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dAlembertian_series_sol

  

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

 

Calling Sequence

Parameters

Description

Options

Examples

Calling Sequence

dAlembertian_series_sol(ode,var,opts)

dAlembertian_series_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_series_sol command returns one formal power series solution or a set of formal power series solutions with d'Alembertian coefficients for 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 linear space of formal power series solutions n=0vnPnx where P[n]x is one of xan, xann!, 1xn, or 1xnn!, a is the expansion point, and the sequence vn satisfies a homogeneous linear recurrence.

• 

The routine selects such formal power series solutions where vn is a d'Alembertian sequence, that is, vn 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

• 

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:

odex2+x2ⅆ2ⅆx2yx+x2xⅆⅆxyx6x2+7xyx

ode:=x2+x2ⅆ2ⅆx2yx+x2xⅆⅆxyx6x2+7xyx

(1)

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

12_C0n=1∞2nx1nΓn,3158n=7∞2nn6x+2nΓn+14095415754x315x+223152x+231052x+24212x+25_C0+17143n=7∞2nn6n1=7n116n1Γn1+1n15n16x+2nΓn+1409n=7∞2nn6n1=7n116n1Γn1+1n2=7n11n259n2Γn2n15n16x+2nΓn+1+997295486+383575486x+153430243x+22+77435243x+23+73295729x+24+18835729x+252540729x+26_C1,n=9∞1512n82nx+3nn!252567256x3128x+325384x+331192x+341640x+3512880x+36120160x+37_C0+5282992198016n=9∞10242835n82nn1=9n118n1Γn1+1n178+n1x+3nn!+n=9∞219683n82nn1=9n118n1Γn1+1n2=9n11n2712n2I112n27I11+2n2+7n2+1Γn2+1n178+n1x+3nI1125I11+25n!+869574607273468358656+243522833273468358656x+38539995064228864x+32+366367859963392x+33+12699897596343296x+34+11983373227890298880x+35+1166693113945149440x+36+572503265872015360x+3793781398808023040x+38_C1

(2)

See Also

LinearOperators

LODEstruct

Slode

Slode[hypergeom_series_sol]

Slode[polynomial_series_sol]

Slode[rational_series_sol]

SumTools[Hypergeometric]

 


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