PDEtools
ToMissingDependentVariable
Transform a PDE into another one missing the dependent variable
Calling Sequence
Parameters
Description
Examples
Compatibility
ToMissingDependentVariable(PDE, U, v)
PDE
-
differential expression
U
the dependent variable, that is an unknown function of one or more independent variables (names)
v
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.
with⁡PDEtools:
Consider this PDE, out of the scope of pdsolve in Maple 2015 and its previous releases
x⁢∂∂x⁢m⁡x,y3+∂∂y⁢m⁡x,y3=m⁡x,y⁢∂∂x⁢m⁡x,y
x⁢∂∂xm⁡x,y3+∂∂ym⁡x,y3=m⁡x,y⁢∂∂xm⁡x,y
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
ToMissingDependentVariable⁡,m⁡x,y,v
x⁢−∂∂xv⁡x,y,m3−∂∂yv⁡x,y,m3∂∂mv⁡x,y,m3=−m⁢∂∂xv⁡x,y,m∂∂mv⁡x,y,m,v⁡x,y,m
The returned PDE is within the scope of pdsolve in all Maple releases
pdsolve⁡,build
v⁡x,y,m=1213⁢∫−9⁢x3⁢_c23+x2⁢3⁢27⁢x2⁢_c26+4⁢_c13x13xⅆx6−_c1⁢1223⁢∫1−9⁢x3⁢_c23+x2⁢3⁢27⁢x2⁢_c26+4⁢_c13x13ⅆx6+c__1+_c2⁢y+c__2−2⁢−m⁢_c1+c__3
Equate the right-hand-side to a constant and you have the solution of the PDE (1) passed to ToMissingDependentVariable
α=subs⁡m=m⁡x,y,rhs⁡
α=1213⁢∫−9⁢x3⁢_c23+x2⁢3⁢27⁢x2⁢_c26+4⁢_c13x13xⅆx6−_c1⁢1223⁢∫1−9⁢x3⁢_c23+x2⁢3⁢27⁢x2⁢_c26+4⁢_c13x13ⅆx6+c__1+_c2⁢y+c__2−2⁢−m⁡x,y⁢_c1+c__3
pdetest⁡,
0
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
isolate⁡,m⁡x,y
m⁡x,y=−−α2+1213⁢∫−9⁢x3⁢_c23+x2⁢3⁢27⁢x2⁢_c26+4⁢_c13x13xⅆx12−_c1⁢1223⁢∫1−9⁢x3⁢_c23+x2⁢3⁢27⁢x2⁢_c26+4⁢_c13x13ⅆx12+c__12+_c2⁢y2+c__22+c__322_c1
The PDEtools[ToMissingDependentVariable] command was introduced in Maple 2016.
For more information on Maple 2016 changes, see Updates in Maple 2016.
See Also
isolate
pdetest
PDEtools,build
pdsolve
rhs
solve
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