ToMissingDependentVariable - Maple Help

PDEtools

 ToMissingDependentVariable
 Transform a PDE into another one missing the dependent variable

 Calling Sequence ToMissingDependentVariable(PDE, U, v)

Parameters

 PDE - differential expression U - the dependent variable, that is an unknown function of one or more independent variables (names) v - the name to be used for the new dependent variable entering the returned PDE

Description

 • ToMissingDependentVariable receives a a partial differential equation (PDE), typically depending explicitly on the dependent variable U - say $u\left(x,y,\mathrm{...}\right)$, where the independent variables are $\left(x,y,\mathrm{...}\right)=X$, and returns another PDE for a a new dependent variable $v\left(X,u\right)$, that depend on $v\left(X,u\right)$ only through its derivatives with respect to $X,u$. The output actually consists of a sequence of two objects, the first being the PDE in $v\left(X,u\right)$, the second being $v\left(X,u\right)$ itself.
 • The relevance of this command is in that from the knowledge of the solution of the PDE for $v\left(X,u\right)$ one can write, directly, the solution to the original PDE for $u\left(X\right)$, as shown below in the Examples section.

Examples

Consider the following expression.

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

Consider this PDE, out of the scope of pdsolve in Maple 2015 and its previous releases

 > $x\left({\mathrm{diff}\left(m\left(x,y\right),x\right)}^{3}+{\mathrm{diff}\left(m\left(x,y\right),y\right)}^{3}\right)=m\left(x,y\right)\mathrm{diff}\left(m\left(x,y\right),x\right)$
 ${x}{}\left({\left(\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{m}{}\left({x}{,}{y}\right)\right)}^{{3}}{+}{\left(\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{m}{}\left({x}{,}{y}\right)\right)}^{{3}}\right){=}{m}{}\left({x}{,}{y}\right){}\left(\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{m}{}\left({x}{,}{y}\right)\right)$ (1)

This PDE depends on $m\left(x,y\right)$ explicitly, not just through its derivatives with respect to $x$ and $y$. In Maple 2016 this PDE is solved by first transforming it into another one missing the dependent variable using ToMissingDependentVariable

 > $\mathrm{ToMissingDependentVariable}\left(,m\left(x,y\right),v\right)$
 ${-}\frac{{x}{}\left({\left(\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{v}{}\left({x}{,}{y}{,}{m}\right)\right)}^{{3}}{+}{\left(\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{v}{}\left({x}{,}{y}{,}{m}\right)\right)}^{{3}}\right)}{{\left(\frac{{\partial }}{{\partial }{m}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{v}{}\left({x}{,}{y}{,}{m}\right)\right)}^{{3}}}{=}{-}\frac{{m}{}\left(\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{v}{}\left({x}{,}{y}{,}{m}\right)\right)}{\frac{{\partial }}{{\partial }{m}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{v}{}\left({x}{,}{y}{,}{m}\right)}{,}{v}{}\left({x}{,}{y}{,}{m}\right)$ (2)

The returned PDE is within the scope of pdsolve in all Maple releases

 > $\mathrm{pdsolve}\left(,\mathrm{build}\right)$
 ${v}{}\left({x}{,}{y}{,}{m}\right){=}\frac{{{12}}^{{1}}{{3}}}{}\left({\int }\frac{{\left({-}{9}{}{{x}}^{{3}}{}{{\mathrm{_c}}}_{{2}}^{{3}}{+}{{x}}^{{2}}{}\sqrt{{3}}{}\sqrt{{27}{}{{x}}^{{2}}{}{{\mathrm{_c}}}_{{2}}^{{6}}{+}\frac{{4}{}{{\mathrm{_c}}}_{{1}}^{{3}}}{{x}}}\right)}^{{1}}{{3}}}}{{x}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\right)}{{6}}{-}\frac{{{\mathrm{_c}}}_{{1}}{}{{12}}^{{2}}{{3}}}{}\left({\int }\frac{{1}}{{\left({-}{9}{}{{x}}^{{3}}{}{{\mathrm{_c}}}_{{2}}^{{3}}{+}{{x}}^{{2}}{}\sqrt{{3}}{}\sqrt{{27}{}{{x}}^{{2}}{}{{\mathrm{_c}}}_{{2}}^{{6}}{+}\frac{{4}{}{{\mathrm{_c}}}_{{1}}^{{3}}}{{x}}}\right)}^{{1}}{{3}}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\right)}{{6}}{+}{\mathrm{_C1}}{+}{{\mathrm{_c}}}_{{2}}{}{y}{+}{\mathrm{_C2}}{-}{2}{}\sqrt{{-}{m}{}{{\mathrm{_c}}}_{{1}}}{+}{\mathrm{_C3}}$ (3)

