(* Content-type: application/mathematica *)

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(* Beginning of Notebook Content *)
Notebook[{
Cell[TextData[StyleBox["In this example we use the package IntroToSymmetry.m \
to work out the point group of the 3-D incompressible Euler equations,\n\n\
Ut+U*Ux+V*Uy+W*Uz+Px=0,\nVt+U*Vx+V*Vy+W*Vz+Py=0,\nWt+U*Wx+V*Wy+W*Wz+Py=0\n\
Ux+Vy=0. \n\nP is the kinematic pressure; pressure/density.",
 FontSize->14,
 FontWeight->"Plain"]], "Text",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell["\<\
This next command turns off spurious spelling error warnings.\
\>", "Text"],

Cell[BoxData[
 RowBox[{"Off", "[", 
  RowBox[{"General", "::", "spell"}], "]"}]], "Input"],

Cell["Clear all symbols in the current context.", "Text",
 CellChangeTimes->{{3.4715644485587473`*^9, 3.47156448605101*^9}}],

Cell[BoxData[
 RowBox[{"ClearAll", "[", 
  RowBox[{"Evaluate", "[", 
   RowBox[{
    RowBox[{"Context", "[", "]"}], "<>", "\"\<*\>\""}], "]"}], "]"}]], "Input"],

Cell[TextData[{
 StyleBox["First read in the package which is located in User Home \
Folder/Library/",
  FontWeight->"Plain"],
 StyleBox["Mathematica",
  FontWeight->"Plain",
  FontSlant->"Italic"],
 StyleBox["/Applications/SymmetryAnalysis.",
  FontWeight->"Plain"]
}], "Text",
 CellChangeTimes->{{3.472046215957556*^9, 3.472046255573135*^9}, {
  3.472046289795615*^9, 3.472046308115048*^9}, {3.4720466725423717`*^9, 
  3.472046673357225*^9}},
 ImageRegion->{{0, 1}, {0, 1}}],

Cell["Needs[\"SymmetryAnalysis`IntroToSymmetry`\"]", "Input",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell[TextData[StyleBox["Enter the list of independent variables.",
 FontWeight->"Plain"]], "Text",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"independentvariables", "=", 
   RowBox[{"{", 
    RowBox[{
    "\"\<x\>\"", ",", "\"\<y\>\"", ",", "\"\<z\>\"", ",", "\"\<t\>\""}], 
    "}"}]}], ";"}]], "Input"],

Cell[TextData[StyleBox["Enter the list of dependent variables.",
 FontWeight->"Plain"]], "Text",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"dependentvariables", "=", 
   RowBox[{"{", 
    RowBox[{
    "\"\<U\>\"", ",", "\"\<V\>\"", ",", "\"\<W\>\"", ",", "\"\<P\>\""}], 
    "}"}]}], ";"}]], "Input"],

Cell[TextData[StyleBox["Enter the list of function and/or constant names that \
must be preserved when the equations are expressed in generic variables.",
 FontWeight->"Plain"]], "Text",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"frozennames", "=", 
   RowBox[{"{", "\"\<\>\"", "}"}]}], ";"}]], "Input"],

Cell[TextData[StyleBox["Enter the maximum derivative order of the input \
equation(s).",
 FontWeight->"Plain"]], "Text",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"porder", "=", "1"}], ";"}]], "Input"],

Cell[TextData[StyleBox["The maximum derivative order that the infinitesimals \
are assumed to depend on is specified by the input parameter rorder. This \
parameter is only nonzero when the user is looking for Lie contact groups or \
Lie-Backlund groups. For the usual case where one is searching for point \
groups set r=0.",
 FontWeight->"Plain"]], "Text",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"rorder", "=", "0"}], ";"}]], "Input"],

Cell[TextData[StyleBox["When searching for Lie-B\[ADoubleDot]cklund groups \
(rorder=1 or greater) one can, without loss of generality, leave the \
independent variables untransformed. The corresponding infinitesimals (the \
xse's) are set to zero by setting xseon=0. If one is searching for point \
groups then set xseon=1. The choice xseon=1 is also an option when looking \
for Lie-Backlund groups and this can be useful when looking for contact \
symmetries.",
 FontWeight->"Plain"]], "Text",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"xseon", "=", "1"}], ";"}]], "Input"],

