Ore_to_DESol - Maple Help

Ore_algebra

 Ore_to_diff
 convert a differential operator to a differential equation
 Ore_to_shift
 convert a shift operator to a recurrence equation
 Ore_to_DESol
 convert a differential operator to a DESol structure
 Ore_to_RESol
 convert a shift operator to an RESol structure

 Calling Sequence Ore_to_diff(G, f, A) Ore_to_diff(G, f, A, 'D') Ore_to_shift(G, u, A) Ore_to_shift(G, u, A, 'indexed') Ore_to_DESol(P, f, A) Ore_to_RESol(P, u, A)

Parameters

 G - list of operators of the Ore algebra A P - operator of the Ore algebra A f - expression denoting a mathematical function A - Ore algebra table

Description

 • The Ore_to_diff command converts a differential operator or a list of differential operators of the skew algebra A into a differential equation or a list of differential equations in the function f.  The output is expressed in terms of the diff function by default, or in terms of the D function when the optional parameter is set.
 • The Ore_to_DESol command converts a single differential operator of the skew algebra A into a DESol structure in the function f.
 • The Ore_to_shift command converts a shift operator or a list of shift operators of the skew algebra A into a recurrence equation or a list of recurrence equations in the sequence u.  The output is expressed in functional notation ( $u\left(n\right),...$ ) by default, or in the indexed notation ( ${u}_{n},...$ ) when the optional argument is set.
 • The Ore_to_RESol command converts a single recurrence operator of the skew algebra A into an RESol structure in the sequence u.

Examples

 > $\mathrm{with}\left(\mathrm{Ore_algebra}\right):$

Differential case.

 > $A≔\mathrm{diff_algebra}\left(\left[\mathrm{Dx},x\right],\left[\mathrm{comm},\mathrm{\mu }\right]\right):$
 > $P≔{x}^{2}{\mathrm{Dx}}^{2}+x\mathrm{Dx}+{x}^{2}-{\mathrm{\mu }}^{2}:$
 > $\mathrm{Ore_to_diff}\left(P,f,A\right)$
 ${{x}}^{{2}}{}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right){+}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right){+}\left({-}{{\mathrm{\mu }}}^{{2}}{+}{{x}}^{{2}}\right){}{f}{}\left({x}\right)$ (1)
 > $\mathrm{Ore_to_diff}\left(P,f,A,\mathrm{D}\right)$
 ${{x}}^{{2}}{}{{\mathrm{D}}}^{\left({2}\right)}{}\left({f}\right){}\left({x}\right){+}{x}{}{\mathrm{D}}{}\left({f}\right){}\left({x}\right){+}\left({-}{{\mathrm{\mu }}}^{{2}}{+}{{x}}^{{2}}\right){}{f}{}\left({x}\right)$ (2)
 > $\mathrm{Ore_to_DESol}\left(P,f,A\right)$
 ${\mathrm{DESol}}{}\left(\left\{{{x}}^{{2}}{}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right){+}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right){+}\left({-}{{\mathrm{\mu }}}^{{2}}{+}{{x}}^{{2}}\right){}{f}{}\left({x}\right)\right\}{,}\left\{{f}{}\left({x}\right)\right\}\right)$ (3)
 > $\mathrm{normal}\left(\mathrm{applyopr}\left(P,,A\right)\right)$
 ${0}$ (4)

Euler case.

 > $A≔\mathrm{skew_algebra}\left(\mathrm{euler}=\left[\mathrm{Tx},x\right],\mathrm{comm}=\mathrm{\mu }\right):$
 > $P≔{\mathrm{Tx}}^{2}+{x}^{2}-{\mathrm{\mu }}^{2}:$
 > $\mathrm{Ore_to_diff}\left(P,f,A\right)$
 ${x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right){+}{x}{}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right)\right){+}\left({-}{{\mathrm{\mu }}}^{{2}}{+}{{x}}^{{2}}\right){}{f}{}\left({x}\right)$ (5)

Recurrence case.

 > $A≔\mathrm{shift_algebra}\left(\left[\mathrm{Sn},n\right],\left[\mathrm{comm},\mathrm{\alpha }\right]\right):$
 > $P≔{\mathrm{Sn}}^{2}+\mathrm{\alpha }\mathrm{Sn}+1:$
 > $\mathrm{Ore_to_shift}\left(P,u,A\right)$
 ${u}{}\left({n}{+}{2}\right){+}{\mathrm{\alpha }}{}{u}{}\left({n}{+}{1}\right){+}{u}{}\left({n}\right)$ (6)
 > $\mathrm{Ore_to_shift}\left(P,u,A,\mathrm{indexed}\right)$
 ${\mathrm{\alpha }}{}{{u}}_{{n}{+}{1}}{+}{{u}}_{{n}}{+}{{u}}_{{n}{+}{2}}$ (7)
 > $\mathrm{Ore_to_RESol}\left(P,u,A\right)$
 ${\mathrm{RESol}}{}\left(\left\{{u}{}\left({n}{+}{2}\right){+}{\mathrm{\alpha }}{}{u}{}\left({n}{+}{1}\right){+}{u}{}\left({n}\right){=}{0}\right\}{,}\left\{{u}{}\left({n}\right)\right\}{,}\left\{{u}{}\left({0}\right){=}{u}{}\left({0}\right){,}{u}{}\left({1}\right){=}{u}{}\left({1}\right)\right\}{,}{\mathrm{INFO}}\right)$ (8)

Multivariate differential case.

