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msparse_series_sol

  

formal m-sparse power series solutions for a linear ODE

 

Calling Sequence

Parameters

Description

Options

Examples

Calling Sequence

msparse_series_sol(ode, var, vn, opts)

msparse_series_sol(LODEstr, vn, opts)

Parameters

ode

-

linear ODE with polynomial coefficients

var

-

dependent variable, for example y(x)

vn

-

new function in the form vn

opts

-

optional arguments of the form keyword=value

LODEstr

-

LODEstruct data structure

Description

• 

The msparse_series_sol command returns a set of m-sparse power series solutions of the given 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.

• 

This routine selects such formal power series solutions where for an integer m2 there is an integer i such that

– 

vn0 only if nimodm=0, and

– 

vn+1m+i=pnvmn+i for all sufficiently large n, where pn is a rational function.

• 

The m-sparse power series is represented by an FPSstruct data-structure (see Slode[FPseries]):

FPSstructv0+v1P1x+...+vMPMx+n=M+1vmn+NPmn+Nx,Rvmn+N;

  

where

– 

v0,...,vM are expressions, the initial series coefficients,

– 

M is a nonnegative integer, and

– 

s is an integer such that M+1ms+N.

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 .

  

If this option is given, then the command returns a set of m-sparse power series solutions at the given point a. Otherwise, it returns a set of m-sparse power series solutions for all possible points that are determined by Slode[candidate_mpoints](ode,var).

• 

'sparseorder'=m0

  

Specifies an integer m0. If this option is given, then the procedure computes a set of m-sparse power series solutions with m=m0 only. Otherwise, it returns a set of m-sparse power series solution for all possible values of m.

  

If both an expansion point and a sparse order are given, then the command can also compute a set of m-sparse series solutions for an inhomogeneous equation with polynomial coefficients and a right-hand side that is rational in the independent variable x. Otherwise, the equation has to be homogeneous.

• 

'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.

Examples

withSlode:

ode2x32x23+1x9ⅆ3ⅆx3yx9+29x24x+13ⅆ2ⅆx2yx9+218x4ⅆⅆxyx9+4yx3

ode:=29x323x2+19xⅆ3ⅆx3yx+299x24x+13ⅆ2ⅆx2yx+2918x4ⅆⅆxyx+43yx

(1)

msparse_series_solode,yx,vn

FPSstruct_C0+n=1∞v2nx162n,36v2n2+v2n,FPSstruct_C1x16+n=1∞v2n+1x162n+1,36v2n1+v2n+1

(2)

Inhomogeneous equations are handled:

ode1z2ⅆ2ⅆz2yz+3zⅆⅆzyz+z2+1n2yz=1

ode1:=z2ⅆ2ⅆz2yz+3zⅆⅆzyz+n2+z2+1yz=1

(3)

msparse_series_solode1,yz,vk,z=∞,'sparseorder'=2

FPSstruct1z2+k=2∞v2kz2k,v2k+4k2n212k+9v2k2

(4)

See Also

LODEstruct

Slode

Slode[candidate_mpoints]

Slode[FPseries]

 


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