Equate the right-hand-side to a constant and you have the solution of the PDE (1) passed to ToMissingDependentVariable

 > $\mathrm{\alpha }=\mathrm{subs}\left(m=m\left(x,y\right),\mathrm{rhs}\left(\right)\right)$
 ${\mathrm{\alpha }}{=}\frac{{{12}}^{{1}}{{3}}}{}\left({\int }\frac{{\left({-}{9}{}{{x}}^{{3}}{}{{\mathrm{_c}}}_{{2}}^{{3}}{+}{{x}}^{{2}}{}\sqrt{{3}}{}\sqrt{{27}{}{{x}}^{{2}}{}{{\mathrm{_c}}}_{{2}}^{{6}}{+}\frac{{4}{}{{\mathrm{_c}}}_{{1}}^{{3}}}{{x}}}\right)}^{{1}}{{3}}}}{{x}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\right)}{{6}}{-}\frac{{{\mathrm{_c}}}_{{1}}{}{{12}}^{{2}}{{3}}}{}\left({\int }\frac{{1}}{{\left({-}{9}{}{{x}}^{{3}}{}{{\mathrm{_c}}}_{{2}}^{{3}}{+}{{x}}^{{2}}{}\sqrt{{3}}{}\sqrt{{27}{}{{x}}^{{2}}{}{{\mathrm{_c}}}_{{2}}^{{6}}{+}\frac{{4}{}{{\mathrm{_c}}}_{{1}}^{{3}}}{{x}}}\right)}^{{1}}{{3}}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\right)}{{6}}{+}{\mathrm{_C1}}{+}{{\mathrm{_c}}}_{{2}}{}{y}{+}{\mathrm{_C2}}{-}{2}{}\sqrt{{-}{m}{}\left({x}{,}{y}\right){}{{\mathrm{_c}}}_{{1}}}{+}{\mathrm{_C3}}$ (4)
 > $\mathrm{pdetest}\left(,\right)$
 ${0}$ (5)

As seen above, the solution for the original dependent variable ($m\left(x,y\right)$) appears in implicit form. This solution can frequently be made explicit by just solving for the dependent variable, using solve or isolate

 > $\mathrm{isolate}\left(,m\left(x,y\right)\right)$
 ${m}{}\left({x}{,}{y}\right){=}{-}\frac{{\left({-}\frac{{\mathrm{\alpha }}}{{2}}{+}\frac{{{12}}^{{1}}{{3}}}{}\left({\int }\frac{{\left({-}{9}{}{{x}}^{{3}}{}{{\mathrm{_c}}}_{{2}}^{{3}}{+}{{x}}^{{2}}{}\sqrt{{3}}{}\sqrt{{27}{}{{x}}^{{2}}{}{{\mathrm{_c}}}_{{2}}^{{6}}{+}\frac{{4}{}{{\mathrm{_c}}}_{{1}}^{{3}}}{{x}}}\right)}^{{1}}{{3}}}}{{x}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\right)}{{12}}{-}\frac{{{\mathrm{_c}}}_{{1}}{}{{12}}^{{2}}{{3}}}{}\left({\int }\frac{{1}}{{\left({-}{9}{}{{x}}^{{3}}{}{{\mathrm{_c}}}_{{2}}^{{3}}{+}{{x}}^{{2}}{}\sqrt{{3}}{}\sqrt{{27}{}{{x}}^{{2}}{}{{\mathrm{_c}}}_{{2}}^{{6}}{+}\frac{{4}{}{{\mathrm{_c}}}_{{1}}^{{3}}}{{x}}}\right)}^{{1}}{{3}}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\right)}{{12}}{+}\frac{{\mathrm{_C1}}}{{2}}{+}\frac{{{\mathrm{_c}}}_{{2}}{}{y}}{{2}}{+}\frac{{\mathrm{_C2}}}{{2}}{+}\frac{{\mathrm{_C3}}}{{2}}\right)}^{{2}}}{{{\mathrm{_c}}}_{{1}}}$ (6)
 > $\mathrm{pdetest}\left(,\right)$
 ${0}$ (7)

Compatibility

 • The PDEtools[ToMissingDependentVariable] command was introduced in Maple 2016.