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adjoint

  

return the adjoint of a differential operator

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

adjoint(L, domain)

adjoint(eqn, dvar)

Parameters

L

-

differential operator

domain

-

list containing two names

eqn

-

homogeneous linear differential equation

dvar

-

dependent variable

Description

• 

If the input L is the differential operator a0Dx0+...+anDxn, then the output is 1nmultDx0,a0+...+multDxn,an.

• 

The adjoint has the following properties: adjoint(adjoint(L)) = L and mult(adjoint(L), adjoint(M)) = adjoint(mult(M,L)). So applying the adjoint results in switching the order of multiplication; the adjoint is an anti-automorphism of C(x)[Dx], multiplied by 1n.

• 

The argument domain describes the differential algebra. If this argument is the list Dx,x, then the differential operators are notated with the symbols Dx and x. They are viewed as elements of the differential algebra C(x)[Dx] where C is the field of constants.

• 

If the argument domain is omitted, then the differential specified by the environment variable _Envdiffopdomain is used. If this environment variable is not set then the argument domain may not be omitted.

• 

Instead of a differential operator, the input can also be a linear homogeneous differential equation eqn. In this case the second argument dvar must be the dependent variable.

• 

This function is part of the DEtools package, and so it can be used in the form adjoint(..) only after executing the command with(DEtools). However, it can always be accessed through the long form of the command by using DEtools[adjoint](..).

Examples

withDEtools:

_EnvdiffopdomainDx,x

_EnvdiffopdomainDx,x

(1)

LDx5+2x3Dx3+6x2Dx2+6xDx+x6Dx+1

LDxx6+2Dx3x3+Dx5+6Dx2x2+6Dxx+1

(2)

adjointL

Dx5+2x3Dx3+12x2Dx2+x6Dx+18xDx+6x5+5

(3)

adjoint

Dx5+2x3Dx3+6x2Dx2+x6Dx+6xDx+1

(4)

adjointⅆⅆxyx+yx,yx

yx+ⅆⅆxyx

(5)

See Also

DEtools[mult]

diffop