compute d'Alembertian solution of functional linear inhomogeneous equation with d'Alembertian right hand side
right hand side of the equation which is a d'Alembertian term
name of the independent variable
parameter indicating the case of the equation ('differential' or 'shift')
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+x⁢D+x+1⁢D2+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.
L := OrePoly(2,0,0,2*x,x^2); b := x^2;
L := OrePoly(-x,0,1); b := (4*x^3+1)*ln(x)/(x*sqrt(x));
L := OrePoly(1,n,1); b := GAMMA(n+2)+n*GAMMA(n+1)+GAMMA(n);
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.
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