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ToMissingDependentVariable

  

Transform a PDE into another one missing the dependent variable

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

ToMissingDependentVariable(PDE, U, v)

Parameters

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

Description

• 

ToMissingDependentVariable receives a a partial differential equation (PDE), typically depending explicitly on the dependent variable U - say ux,y,..., where the independent variables are x,y,...=X, and returns another PDE for a a new dependent variable vX,u, that depend on vX,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 vX,u, the second being vX,u itself.

• 

The relevance of this command is in that from the knowledge of the solution of the PDE for vX,u one can write, directly, the solution to the original PDE for uX, as shown below in the Examples section.

Examples

Consider the following expression.

withPDEtools:

Consider this PDE, out of the scope of pdsolve in Maple 2015 and its previous releases

xxmx,y3+ymx,y3=mx,yxmx,y

xxmx,y3+ymx,y3=mx,yxmx,y

(1)

This PDE depends on mx,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,mx,y,v

xxvx,y,m3+yvx,y,m3mvx,y,m3=mxvx,y,mmvx,y,m,vx,y,m

(2)

The returned PDE is within the scope of pdsolve in all Maple releases

pdsolve,build

vx,y,m=12139x3_c23+x2327x2_c26+4_c13x13xⅆx6_c1122319x3_c23+x2327x2_c26+4_c13x13ⅆx6+_C1+y_c2+_C22m_c1+_C3

(3)

Equate the right-hand-side to a constant and you have the solution of the PDE (1) passed to ToMissingDependentVariable

α=subsm=mx,y,rhs

α=12139x3_c23+x2327x2_c26+4_c13x13xⅆx6_c1122319x3_c23+x2327x2_c26+4_c13x13ⅆx6+_C1+y_c2+_C22mx,y_c1+_C3

(4)

pdetest,

0

(5)

As seen above, the solution for the original dependent variable (mx,y) appears in implicit form. This solution can frequently be made explicit by just solving for the dependent variable, using solve or isolate

isolate,mx,y

mx,y=α2+12139x3_c23+x2327x2_c26+4_c13x13xⅆx12_c1122319x3_c23+x2327x2_c26+4_c13x13ⅆx12+_C12+y_c22+_C22+_C322_c1

(6)

pdetest,

0

(7)

Compatibility

• 

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

PDEtools,build

pdsolve

rhs

solve