return the PDE for the coefficients of the symmetry generator - Maple Help

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DEtools[odepde] - return the PDE for the coefficients of the symmetry generator

Calling Sequence

odepde(ODE, sym)

odepde(ODE, chi)

odepde(ODE, C)

Parameters

ODE

-

ODE (ordinary differential equation)

sym

-

symmetry from which to build a PDE

chi

-

(optional) different scheme to obtain the coefficients of the infinitesimal that generate the PDE

C

-

obtain the ODE generated by applying a differential operator to C

Description

• 

Given a first order ODE

diff(y(x),x) = F(x, y(x));

ⅆⅆxyx=Fx,yx

(1)
  

that is assumed to be invariant under the action of the one-parameter Lie group with infinitesimal generator

f -> xi(x, y)*diff(f,x) + eta(x, y) *diff(f,y);

f→ξx,yxf+ηx,yyf

(2)
  

or a second order ODE

diff(y(x),x,x) = F(x, y(x), diff(y(x),x));

ⅆ2ⅆx2yx=Fx,yx,ⅆⅆxyx

(3)
  

assumed to be invariant under

f -> xi(x, y, _y1)*diff(f,x) + eta(x, y, _y1) *diff(f,y);

f→ξx,y,_y1xf+ηx,y,_y1yf

(4)
  

or an nth order ODE

yn=Fx,y,y' ,y'',...,yn1

  

assumed to be invariant under

f -> xi(x, y, _y1,`...`,_yn)*diff(f,x) + eta(x, y, _y1,`...`,_yn) *diff(f,y);

f→ξx,y,_y1,...,_ynxf+ηx,y,_y1,...,_ynyf

(5)
  

the odepde command builds the PDE that must be satisfied by the infinitesimals xi and eta. This command is context sensitive; it returns the appropriate PDE depending on the differential order of the given ODE.

• 

As a convention, _y1 represents the first derivative of the indeterminate function of the given ODE, _y2 represents the second, and so on.

• 

In general, ξ=G, η=_y1G is always a (trivial) solution to the PDE, where G is an arbitrary function of x,y,_y1,,_yn, and _yn is replaced by the right hand side of the ODE.

  

Thus, any solution ξ=A,η=B, gives rise to a solution with xi=0, η=A_y1+B. This means that we can always set xi=0 without any loss of generality. To obtain the PDE for eta when xi=0, call odepde directly with the extra argument chi.

• 

To any nth order ODE, _yn=Fx,y,_y1,,yn1, there is a corresponding differential operator, A,

A := C -> diff(C,x) + _y1*diff(C,y)+ `...` + `_y(n-1)`*diff(C,`_y(n-2)`) + F*diff(C,`_y(n-1)`);

A:=C→xC+_y1yC+...+_y(n-1)_y(n-2)C+F_y(n-1)C

(6)
  

with the property that, if AC=0, then Cx,y,... is a first integral of the ODE. If the option C is specified, then odepde returns the ODE that is generated by applying A to Cx,y,....

• 

If you are not directly interested in the PDE for xi and eta, but in the PDE for "a restricted form" of these infinitesimals, you can specify this restricted form at first, as an extra argument, and odepde builds the PDE starting from it. This is more efficient than generating the most general PDE and then substituting particular values for xi and eta afterwards, since for ODEs of degree three or higher, the size of the corresponding PDE grows exponentially as the order increases.

• 

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

Examples

withDEtools:

ODE:=ⅆⅆxyx=Φx,yx

ODE:=ⅆⅆxyx=Φx,yx

(7)

PDE_xi_eta:=odepdeODE

PDE_xi_eta:=x_ηx,y+y_ηx,yx_ξx,yΦx,yy_ξx,yΦx,y2_ξx,yxΦx,y_ηx,yyΦx,y

(8)

PDE_chi:=odepdeODE,χ

PDE_chi:=x_χx,y+y_χx,yΦx,y_χx,yyΦx,y

(9)

A concrete example of a second order ODE

ODE:=ⅆ2ⅆx2yx=2ayx+2xa

ODE:=ⅆ2ⅆx2yx=2ayx+2xa

(10)

PDE_chi:=odepdeODE,χ

PDE_chi:=2_y1_y1_χx,y,_y1_χx,y,_y1a2axy2+42_y12_χx,y,_y1a22axy2+22_y12y_y1_χx,y,_y1+22x_y1_χx,y,_y1+y_χx,y,_y1a2axy4_y1_χx,y,_y1a22axy2+2x2_χx,y,_y1+2y2_χx,y,_y1_y12+22yx_χx,y,_y1_y1

(11)

What some of symgen's routines (way=2 and way=5) actually do is to look for particular solutions for PDE_chi and translate the result into an expression for _xi and _eta using xi = F*eta + chi explained above. For example:

symgenODE,way=2

_ξ=1,_η=2a,_ξ=x,_η=y

(12)

The size of the PDE grows exponentially as the order of the ODE increases:

ODE2:=ⅆ2ⅆx2yx=Fx,yx,ⅆⅆxyx

ODE2:=ⅆ2ⅆx2yx=Fx,yx,ⅆⅆxyx

(13)

ODE3:=ⅆ3ⅆx3yx=Fx,yx,ⅆⅆxyx,ⅆ2ⅆx2yx

ODE3:=ⅆ3ⅆx3yx=Fx,yx,ⅆⅆxyx,ⅆ2ⅆx2yx

(14)

ODE4:=ⅆ4ⅆx4yx=Fx,yx,ⅆⅆxyx,ⅆ2ⅆx2yx,ⅆ3ⅆx3yx

ODE4:=ⅆ4ⅆx4yx=Fx,yx,ⅆⅆxyx,ⅆ2ⅆx2yx,ⅆ3ⅆx3yx

(15)

lengthodepdeODE2;lengthodepdeODE3;lengthodepdeODE4

1218

15220

147211

(16)

However, the expressions are not so large when you work with a restricted form of the infinitesimals. For example:

odepdeODE4,_ξ=0,_η=ηx

ηxyFx,y,_y1,_y2,_y3ⅆⅆxηx_y1Fx,y,_y1,_y2,_y3ⅆ2ⅆx2ηx_y2Fx,y,_y1,_y2,_y3ⅆ3ⅆx3ηx_y3Fx,y,_y1,_y2,_y3+ⅆ4ⅆx4ηx

(17)

See Also

buildsym, DEtools, dsolve,Lie, equinv, PDEtools, pdsolve, symgen


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