un,u0=u0,u1=u1,INFO

(8)

Multivariate differential case.

A:=diff_algebraDx,x,Dy,y,comm,μ:

G:=2Dxx+Dyy,4μxy2μ+xy2+2Dxx+y2Dy2,2μxy2μ+xy2+DyDxyx,μxy2μ+xy2+Dxx+Dx2x2:

These are operators for BesselJ(mu,x*y^2).

Ore_to_diffG,f,A

2xxfx,y+yyfx,y,4xy2+μxy2+μfx,y+y22y2fx,y+2xxfx,y,2xy2+μxy2+μfx,y+yx2yxfx,y,xy2+μxy2+μfx,y+x22x2fx,y+xxfx,y

(9)

Ore_to_diffG,f,A,D

2xD1fx,y+yD2fx,y,4xy2+μxy2+μfx,y+y2D2,2fx,y+2xD1fx,y,2xy2+μxy2+μfx,y+yxD1,2fx,y,xy2+μxy2+μfx,y+x2D1,1fx,y+xD1fx,y

(10)

No conversion is available to a multivariate DESol.

Ore_to_DESolG,f,A

Error, invalid input: Ore_algebra:-Ore_to_DESol expects its 1st argument, poly, to be of type polynom, but received [-2*Dx*x+Dy*y, -4*(-x*y^2+mu)*(x*y^2+mu)+2*x*Dx+y^2*Dy^2, -2*(-x*y^2+mu)*(x*y^2+mu)+Dy*Dx*y*x, -(-x*y^2+mu)*(x*y^2+mu)+x*Dx+x^2*Dx^2]

See Also

Ore_algebra, Ore_algebra/applyopr, Ore_algebra/skew_algebra


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