Cell[TextData[StyleBox["When searching for Lie-B\[ADoubleDot]cklund groups it \
is necessary to differentiate the input equations with respect to all the \
independent variables producing derivatives of order porder+rorder. These \
higher order differential consequences are appended to the set of rules \
applied to the invariance condition. This process is carried out \
automatically when internalrules=1. For point groups the equation or equation \
system is the only rule or set of rules needed and one sets internalrules=0.",
 
 FontWeight->"Plain"]], "Text",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"internalrules", "=", "0"}], ";"}]], "Input"],

Cell[TextData[StyleBox["Construct the array of rules to be applied to the \
invariance condition.",
 FontWeight->"Plain",
 FontVariations->{"CompatibilityType"->0}]], "Text",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"rulesarray", "=", 
   RowBox[{"{", "\[IndentingNewLine]", 
    RowBox[{
    "\"\<D[U[x,y,z,t],t]\[Rule]-\[IndentingNewLine]U[x,y,z,t]*D[U[x,y,z,t],x]-\
\[IndentingNewLine]V[x,y,z,t]*D[U[x,y,z,t],y]-\[IndentingNewLine]W[x,y,z,t]*D[\
U[x,y,z,t],z]-\[IndentingNewLine]D[P[x,y,z,t],x]\>\"", ",", 
     "\[IndentingNewLine]", 
     "\"\<D[V[x,y,z,t],t]\[Rule]-\[IndentingNewLine]U[x,y,z,t]*D[V[x,y,z,t],x]\
-\[IndentingNewLine]V[x,y,z,t]*D[V[x,y,z,t],y]-\[IndentingNewLine]W[x,y,z,t]*\
D[V[x,y,z,t],z]-\[IndentingNewLine]D[P[x,y,z,t],y]\>\"", ",", 
     "\[IndentingNewLine]", 
     "\"\<D[W[x,y,z,t],t]\[Rule]-\[IndentingNewLine]U[x,y,z,t]*D[W[x,y,z,t],x]\
-\[IndentingNewLine]V[x,y,z,t]*D[W[x,y,z,t],y]-\[IndentingNewLine]W[x,y,z,t]*\
D[W[x,y,z,t],z]-\[IndentingNewLine]D[P[x,y,z,t],z]\>\"", ",", 
     "\[IndentingNewLine]", 
     "\"\<D[U[x,y,z,t],x]\[Rule]-\[IndentingNewLine]D[V[x,y,z,t],y]-\
\[IndentingNewLine]D[W[x,y,z,t],z]\>\""}], "}"}]}], ";"}]], "Input"],

Cell["Enter the x-momentum equation as a string.", "Text"],

Cell["\<\
inputequation1=
\"D[U[x,y,z,t],t]+
        U[x,y,z,t]*D[U[x,y,z,t],x]+
    V[x,y,z,t]*D[U[x,y,z,t],y]+
    W[x,y,z,t]*D[U[x,y,z,t],z]+
    D[P[x,y,z,t],x]\";\
\>", "Input",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell["Enter the y-momentum equation as a string. ", "Text"],

Cell["\<\
inputequation2=
\"D[V[x,y,z,t],t]+
        U[x,y,z,t]*D[V[x,y,z,t],x]+
    V[x,y,z,t]*D[V[x,y,z,t],y]+
    W[x,y,z,t]*D[V[x,y,z,t],z]+
    D[P[x,y,z,t],y]\";\
\>", "Input",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell["Enter the z-momentum equation as a string. ", "Text"],

Cell["\<\
inputequation3=
\"D[W[x,y,z,t],t]+
        U[x,y,z,t]*D[W[x,y,z,t],x]+
    V[x,y,z,t]*D[W[x,y,z,t],y]+
    W[x,y,z,t]*D[W[x,y,z,t],z]+
    D[P[x,y,z,t],z]\";\
\>", "Input",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell["Enter the continuity equation as a string. ", "Text"],

Cell["\<\
inputequation4=
\"D[U[x,y,z,t],x]+
      D[V[x,y,z,t],y]+
    D[W[x,y,z,t],z]\";\
\>", "Input",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell["\<\
Now work out the determining equations of the Lie point group which leaves \
equation1 invariant.\
\>", "Text"],

