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LinearOperators

  

dAlembertianSolver

  

compute d'Alembertian solution of functional linear inhomogeneous equation with d'Alembertian right hand side

 

Calling Sequence

Parameters

Description

Examples

References

Calling Sequence

dAlembertianSolver(L,b,x,case)

Parameters

L

-

Ore operator

b

-

right hand side of the equation which is a d'Alembertian term

x

-

name of the independent variable

case

-

parameter indicating the case of the equation ('differential' or 'shift')

Description

• 

The LinearOperators[dAlembertianSolver] function returns a d'Alembertian solution of the given inhomogeneous linear functional equation with a d'Alembertian right hand side if such a solution exists. Otherwise, it returns FAIL.

• 

An Ore operator is a structure that consists of the keyword OrePoly with a sequence of coefficients starting with the one of degree zero. The coefficients must be rational functions in x. For example, in the differential case with the differential operator D, OrePoly(2/x, x, x+1, 1) represents the operator 2x+xD+x+1D2+D3.

• 

The right hand side b must be a d'Alembertian term. The main property of a d'Alembertian term is that it is annihilated by a linear operator that can be written as a composition of operators of the first degree. The set of d'Alembertian terms has a ring structure. The package recognizes some basic d'Alembertian terms and their ring-operation closure terms. The result of the substitution of a rational term for the independent variable in the d'Alembertian term is also a d'Alembertian term.

• 

The routine returns an error message if the right hand side is not d'Alembertian.

Examples

LOrePoly2,0,0,2x,x2;bx2

L:=OrePoly2,0,0,2x,x2

b:=x2

(1)

LinearOperators[dAlembertianSolver]L,b,x,'differential'

12x2

(2)

LOrePolyx,0,1;b4x3+1lnxxx

L:=OrePolyx,0,1

b:=4x3+1lnxx3/2

(3)

LinearOperators[dAlembertianSolver]L,b,x,'differential'

4xlnx

(4)

LOrePoly1,n,1;bΓn+2+nΓn+1+Γn

L:=OrePoly1,n,1

b:=Γn+2+nΓn+1+Γn

(5)

LinearOperators[dAlembertianSolver]L,b,n,'shift'

Γn

(6)

References

  

Abramov, S. A., and Zima, E. V. "D'Alembertian Solutions of Inhomogeneous Equations (differential, difference, and some other)." In Proceedings ISSAC '96, pp. 232-240. Edited by Y. N. Lakshman. New York: ACM Press, 1996.

See Also

LinearOperators

 


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