construct a LHPDE object
LHPDE( sys, options)
LHPDE( rifTable, options)
LHPDE( str, dep = vars, options)
LHPDE( , dep = vars, options)
a list or set of linear homogeneous PDEs or ODEs or expressions
a table as returned by DEtools[rifsimp]
a string: either "trivial" or "universal"
a list of dependent variables as functions
optional equations controlling details of the first input argument
dep = deps
This option specifies the dependent variables of the DEs system sys, as a list of functions (or a list of names if the functional dependencies are apparent from the system)
indep = indeps
This option specifies the independent variables of the DEs system sys, as a list of names
inRifReducedForm = true or inRifReducedForm = false
This option indicates whether the DEs system sys is in rif-reduced form or not
ranking = rk
This option specifies the ranking that is used on the rif-reduced DEs system sys; as a list (or list of lists) of dependent variable names. The specification of this option is consistent with the ranking used by the DEtools[rifsimp] command. See ranking for more detail. For the first calling sequence, this option requires that the option inRifReducedForm = true be specified.
The command LHPDO(...) is for constructing a LHPDO object. It returns a LHPDO object if successful. A valid LHPDO object has access to various methods which allow it to be manipulated and its contents queried. For more detail, see Overview of the LHPDO object.
A LHPDEs system E consists of independent variables x=x1,x2,…,xn, dependent variables u=u1,u2,..,um and a collection of linear homogeneous PDEs f=f1,f2,..,fs . Here, each fi is an equation that depends on the ui and derivatives of ui of order up to k, such that fi is linear homogeneous in each ui.
In the first calling sequence, the items in the input argument sys must be linear homogeneous with respect to the dependent variables. If sys is a list of expressions then any expression f will be turned into an equation f=0.
In the second calling sequence, the input argument rifTable must be a table as returned by DEtools[rifsimp]. The rifTable R must only contain a single case and include no more than "Solved", "Pivots", and "Case" indices (see DEtools[rifsimp] Algorithm Output for more detail). The equations in R[Solved] will used as the DEs system for constructing the LHPDE object. A LHPDE object that is constructed from a rifTable is automatically marked as a rif-reduced LHPDE object (see IsRifReduced).
The third calling sequence is a special constructor for either a trivial LHPDE object or a universal LHPDE object. A trivial LHPDE has only the zero solution. A universal LHPDE has empty system (i.e. no restriction on solutions). The second input argument dep = vars must be provided. A LHPDE object that is constructed using this calling sequence is automatically marked as a rif-reduced LHPDE object.
The fourth calling sequence is a special constructor for a universal LHPDE object. The second input argument dep = vars must be provided. A LHPDE object that is constructed using this calling sequence is automatically marked as a rif-reduced LHPDE object.
The options (dep, indep, inRifReducedForm, and ranking) are used for fully specifying properties of the DEs system.
This command is part of the LieAlgebrasOfVectorFields package. For more detail, see Overview of the LieAlgebrasOfVectorFields package.
This command can be used in the form LHPDE(...) only after executing the command with(LieAlgebrasOfVectorFields), but can always be used in the form LieAlgebrasOfVectorFields:-LHPDE(...).
detSys ≔ ∂2∂y2⁢ξ⁡x,y=0,∂∂x⁢η⁡x,y+∂∂y⁢ξ⁡x,y=0,∂∂y⁢η⁡x,y=0,∂∂x⁢ξ⁡x,y=0
S ≔ LHPDE⁡detSys
S ≔ LHPDE⁡detSys,dep=ξ,η,indep=x,y
Rif-table as input argument, the LHPDE object is automatically marked as rif-reduced.
R ≔ DEtoolsrifsimp⁡detSys,ξ,η
S1 ≔ LHPDE⁡R,ranking=ξ,η
Constructing a trivial LHPDE and a universal LHPDE.
T ≔ LHPDE⁡trivial,dep=α⁡x,β⁡y
U ≔ LHPDE⁡universal,dep=α⁡x,β⁡y
The LieAlgebrasOfVectorFields[LHPDE] command was introduced in Maple 2020.
For more information on Maple 2020 changes, see Updates in Maple 2020.
LieAlgebrasOfVectorFields (Package overview)
LHPDE (Object overview)
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