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LinearOperators

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

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

Examples

 > $L≔\mathrm{OrePoly}\left(2,0,0,2x,{x}^{2}\right);$$b≔{x}^{2}$
 ${L}{≔}{\mathrm{OrePoly}}{}\left({2}{,}{0}{,}{0}{,}{2}{}{x}{,}{{x}}^{{2}}\right)$
 ${b}{≔}{{x}}^{{2}}$ (1)
 > $\mathrm{LinearOperators}\left[\mathrm{dAlembertianSolver}\right]\left(L,b,x,'\mathrm{differential}'\right)$
 $\frac{{{x}}^{{2}}}{{2}}$ (2)
 > $L≔\mathrm{OrePoly}\left(-x,0,1\right);$$b≔\frac{\left(4{x}^{3}+1\right)\mathrm{ln}\left(x\right)}{x\mathrm{sqrt}\left(x\right)}$
 ${L}{≔}{\mathrm{OrePoly}}{}\left({-}{x}{,}{0}{,}{1}\right)$
 ${b}{≔}\frac{\left({4}{}{{x}}^{{3}}{+}{1}\right){}{\mathrm{ln}}{}\left({x}\right)}{{{x}}^{{3}}{{2}}}}$ (3)
 > $\mathrm{LinearOperators}\left[\mathrm{dAlembertianSolver}\right]\left(L,b,x,'\mathrm{differential}'\right)$
 ${-}{4}{}\sqrt{{x}}{}{\mathrm{ln}}{}\left({x}\right)$ (4)
 > $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)$
 ${L}{≔}{\mathrm{OrePoly}}{}\left({1}{,}{n}{,}{1}\right)$
 ${b}{≔}{\mathrm{\Gamma }}{}\left({n}{+}{2}\right){+}{n}{}{\mathrm{\Gamma }}{}\left({n}{+}{1}\right){+}{\mathrm{\Gamma }}{}\left({n}\right)$ (5)
 > $\mathrm{LinearOperators}\left[\mathrm{dAlembertianSolver}\right]\left(L,b,n,'\mathrm{shift}'\right)$
 ${\mathrm{\Gamma }}{}\left({n}\right)$ (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.