 Builtins - Maple Help

overview of overloaded builtins for LHPDE object. Description

 • The functionalities of some Maple builtin commands are extended for use on LHPDE object.
 • The following builtins have been overloaded for this purpose: normal, expand, simplify, indets, has, type, hastype, convert
 • The normal, expand, simplify builtin commands accept a LHPDE object and apply their methods onto its DEs system. Then they return a LHPDE object with new DEs system.
 • Let S be a LHPDE object.
 • (i) The call type(S, t) returns true if t is any of the following types: module, object, anything, and LHPDE. See examples below.
 • (ii) The call type(S, dependent(x)) and type(S, freeof(x)) respectively return true if the DEs system, the independent variables, and the dependent variables of S contain (respectively don't contain) x. See example below.
 • The indets, has, hastype builtin commands accept a LHPDE object and apply their methods onto the DEs system, the independent variables, and the dependent variables of the object.
 • The convert builtin command can convert a LHPDE object S into a LHPDO object.
 • These overloaded builtins are associated with the LHPDE object. For more detail, see Overview of the LHPDE object. Examples

 > $\mathrm{with}\left(\mathrm{LieAlgebrasOfVectorFields}\right):$
 > $\mathrm{Typesetting}:-\mathrm{Settings}\left(\mathrm{userep}=\mathrm{true}\right):$
 > $\mathrm{Typesetting}:-\mathrm{Suppress}\left(u\left(x,y\right)\right):$ normal, expand, simplify

 > $S≔\mathrm{LHPDE}\left(\left[\left(x\left(x-1\right)-{x}^{2}\right)\left(\frac{{\partial }^{2}}{\partial {x}^{2}}u\left(x,y\right)\right)+\left({\mathrm{cos}\left(y\right)}^{2}+{\mathrm{sin}\left(y\right)}^{2}\right)\left(\frac{{\partial }^{2}}{\partial {y}^{2}}u\left(x,y\right)\right)=0\right]\right)$
 ${S}{≔}\left[\left({x}{}\left({x}{-}{1}\right){-}{{x}}^{{2}}\right){}\left({{u}}_{{x}{,}{x}}\right){+}\left({{\mathrm{cos}}{}\left({y}\right)}^{{2}}{+}{{\mathrm{sin}}{}\left({y}\right)}^{{2}}\right){}\left({{u}}_{{y}{,}{y}}\right){=}{0}\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}\right]{,}{\mathrm{dep}}{=}\left[{u}\right]$ (1)
 > $\mathrm{normal}\left(S\right)$
 $\left[{{\mathrm{cos}}{}\left({y}\right)}^{{2}}{}\left({{u}}_{{y}{,}{y}}\right){+}{{\mathrm{sin}}{}\left({y}\right)}^{{2}}{}\left({{u}}_{{y}{,}{y}}\right){-}\left({{u}}_{{x}{,}{x}}\right){}{x}{=}{0}\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}\right]{,}{\mathrm{dep}}{=}\left[{u}\right]$ (2)
 > $\mathrm{expand}\left(S\right)$
 $\left[{{\mathrm{cos}}{}\left({y}\right)}^{{2}}{}\left({{u}}_{{y}{,}{y}}\right){+}{{\mathrm{sin}}{}\left({y}\right)}^{{2}}{}\left({{u}}_{{y}{,}{y}}\right){-}\left({{u}}_{{x}{,}{x}}\right){}{x}{=}{0}\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}\right]{,}{\mathrm{dep}}{=}\left[{u}\right]$ (3)
 > $\mathrm{simplify}\left(S\right)$
 $\left[{-}\left({{u}}_{{x}{,}{x}}\right){}{x}{+}{{u}}_{{y}{,}{y}}{=}{0}\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}\right]{,}{\mathrm{dep}}{=}\left[{u}\right]$ (4) type

 > $\left\{\mathrm{type}\left(S,'\mathrm{LHPDE}'\right),\mathrm{type}\left(S,'\mathrm{object}'\right),\mathrm{type}\left(S,'\mathrm{module}'\right)\right\}$
 $\left\{{\mathrm{true}}\right\}$ (5)

The LHPDE object contains x

 > $\mathrm{type}\left(S,\mathrm{dependent}\left(x\right)\right)$
 ${\mathrm{true}}$ (6)
 > $\mathrm{type}\left(S,\mathrm{freeof}\left(x\right)\right)$
 ${\mathrm{false}}$ (7) indets, has, hastype

 > $\mathrm{indets}\left(S\right)$
 $\left\{{x}{,}{y}{,}{\mathrm{cos}}{}\left({y}\right){,}{{u}}_{{x}{,}{x}}{,}{{u}}_{{y}{,}{y}}{,}{{u}}_{{x}}{,}{{u}}_{{y}}{,}{\mathrm{sin}}{}\left({y}\right){,}{u}\right\}$ (8)
 > $\mathrm{has}\left(S,z\right)$
 ${\mathrm{false}}$ (9)
 > $\mathrm{hastype}\left(S,\mathrm{scalar}\right)$
 ${\mathrm{true}}$ (10)
 > $\mathrm{hastype}\left(S,\mathrm{float}\right)$
 ${\mathrm{false}}$ (11) convert

 > $\mathrm{convert}\left(S,'\mathrm{LHPDO}'\right)$
 ${u}{→}\left[\left({x}{}\left({x}{-}{1}\right){-}{{x}}^{{2}}\right){}\left(\frac{{\partial }}{{\partial }{x}}{}\left(\frac{{\partial }}{{\partial }{x}}{}{u}\right)\right){+}\left({{\mathrm{cos}}{}\left({y}\right)}^{{2}}{+}{{\mathrm{sin}}{}\left({y}\right)}^{{2}}\right){}\left(\frac{{\partial }}{{\partial }{y}}{}\left(\frac{{\partial }}{{\partial }{y}}{}{u}\right)\right)\right]$ (12) Compatibility

 • The LHPDE Object Overloaded Builtins command was introduced in Maple 2020.