DEtools - Maple Programming Help

Online Help

All Products    Maple    MapleSim


Home : Support : Online Help : Mathematics : Differential Equations : DEtools : Solving Methods : DEtools/MeijerGsols

DEtools

  

MeijerGsols

  

solutions of a Meijer G type of linear ode

 

Calling Sequence

Parameters

Description

Examples

References

Calling Sequence

MeijerGsols(lode,v)

MeijerGsols(coeff_list,x)

Parameters

lode

-

homogeneous linear differential equation

v

-

dependent variable of the lode

coeff_list

-

list of coefficients of a linear ode

x

-

independent variable of the lode

Description

• 

The MeijerGsols routine returns a basis of the space of solutions of a linear differential equation of Meijer G type.

• 

The classical notation used to represent the MeijerG function relates to the notation used in Maple by

Gpqmnz|a1,...,an,an+1,...,apb1,...,bm,bm+1,...,bq

=MeijerGa1,...,an,an+1,...,ap,b1,...,bm,bm+1,...,bq,z

  

Note: See Prudnikov, Brychkov, and Marichev.

  

The MeijerG function satisfies the following qth-order linear differential equation

1pmnxi=1pDxai+1i=1qDxbiyx=0

  

where 'D'=ddx and p is less than or equal to q.

• 

For example, MeijerG( [[a[1]],[a[p]]], [[b[1]],[b[q]]], x ) satisfies:

PDEtools[declare](y(x), prime=x);

yxwill now be displayed asy

derivatives with respect toxof functions of one variable will now be displayed with '

(1)

DEtools[hyperode]( MeijerG( [[a[1]],[a[p]]], [[b[1]],[b[q]]], x ), y(x) ) = 0;

ap+1a1+ap1x+b1bqy+a1+ap3x2+b1bq+1xy'+x3+x2y''=0

(2)
  

For information about making symbolic experiments with expressions that contain the MeijerG function of different arguments and the differential equation the expression satisfies, see dpolyform.

• 

MeijerGsols accepts two calling sequences. The first argument of the first calling sequence is a linear differential equation in diff or D form, and the second argument is the function in the differential equation.

• 

The first argument in the second calling sequence is the list of coefficients of a linear ode, and the second is the independent variable. This calling sequence can be convenient for programming with the MeijerGsols routine.

• 

This function is part of the DEtools package, and so it can be used in the form MeijerGsols(..) only after executing the command with(DEtools). However, it can always be accessed through the long form of the command by using DEtools[MeijerGsols](..).

Examples

withDEtools:

odeⅆ2ⅆx2yx=42x2ⅆⅆxyxx3x7yx4x44x2

odey''=2x2+4y'x3x7y4x44x2

(3)

MeijerGsolsode

hypergeom74,14,12,1x2,hypergeom54,34,32,1x2x

(4)

These routines for tackling MeijerG type linear ODEs can also be used directly from Maple's dsolve via

dsolveode,MeijerG

y=_C1hypergeom74,14,12,1x2+_C2hypergeom54,34,32,1x2x

(5)

MeijerGsolsx23,x,x2,x

xBesselJ2,x,xBesselY2,x

(6)

AmultxDF14,xDF1,xDF13,DF,xxbmultxDF+23,xDF12,DF,x:

odediffop2deA,yx,DF,x

ode112+13xby+112x76xbxy'+1712x2xbx2y''+x3y'''

(7)

MeijerGsolsode

xhypergeom53b,12b,144b+3b,133b+2b,xbb,x1/4hypergeom14b,1112b,144b3b,11212b1b,xbb,x1/3hypergeom1b,16b,133b2b,11212b+1b,xbb

(8)

References

  

Prudnikov, A. P.; Brychkov, Yu; and Marichev, O. Integrals and Series. Gordon and Breach Science, 1990. Vol. 3: More Special Functions.

See Also

D

DEtools

diff

dpolyform

dsolve

hyperode