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dAlembertian_formal_sol

  

formal solutions with d'Alembertian series coefficients for a linear ODE

 

Calling Sequence

Parameters

Description

Options

Examples

Calling Sequence

dAlembertian_formal_sol(ode, var, opts)

dAlembertian_formal_sol(LODEstr, opts)

Parameters

ode

-

homogeneous linear ODE with polynomial coefficients

var

-

dependent variable, for example y(x)

opts

-

optional arguments of the form keyword=value

LODEstr

-

LODEstruct data structure

Description

• 

The dAlembertian_formal_sol command returns formal solutions with d'Alembertian series coefficients to the given homogeneous linear ordinary differential equation with polynomial coefficients.

• 

If ode is an expression, then it is equated to zero.

• 

The routine returns an error message if the differential equation ode does not satisfy the following conditions.

– 

ode must be homogeneous and linear in var

– 

ode must have polynomial coefficients in the independent variable of var, for example, x

– 

The coefficients of ode must be either rational numbers or depend rationally on one or more parameters.

• 

A homogeneous linear ordinary differential equation with coefficients that are polynomials in x has a basis of formal solutions (see DEtools[formal_sol]). A formal solution contains a finite number of power series n=0vnTn where T is a parameter and the sequence vn satisfies a linear recurrence (homogeneous or inhomogeneous).

• 

This routine selects such formal solutions that contain only series with d'Alembertian coefficients. A sequence is called d'Alembertian if it is annihilated by a linear recurrence operator that can be written as a composition of first-order operators (see LinearOperators).

• 

The routine determines an integer N0 such that vn can be represented in the form of a d'Alembertian term:

vn=h[1]nn1=Nn1h[2]n1n2=Nn11...ns=Nns11h[s+1]ns ( + )

  

for all nN, where h[i]n, 1is+1, is a hypergeometric term (see SumTools[Hypergeometric]):

h[i]n=hNk=Nn1Rk ( ++ )

  

such that Rk=h[i]k+1h[i]k is rational in k for all kN.

Options

• 

'parameter'=T

  

Specifies the name T that is used to denote ax1r where a is a constant and r is called the ramification index. If this option is given, then the routine expresses the formal solutions in terms of T and returns a list of lists each of which is of the form [formal solution, relation between T and x]. Otherwise, it returns the formal solutions in terms of x1r.

• 

x=a or 'point'=a

  

Specifies the expansion point a. The default is a=0. It can be an algebraic number, depending rationally on some parameters, or .

• 

'free'=C

  

Specifies a base name C to use for free variables C[0], C[1], etc. The default is the global name  _C. Note that the number of free variables may be less than the order of the given equation if the expansion point is singular.

• 

'indices'=[n,k]

  

Specifies base names for dummy variables. The default values are the global names _n and _k, respectively. The name n is used as the summation index in the power series. the names n1, n2, etc., are used as summation indices in ( + ). The name k is used as the product index in ( ++ ).

• 

'outputHGT'=name

  

Specifies the form of representation of hypergeometric terms.  The default value is 'inert'.

– 

'inert' - the hypergeometric term ( ++ ) is represented by an inert product, except for k=Nn11, which is simplified to 1.

– 

'rcf1' or 'rcf2' - the hypergeometric term is represented in the first or second minimal representation, respectively (see ConjugateRTerm).

– 

'active' - the hypergeometric term is represented by non-inert products which, if possible, are computed (see product).

• 

'outputDAT'=name

  

Specifies the form of representation of the sums in ( + ). The default is 'inert'.

– 

'inert' - the sums are in the inert form, except for trivial sums of the form k=uv11, which are simplified to vu.

– 

'gosper' - Gosper's algorithm (see Gosper) is used to find a closed form for the sums in ( + ), if possible, starting with the innermost one.

Examples

withSlode:

ode4x2+2xyx+2x3x3x2ⅆⅆxyx+x3x4ⅆ2ⅆx2yx

ode:=x2+2x4yx+3x3x2+2xⅆⅆxyx+x4+x3ⅆ2ⅆx2yx

(1)

dAlembertian_formal_solode,yx,'outputHGT'='active','indices'=n,k

x212n=0∞xn+14n=0∞n1=0n112n1Γn1+3n1+2xn_C0+ⅇ2xn=0∞xn13_C1x

(2)

odex12ⅆ3ⅆx3yxx1x7ⅆ2ⅆx2yx22x5ⅆⅆxyx2yx

ode:=x12ⅆ3ⅆx3yxx1x7ⅆ2ⅆx2yx22x5ⅆⅆxyx2yx

(3)

dAlembertian_formal_solode,yx,x=0,'outputHGT'='inert','indices'=n,k

_C0n=0∞xn+_C1n=0∞nxn+_C2n=0∞n1=0n1n2=0n11k=0n211k+3xn

(4)

See Also

DEtools[formal_sol]

DEtools[translate]

LinearOperators

LODEstruct

Slode

Slode[hypergeom_formal_sol]

Slode[mhypergeom_formal_sol]

SumTools[Hypergeometric]

 


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