Cell[CellGroupData[{

Cell["\<\
Timing[FindDeterminingEquations[
independentvariables,dependentvariables,frozennames,porder,rorder,xseon,
inputequation1,rulesarray,internalrules]]\
\>", "Input",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData[
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memory in use = \\!\\(30426280\\), the time used = \\!\\(9.428106`\\)\"\>",
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Cell[BoxData["\<\"The function FindDeterminingEquations is nearly complete. \
The invariance condition has been created with all rules applied. The final \
step in the generation of the determining equations is to sum together terms \
in the table of invariance condition terms (called infinitesimaltable) that \
are multiplied by the same combination of products of free y derivatives. The \
result is the table infinitesimaltablesums corresponding to matching \
y-derivative expressions. If the invariance condition is long as it often is \
this process could take a long time since it requires sorting through the \
table infinitesimaltable once for each possible combination of y derivative \
products. This is the rate limiting step in the function \
FindDeterminingEquations.Virtually all other steps are quite fast including \
the generation of the extended derivatives of the infinitesimals.\"\>"], \
"Print",
 CellChangeTimes->{3.4715708594310913`*^9, 3.483787614961541*^9}],

Cell[BoxData[
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terms of z-variables, the length of zdeterminingequations = \\!\\(33\\), the \
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  StringForm[
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length of zdeterminingequations = ``, the byte count of zdeterminingequations \
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Cell[BoxData[
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 CellChangeTimes->{3.4715708594310913`*^9, 3.4837876150161047`*^9}],

Cell[BoxData["\<\"FindDeterminingEquations has finished executing. You can \
look at the output in the table zdeterminingequations. Each entry in this \
table is a determining equation in string format expressed in terms of \
z-variables. Rules for converting between z-variables and conventional \
variables are contained in the table ztableofrules. To view the determining \
equations in terms of conventional variables use the command \
ToExpression[zdeterminingequations]/.ztableofrules. There are two other items \
the user may wish to look at; the equation converted to generic \
(x1,x2,...,y1,y2,...) variables is designated equationgenericvariables and \
the various derivatives of the equation that appear in the invariance \
condition can be viewed in the table invarconditiontable. Rules for \
converting between z-variables and generic variables are contained in the \
table ztableofrulesxy.\"\>"], "Print",
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Cell[BoxData[
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Cell["The determining equations of equation1.", "Text"],

Cell["zdeterminingequations1=zdeterminingequations;", "Input",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData["33"], "Output",
 CellChangeTimes->{3.471570859988617*^9, 3.483787615087186*^9}]
}, Open  ]],

Cell["\<\
Now work out the determining equations of the Lie point group which leaves \
equation2 invariant.\
\>", "Text"],

Cell[CellGroupData[{

Cell["\<\
Timing[FindDeterminingEquations[
independentvariables,dependentvariables,frozennames,porder,rorder,xseon,
inputequation2,rulesarray,internalrules]]\
\>", "Input",
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Cell[BoxData[
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Cell[BoxData["\<\"The function FindDeterminingEquations is nearly complete. \
The invariance condition has been created with all rules applied. The final \
step in the generation of the determining equations is to sum together terms \
in the table of invariance condition terms (called infinitesimaltable) that \
are multiplied by the same combination of products of free y derivatives. The \
result is the table infinitesimaltablesums corresponding to matching \
y-derivative expressions. If the invariance condition is long as it often is \
this process could take a long time since it requires sorting through the \
table infinitesimaltable once for each possible combination of y derivative \
products. This is the rate limiting step in the function \
FindDeterminingEquations.Virtually all other steps are quite fast including \
the generation of the extended derivatives of the infinitesimals.\"\>"], \
"Print",
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Cell[BoxData[
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terms of z-variables, the length of zdeterminingequations = \\!\\(32\\), the \
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\\!\\(31164984\\), the time used = \\!\\(10.445267999999999`\\)\"\>",
  StringForm[
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length of zdeterminingequations = ``, the byte count of zdeterminingequations \
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Cell[BoxData["\<\"FindDeterminingEquations has finished executing. You can \
look at the output in the table zdeterminingequations. Each entry in this \
table is a determining equation in string format expressed in terms of \
z-variables. Rules for converting between z-variables and conventional \
variables are contained in the table ztableofrules. To view the determining \
equations in terms of conventional variables use the command \
ToExpression[zdeterminingequations]/.ztableofrules. There are two other items \
the user may wish to look at; the equation converted to generic \
(x1,x2,...,y1,y2,...) variables is designated equationgenericvariables and \
the various derivatives of the equation that appear in the invariance \
condition can be viewed in the table invarconditiontable. Rules for \
converting between z-variables and generic variables are contained in the \
table ztableofrulesxy.\"\>"], "Print",
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Cell["The determiningequations of equation2.", "Text"],