 > $A≔\mathrm{diff_algebra}\left(\left[\mathrm{Dx},x\right],\left[\mathrm{Dy},y\right],\left[\mathrm{comm},\mathrm{\mu }\right]\right):$
 > $G≔\left[-2\mathrm{Dx}x+\mathrm{Dy}y,-4\left(\mathrm{\mu }-x{y}^{2}\right)\left(\mathrm{\mu }+x{y}^{2}\right)+2\mathrm{Dx}x+{y}^{2}{\mathrm{Dy}}^{2},-2\left(\mathrm{\mu }-x{y}^{2}\right)\left(\mathrm{\mu }+x{y}^{2}\right)+\mathrm{Dy}\mathrm{Dx}yx,-\left(\mathrm{\mu }-x{y}^{2}\right)\left(\mathrm{\mu }+x{y}^{2}\right)+\mathrm{Dx}x+{\mathrm{Dx}}^{2}{x}^{2}\right]:$

These are operators for BesselJ(mu,x*y^2).

 > $\mathrm{Ore_to_diff}\left(G,f,A\right)$
 $\left[{-}{2}{}{x}{}\left(\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}{,}{y}\right)\right){+}{y}{}\left(\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}{,}{y}\right)\right){,}{-}{4}{}\left({-}{x}{}{{y}}^{{2}}{+}{\mathrm{\mu }}\right){}\left({x}{}{{y}}^{{2}}{+}{\mathrm{\mu }}\right){}{f}{}\left({x}{,}{y}\right){+}{{y}}^{{2}}{}\left(\frac{{{\partial }}^{{2}}}{{\partial }{{y}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}{,}{y}\right)\right){+}{2}{}{x}{}\left(\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}{,}{y}\right)\right){,}{-}{2}{}\left({-}{x}{}{{y}}^{{2}}{+}{\mathrm{\mu }}\right){}\left({x}{}{{y}}^{{2}}{+}{\mathrm{\mu }}\right){}{f}{}\left({x}{,}{y}\right){+}{y}{}{x}{}\left(\frac{{{\partial }}^{{2}}}{{\partial }{x}{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}{,}{y}\right)\right){,}{-}\left({-}{x}{}{{y}}^{{2}}{+}{\mathrm{\mu }}\right){}\left({x}{}{{y}}^{{2}}{+}{\mathrm{\mu }}\right){}{f}{}\left({x}{,}{y}\right){+}{{x}}^{{2}}{}\left(\frac{{{\partial }}^{{2}}}{{\partial }{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}{,}{y}\right)\right){+}{x}{}\left(\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}{,}{y}\right)\right)\right]$ (9)
 > $\mathrm{Ore_to_diff}\left(G,f,A,\mathrm{D}\right)$
 $\left[{-}{2}{}{x}{}{{\mathrm{D}}}_{{1}}{}\left({f}\right){}\left({x}{,}{y}\right){+}{y}{}{{\mathrm{D}}}_{{2}}{}\left({f}\right){}\left({x}{,}{y}\right){,}{-}{4}{}\left({-}{x}{}{{y}}^{{2}}{+}{\mathrm{\mu }}\right){}\left({x}{}{{y}}^{{2}}{+}{\mathrm{\mu }}\right){}{f}{}\left({x}{,}{y}\right){+}{{y}}^{{2}}{}{{\mathrm{D}}}_{{2}{,}{2}}{}\left({f}\right){}\left({x}{,}{y}\right){+}{2}{}{x}{}{{\mathrm{D}}}_{{1}}{}\left({f}\right){}\left({x}{,}{y}\right){,}{-}{2}{}\left({-}{x}{}{{y}}^{{2}}{+}{\mathrm{\mu }}\right){}\left({x}{}{{y}}^{{2}}{+}{\mathrm{\mu }}\right){}{f}{}\left({x}{,}{y}\right){+}{y}{}{x}{}{{\mathrm{D}}}_{{1}{,}{2}}{}\left({f}\right){}\left({x}{,}{y}\right){,}{-}\left({-}{x}{}{{y}}^{{2}}{+}{\mathrm{\mu }}\right){}\left({x}{}{{y}}^{{2}}{+}{\mathrm{\mu }}\right){}{f}{}\left({x}{,}{y}\right){+}{{x}}^{{2}}{}{{\mathrm{D}}}_{{1}{,}{1}}{}\left({f}\right){}\left({x}{,}{y}\right){+}{x}{}{{\mathrm{D}}}_{{1}}{}\left({f}\right){}\left({x}{,}{y}\right)\right]$ (10)

No conversion is available to a multivariate DESol.

 > $\mathrm{Ore_to_DESol}\left(G,f,A\right)$