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Calling Sequence
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LHPDE( sys, options)
LHPDE( rifTable, options)
LHPDE( str, dep = vars, options)
LHPDE( [], dep = vars, options)
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Parameters
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sys
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a list or set of linear homogeneous PDEs or ODEs or expressions
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rifTable
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a table as returned by DEtools[rifsimp]
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str
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a string: either "trivial" or "universal"
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vars
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a list of dependent variables as functions
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options
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optional equations controlling details of the first input argument
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Options
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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)
This option specifies the independent variables of the DEs system sys, as a list of names
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inRifReducedForm = true or inRifReducedForm = false
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This option indicates whether the DEs system sys is in rif-reduced form or not
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.
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Description
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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.
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A LHPDEs system consists of independent variables , dependent variables and a collection of linear homogeneous PDEs . Here, each is an equation that depends on the and derivatives of of order up to , such that is linear homogeneous in each .
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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 will be turned into an equation .
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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).
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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.
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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.
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The options (dep, indep, inRifReducedForm, and ranking) are used for fully specifying properties of the DEs system.
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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(...).
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Examples
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Rif-table as input argument, the LHPDE object is automatically marked as rif-reduced.
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Constructing a trivial LHPDE and a universal LHPDE.
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Compatibility
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The LieAlgebrasOfVectorFields[LHPDE] command was introduced in Maple 2020.
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