Cell["zdeterminingequations2=zdeterminingequations;", "Input",
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Cell[CellGroupData[{

Cell[BoxData[
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Cell["\<\
Now work out the determining equations of the Lie point group which leaves \
equation3 invariant.\
\>", "Text"],

Cell[CellGroupData[{

Cell["\<\
Timing[FindDeterminingEquations[
independentvariables,dependentvariables,frozennames,porder,rorder,xseon,
inputequation3,rulesarray,internalrules]]\
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Cell[BoxData[
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Cell[BoxData["\<\"The function FindDeterminingEquations is nearly complete. \
The invariance condition has been created with all rules applied. The final \
step in the generation of the determining equations is to sum together terms \
in the table of invariance condition terms (called infinitesimaltable) that \
are multiplied by the same combination of products of free y derivatives. The \
result is the table infinitesimaltablesums corresponding to matching \
y-derivative expressions. If the invariance condition is long as it often is \
this process could take a long time since it requires sorting through the \
table infinitesimaltable once for each possible combination of y derivative \
products. This is the rate limiting step in the function \
FindDeterminingEquations.Virtually all other steps are quite fast including \
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Cell[BoxData[
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terms of z-variables, the length of zdeterminingequations = \\!\\(32\\), the \
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Cell[BoxData[
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Cell[BoxData["\<\"FindDeterminingEquations has finished executing. You can \
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table is a determining equation in string format expressed in terms of \
z-variables. Rules for converting between z-variables and conventional \
variables are contained in the table ztableofrules. To view the determining \
equations in terms of conventional variables use the command \
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the user may wish to look at; the equation converted to generic \
(x1,x2,...,y1,y2,...) variables is designated equationgenericvariables and \
the various derivatives of the equation that appear in the invariance \
condition can be viewed in the table invarconditiontable. Rules for \
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Cell["The determiningequations of equation3.", "Text"],

Cell["zdeterminingequations3=zdeterminingequations;", "Input",
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Cell[CellGroupData[{

Cell[BoxData[
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Cell["\<\
Now work out the determining equations of the Lie point group which leaves \
equation3 invariant.\
\>", "Text"],

Cell[CellGroupData[{

Cell["\<\
Timing[FindDeterminingEquations[
independentvariables,dependentvariables,frozennames,porder,rorder,xseon,
inputequation4,rulesarray,internalrules]]\
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Cell[BoxData[
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Cell[BoxData["\<\"The function FindDeterminingEquations is nearly complete. \
The invariance condition has been created with all rules applied. The final \
step in the generation of the determining equations is to sum together terms \
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are multiplied by the same combination of products of free y derivatives. The \
result is the table infinitesimaltablesums corresponding to matching \
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this process could take a long time since it requires sorting through the \
table infinitesimaltable once for each possible combination of y derivative \
products. This is the rate limiting step in the function \
FindDeterminingEquations.Virtually all other steps are quite fast including \
the generation of the extended derivatives of the infinitesimals.\"\>"], \
"Print",
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Cell[BoxData[
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terms of z-variables, the length of zdeterminingequations = \\!\\(36\\), the \
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Cell[BoxData[
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Cell[BoxData["\<\"FindDeterminingEquations has finished executing. You can \
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variables are contained in the table ztableofrules. To view the determining \
equations in terms of conventional variables use the command \
ToExpression[zdeterminingequations]/.ztableofrules. There are two other items \
the user may wish to look at; the equation converted to generic \
(x1,x2,...,y1,y2,...) variables is designated equationgenericvariables and \
the various derivatives of the equation that appear in the invariance \
condition can be viewed in the table invarconditiontable. Rules for \
converting between z-variables and generic variables are contained in the \
table ztableofrulesxy.\"\>"], "Print",
 CellChangeTimes->{3.471570861199567*^9, 3.483787616682423*^9}],

