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 $\frac{2}{x}+x\mathrm{D}+\left(x+1\right){\mathrm{D}}^2+{\mathrm{D}}^3$.

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≔\mathrm{OrePoly}\left(2\,0\,0\,2x\,x^2\right)\;$$b≔x^2$

$L≔\mathrm{OrePoly}\left(2\,0\,0\,2x\,x^2\right)$

$b≔x^2$

$\mathrm{LinearOperators}\left[\mathrm{dAlembertianSolver}\right]\left(L\,b\,x\,'\mathrm{differential}'\right)$

$\frac{x^2}{2}$

$L≔\mathrm{OrePoly}\left(-x\,0\,1\right)\;$$b≔\frac{\left(4x^3+1\right)\ln \left(x\right)}{x\mathrm{sqrt}\left(x\right)}$

$L≔\mathrm{OrePoly}\left(-x\,0\,1\right)$

$b≔\frac{\left(4x^3+1\right)\ln \left(x\right)}{x^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}$

$\mathrm{LinearOperators}\left[\mathrm{dAlembertianSolver}\right]\left(L\,b\,x\,'\mathrm{differential}'\right)$

$-4\sqrt{x}\ln \left(x\right)$

$L≔\mathrm{OrePoly}\left(1\,n\,1\right)\;$$b≔\mathrm{\Gamma }\left(n+2\right)+n\mathrm{\Gamma }\left(n+1\right)+\mathrm{\Gamma }\left(n\right)$

$L≔\mathrm{OrePoly}\left(1\,n\,1\right)$

$b≔\mathrm{\Gamma }\left(n+2\right)+n\mathrm{\Gamma }\left(n+1\right)+\mathrm{\Gamma }\left(n\right)$

$\mathrm{LinearOperators}\left[\mathrm{dAlembertianSolver}\right]\left(L\,b\,n\,'\mathrm{shift}'\right)$

$\mathrm{\Gamma }\left(n\right)$

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.