computes the group invariant (point symmetry) solutions for a given PDE system - Maple Help

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PDEtools[SymmetrySolutions] - computes the group invariant (point symmetry) solutions for a given PDE system

Calling Sequence

SymmetrySolutions(Solution, S, 'options'='value')

Parameters

Solution

-

an equation, a set (list) of equations, or a set (list) of sets (lists) of equations, representing the solution to some system of equations (itself not required)

S

-

the infinitesimals of a symmetry generator or the corresponding infinitesimal generator operator, or a list of n of them representing an n-dimensional group of symmetries

simplifier = ...

-

optional - indicates the simplifier to be used instead of the default simplify/size

Description

• 

Given the Solution to a problem, as an equation or a set (list) of equations, or given a set (list) of these solutions, and given the infinitesimal generators of a symmetry S, or a list of them, SymmetrySolutions computes other solutions obtained by exploring the given symmetries. This command complements Infinitesimals and InvariantSolutions, which respetively compute the symmetry infinitesimals and the group (symmetry) invariant solutions of a PDE system.

• 

The dependent variables of the problem do not need to be indicated: SymmetrySolutions will automatically take the left-hand-sides of the given solutions as dependent variables.

• 

What SymmetrySolutions does can be decomposed into two steps that can be reproduced interactively:

1. 

compute the finite form of the transformation associated to all the given symmetries (can be performed using SymmetryTransformation);

2. 

apply this transformation to each of the given solutions (can be performed using dchange).

  

You can optionally specify a simplifier to be applied to the result using the optional argument simplifier = ...

• 

To avoid having to remember the optional keywords, if you type the keyword misspelled, or just a portion of it, a matching against the correct keywords is performed, and when there is only one match, the input is automatically corrected.

Examples

Consider the wave equation in four dimensions to avoid redundant typing on input and on the display use diff_table and declare

withPDEtools,SymmetrySolutions,Infinitesimals:

Consider for instance the wave equation in two dimensions

pde1:=2x2ux,y,t+2y2ux,y,t=0

pde1:=2x2ux,y,t+2y2ux,y,t=0

(1)

A solution for pde[1] obtained separating the variables by product

sol1:=pdsolvepde1,HINT=`*`,build

sol1:=ux,y,t=_F4t_C1ⅇ_c1x_C3sin_c1y+_F4t_C1ⅇ_c1x_C4cos_c1y+_F4t_C2_C3sin_c1yⅇ_c1x+_F4t_C2_C4cos_c1yⅇ_c1x

(2)

Some Infintesimals of polynomial type for pde[1] are

S1:=Infinitesimalspde1,typeofsymmetry=polynomial,displayfunctionality=false2..1

S1:=_ξx=2yx,_ξy=x2y2,_ξt=0,_ηu=0,_ξx=12x212y2,_ξy=yx,_ξt=0,_ηu=0

(3)

New solutions for pde[1] exploring these infinitesimals can be computed using SymmetrySolutions as follows

newsol1:=SymmetrySolutionssol1,S1

newsol1:=ux,y,t=_F4tcos_c1_εx2+y2y1+x2+y2_ε22y_ε_C4sin_c1_εx2+y2y1+x2+y2_ε22y_ε_C3ⅇ2x_c11+x2+y2_ε22y_ε_C1+_C2ⅇx_c11+x2+y2_ε22y_ε,ux,y,t=_F4tⅇ4_c1_εx2+y2+2x4+x2+y2_ε2+4_εx_C1+_C2cos4_c1y4+x2+y2_ε2+4_εx_C4+sin4_c1y4+x2+y2_ε2+4_εx_C3ⅇ2_c1_εx2+y2+2x4+x2+y2_ε2+4_εx

(4)

Note the introduction of the Lie group parameter _ϵ; the new solutions are valid for any value (not depending on x, y or t) of it. These solutions can be verified using pdetest

mappdetest,newsol1,pde1

0,0

(5)

See Also

dchange, Infinitesimals, InvariantSolutions, InvariantTransformation, pdetest, PDEtools, pdsolve, PolynomialSolutions, SimilaritySolutions, SimilarityTransformation, SymmetryTransformation


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