Transform a PDE into another one missing the dependent variable
ToMissingDependentVariable(PDE, U, v)
the dependent variable, that is an unknown function of one or more independent variables (names)
the name to be used for the new dependent variable entering the returned PDE
ToMissingDependentVariable receives a a partial differential equation (PDE), typically depending explicitly on the dependent variable U - say u⁡x,y,..., where the independent variables are x,y,...=X, and returns another PDE for a a new dependent variable v⁡X,u, that depend on v⁡X,u 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⁡X,u, the second being v⁡X,u itself.
The relevance of this command is in that from the knowledge of the solution of the PDE for v⁡X,u one can write, directly, the solution to the original PDE for u⁡X, as shown below in the Examples section.
Consider the following expression.
Consider this PDE, out of the scope of pdsolve in Maple 2015 and its previous releases
This PDE depends on m⁡x,y 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
The returned PDE is within the scope of pdsolve in all Maple releases
Equate the right-hand-side to a constant and you have the solution of the PDE (1) passed to ToMissingDependentVariable
As seen above, the solution for the original dependent variable (m⁡x,y) appears in implicit form. This solution can frequently be made explicit by just solving for the dependent variable, using solve or isolate
The PDEtools[ToMissingDependentVariable] command was introduced in Maple 2016.
For more information on Maple 2016 changes, see Updates in Maple 2016.
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