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Cell["The determiningequations of equation3.", "Text"],

Cell["zdeterminingequations4=zdeterminingequations;", "Input",
 ImageRegion->{{0, 1}, {0, 1}}],

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Cell[CellGroupData[{

Cell["\<\
zdeterminingequations5=
Join[zdeterminingequations1,
zdeterminingequations2,zdeterminingequations3,zdeterminingequations4];\
\>", "Input",
 ImageRegion->{{0, 1}, {0, 1}}],

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Cell[BoxData["133"], "Output",
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Cell["\<\
Here is the correspondence between z-variables and conventional variables.\
\>", "Text"],

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Cell["ztableofrules", "Input",
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Cell["\<\
Now solve the determining equations in terms of multivariable polynomials. \
The Mathematica function Solve uses Gaussian elimination to solve a large \
number of linear equations for the polynomial coefficients. The time roughly \
follows

time/timeref=((number of equations)/(number of equationsref))^n

where the exponent is between 2.4 and 2.7. The Mathematica function Timing \
outputs the time required for the SolveDeterminingEquations function to \
execute. Running the notebook for, say, order 1and order 2 can provide a \
reasonably accurate estimate for the time required to run for third and \
higher order.\
\>", "Text"],

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Cell["\<\
Timing[SolveDeterminingEquations[
independentvariables,dependentvariables,rorder,xseon,zdeterminingequations5,\
polyorder=3]]\
\>", "Input",
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Cell[BoxData[
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required for each multivariate polynomial used for the infinitesimals. This \
number is determined by the choice of polynomial order and the number of \
zvariables. The time needed to solve the determining equations increases as \
powertable increases. powertablelength = \\!\\(165\\)\"\>",
  StringForm[
  "The variable powertablelength is the number of terms required for each \
multivariate polynomial used for the infinitesimals. This number is \
determined by the choice of polynomial order and the number of zvariables. \
The time needed to solve the determining equations increases as powertable \
increases. powertablelength = ``", 165],
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Cell[BoxData["\<\"The polynomial expansions have been substituted into the \
determining equations. It is now time to collect the coefficients of various \
powers of zvariables into a table called table of coefficientsall. This step \
uses the function CoefficientList and is a fairly time consuming procedure.\"\
\>"], "Print",
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Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData["\<\"Now it we are ready to use the function Solve to find the \
nonzero polynomial coefficients corresponding to the symmetries of the input \
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Cell[BoxData[
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Cell[BoxData["\<\"You can look at the output in the tables xsefunctions and \
etafunctions. Each entry in these tables is an infinitesimal function in \
string format expressed in terms of z-variables and the group parameters. The \
output can also be viewed with the group parameters stripped away by looking \
at the table infinitesimalgroups. In either case you may wish to convert the \
z-variables to conventional variables using the table ztableofrules. \
\\n\\nKeep in mind that this function only finds solutions of the determining \
equations that are of polynomial form. The determining equations may admit \
solutions that involve transcendental functions and/or integrals. Note that \
arbitrary functions may appear in the infinitesimals and that these can be \
detected by running the package function SolveDeterminingEquations for \
several polynomial orders. If terms of ever increasing order appear, then an \
arbitrary function is indicated.\"\>"], "Print",
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Cell["\<\
Here are the infinitesimal transformation functions for the independent \
variables.\
\>", "Text"],

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Cell["and for the dependent variables.", "Text"],

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Cell["\<\
Check the groups. xse's are on the left, eta's are on the right.\
\>", "Text"],

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Cell["\<\
The equation admits two independent dilation groups. Note the increasing \
powers of time that are evidence of arbitrary time dependent translations of \
the spatial coordinates. In addition there is an arbitrary function of time \
that can be added to the pressure. Lets remove these three arbitrary \
functions of time and then form the commutator table of the remaining groups.\
\
\>", "Text"],

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