The LHPDE object is designed and created to represent a collection of linear homogeneous PDEs (LHPDEs) in both rif-reduced or non-rif-reduced form.

There are collections of methods that are available for a LHPDE object, including (i) methods for exploring properties of LHPDEs system such as solution dimension, (ii) utility methods for manipulating DEs system (e.g. rif-reducing the DEs system, solving DEs,..), and (iii) exploring relationship between solution spaces of two LHPDE objects. Some Maple existing builtins are extended for allowing LHPDE object.

All methods of the LHPDE object become available only once a valid LHPDE object is constructed successfully. To construct a LHPDE object, see LieAlgebrasOfVectorFields[LHPDE].

The LHPDE object is the main Maple object exported by the LieAlgebrasOfVectorFields package. See Overview of the LieAlgebrasOfVectorFields package for more detail.

A LHPDE object is mathematically represented by the minimum of three data attributes: the "DEs system", the "independent variables" and the "dependent variables". These data attributes can be accessed via the GetSystem, GetIndependents and GetDependents methods.

To represent a LHPDEs system that is in rif-reduced form with respect to a given ranking, a LHPDE object has two additional data attributes: a boolean variable "inRifReducedForm" and the "ranking". These two attributes can be accessed via the IsRifReduced and GetRanking methods.

After a LHPDE object S is successfully constructed, each method in S can be accessed by either the short form method(S, arguments) or the long form S:-method(S, arguments).

After a LHPDE object is constructed, the following methods are available:

The following Maple builtins functions are extended so that they work for a LHPDE object: type, expand, has, hastype, indets, normal, simplify, convert. See LHPDE Object Overloaded Builtins for more detail.

$\mathrm{with}\left(\mathrm{LieAlgebrasOfVectorFields}\right)\:$

$\mathrm{Typesetting}:-\mathrm{Settings}\left(\mathrm{userep}=\mathrm{true}\right)\:$

$\mathrm{Typesetting}:-\mathrm{Suppress}\left(\left[\mathrm{\xi }\left(x\,y\right)\,\mathrm{\eta }\left(x\,y\right)\,u\left(x\,y\right)\,v\left(x\,y\right)\right]\right)\:$

$S≔\mathrm{LHPDE}\left(\left[\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,y\,y\right)=0\,\mathrm{diff}\left(\mathrm{\eta }\left(x\,y\right)\,x\right)+\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,y\right)=0\,\mathrm{diff}\left(\mathrm{\eta }\left(x\,y\right)\,y\right)=0\,\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,x\right)=0\right]\right)$

$S≔\left[{\mathrm{\xi }}_{y,y}=0\,{\mathrm{\eta }}_x+{\mathrm{\xi }}_y=0\,{\mathrm{\eta }}_y=0\,{\mathrm{\xi }}_x=0\right],\mathrm{indep}=\left[x\,y\right],\mathrm{dep}=\left[\mathrm{\eta }\,\mathrm{\xi }\right]$

$\mathrm{exports}\left(S\,'\mathrm{static}'\right)$

$\mathrm{type},\mathrm{convert},\mathrm{normal},\mathrm{expand},\mathrm{simplify},\mathrm{indets},\mathrm{has},\mathrm{hastype},\mathrm{GetIndependents},\mathrm{GetDependents},\mathrm{GetSystem},\mathrm{GetRanking},\mathrm{SetIDBasis},\mathrm{GetIDBasis},\mathrm{Copy},\mathrm{Augment},\mathrm{SolutionDimension},\mathrm{IsFiniteType},\mathrm{IsTrivial},\mathrm{ParametricDerivatives},\mathrm{OrderOfInvolution},\mathrm{IsRifReduced},\mathrm{IsTotalDegreeRanking},\mathrm{AreSameSpace},\mathrm{AreSame},\mathrm{RifReduce},\mathrm{ReducedForm},\mathrm{AdjustDependencies},\mathrm{Intersection},\mathrm{VectorSpaceSum},\mathrm{EliminationSystem},\mathrm{IsSubspace},\mathrm{DChange},\mathrm{dchange},\mathrm{LHSolve},\mathrm{initialdata},\mathrm{InitialData},\mathrm{ModulePrint},\mathrm{ModuleCopy},\mathrm{ModuleApply}$

$\mathrm{GetSystem}\left(S\right)$

$\left[{\mathrm{\xi }}_{y,y}=0\,{\mathrm{\eta }}_x+{\mathrm{\xi }}_y=0\,{\mathrm{\eta }}_y=0\,{\mathrm{\xi }}_x=0\right]$

$\mathrm{GetIndependents}\left(S\right)$

$\left[x\,y\right]$

$\mathrm{GetDependents}\left(S\right)$

$\left[\mathrm{\eta }\,\mathrm{\xi }\right]$

$Q≔\mathrm{RifReduce}\left(S\right)$

$Q≔\left[{\mathrm{\xi }}_{y,y}=0\,{\mathrm{\eta }}_x=-{\mathrm{\xi }}_y\,{\mathrm{\xi }}_x=0\,{\mathrm{\eta }}_y=0\right],\mathrm{indep}=\left[x\,y\right],\mathrm{dep}=\left[\mathrm{\eta }\,\mathrm{\xi }\right]$

$\mathrm{SolutionDimension}\left(Q\right)$

$3$

$\mathrm{ParametricDerivatives}\left(Q\right)$

$\left[\mathrm{\xi }\,\mathrm{\eta }\,{\mathrm{\xi }}_y\right]$

$R≔\mathrm{Copy}\left(Q\,\left[u\,v\right]\right)$

$R≔\left[v_{y,y}=0\,u_x=-v_y\,v_x=0\,u_y=0\right],\mathrm{indep}=\left[x\,y\right],\mathrm{dep}=\left[u\,v\right]$

$\mathrm{AreSame}\left(Q\,R\,\mathrm{criteria}=''sameOperator''\right)$

$\mathrm{true}$

$\mathrm{R1}≔\mathrm{AdjustDependencies}\left(R\,\mathrm{dep}=''least''\right)$

$\mathrm{R1}≔\left[\frac{{\ⅆ}^2}{\ⅆy^2}v\left(y\right)=0\,\frac{\ⅆ}{\ⅆx}u\left(x\right)=-\frac{\ⅆ}{\ⅆy}v\left(y\right)\right],\mathrm{indep}=\left[x\,y\right],\mathrm{dep}=\left[u\left(x\right)\,v\left(y\right)\right]$

$\mathrm{LHSolve}\left(\mathrm{R1}\right)$

$\left[u\left(x\right)=-\mathrm{c\_\_1}x+\mathrm{c\_\_3}\,v\left(y\right)=\mathrm{c\_\_1}y+\mathrm{c\_\_2}\right]$

$\mathrm{AreSameSpace}\left(S\,Q\,R\,\mathrm{R1}\right)$

$\mathrm{true}$