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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 groups for several second-order ODEs listed in Table 8.1.",
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 FontWeight->"Plain"]], "Text",
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Cell["Clear all symbols in the current context.", "Text",
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Cell[BoxData[
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Cell[TextData[{
 StyleBox["First read in the package which is located in User Home \
Folder/Library/",
  FontWeight->"Plain"],
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  FontWeight->"Plain",
  FontSlant->"Italic"],
 StyleBox["/Applications/SymmetryAnalysis.",
  FontWeight->"Plain"]
}], "Text",
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Cell[BoxData[
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  "]"}]], "Input"],

Cell[TextData[StyleBox["Enter the input equation as a string. Don't include \
the == 0 at the end. The function f[Y,Yx]is arbitrary.",
 FontWeight->"Plain"]], "Text",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData[
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  RowBox[{"inputequation", "=", "\n", "\"\<D[y[x],x,x]-f[x,D[y[x],x]]\>\""}], 
  ";"}]], "Input"],

Cell[TextData[StyleBox["The function y[x] is a solution of the equation and \
this constraint must be applied in the form of a rule to the invariance \
condition. ",
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Cell[BoxData[
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  RowBox[{"rulesarray", "=", "\[IndentingNewLine]", 
   RowBox[{"{", "\"\<D[y[x],x,x]->f[x,D[y[x],x]]\>\"", "}"}]}], 
  ";"}]], "Input"],

Cell[TextData[StyleBox["Enter the list of independent variables.",
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Cell[BoxData[
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   RowBox[{"{", "\"\<x\>\"", "}"}]}], ";"}]], "Input"],

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

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

Cell[TextData[StyleBox["Enter the list of function and constant names that \
need to be preserved when the equation is converted to generic (x1,y1) \
variables.",
 FontWeight->"Plain"]], "Text",
 ImageRegion->{{0, 1}, {0, 1}}],

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

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

Cell[BoxData[
 RowBox[{
  RowBox[{"p", "=", "2"}], ";"}]], "Input"],

Cell[TextData[StyleBox["The maximum derivative order that the infinitesimals \
are assumed to depend on is specified by the input parameter r. 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[{"r", "=", "0"}], ";"}]], "Input"],

Cell[TextData[StyleBox["When searching for Lie-Backlund groups (r=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-Backlund groups it is \
necessary to differentiate the input equation r times producing derivatives \
of order p+r. 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["Now work out the determining equations of the Lie \
point group that leaves the equation invariant. The output is available as a \
table of strings called zdeterminingequations.",
 FontWeight->"Plain"]], "Text",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell[CellGroupData[{

Cell[BoxData[
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     ",", "rulesarray", ",", "internalrules"}], "]"}], "]"}]], "Input"],

Cell[BoxData[
 InterpretationBox["\<\"The function FindDetermining Equations has begun, the \
memory in use = \\!\\(25921168\\), the time used = \\!\\(1.125613`\\)\"\>",
  StringForm[
  "The function FindDetermining Equations has begun, the memory in use = ``, \
the time used = ``", 25921168, 1.125613],
  Editable->False]], "Print",
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  3.471577908869287*^9, 3.483655768437311*^9}],

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->{3.4694033862178793`*^9, 3.471554993408329*^9, 
  3.471577908869287*^9, 3.483655768450588*^9}],

Cell[BoxData[
 InterpretationBox["\<\"The determining equations have been expressed in \
terms of z-variables, the length of zdeterminingequations = \\!\\(10\\), the \
byte count of zdeterminingequations = \\!\\(944\\), the memory in use = \
\\!\\(25939048\\), the time used = \\!\\(1.1652809999999998`\\)\"\>",
  StringForm[
  "The determining equations have been expressed in terms of z-variables, the \
length of zdeterminingequations = ``, the byte count of zdeterminingequations \
= ``, the memory in use = ``, the time used = ``", 10, 944, 25939048, 
   1.1652809999999998`],
  Editable->False]], "Print",
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Cell[BoxData[
 InterpretationBox["\<\"FindDeterminingEquations is done. The memory in use = \
\\!\\(25939792\\), the time used = \\!\\(1.165467`\\)\"\>",
  StringForm[
  "FindDeterminingEquations is done. The memory in use = ``, the time used = \
``", 25939792, 1.165467],
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  3.471577908869287*^9, 3.483655768453186*^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[CellGroupData[{

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Cell[BoxData["\<\"D[y1[x1],x1,x1]-f[x1,D[y1[x1],x1]]\"\>"], "Output",
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Cell[BoxData["ztableofrules"], "Input"],

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Cell[CellGroupData[{

Cell[TextData[StyleBox["Here are the determining equations expressed in terms \
of z-variables. This output may be quite long. The equations in the table can \
be distinguished by the ==0 at the end of each item.",
 FontWeight->"Plain"]], "Text",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData["zdeterminingequations"], "Input"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{"\<\"-xse1[z1, z2] == 0\"\>", 
   ",", "\<\"-3*Derivative[0, 1][xse1][z1, z2] == 0\"\>", 
   ",", "\<\"Derivative[0, 1][xse1][z1, z2] == 0\"\>", 
   ",", "\<\"-Derivative[0, 2][xse1][z1, z2] == 0\"\>", 
   ",", "\<\"-Derivative[1, 0][eta1][z1, z2] == 0\"\>", 
   ",", "\<\"Derivative[0, 1][eta1][z1, z2] - 2*Derivative[1, 0][xse1][z1, \
z2] == 0\"\>", 
   ",", "\<\"-Derivative[0, 1][eta1][z1, z2] + Derivative[1, 0][xse1][z1, z2] \
== 0\"\>", 
   ",", "\<\"Derivative[0, 2][eta1][z1, z2] - 2*Derivative[1, 1][xse1][z1, \
z2] == 0\"\>", ",", "\<\"Derivative[2, 0][eta1][z1, z2] == 0\"\>", 
   ",", "\<\"2*Derivative[1, 1][eta1][z1, z2] - Derivative[2, 0][xse1][z1, \
z2] == 0\"\>"}], "}"}]], "Output",
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}, Open  ]],

Cell["How many determining equations are there?", "Text"],

Cell[CellGroupData[{

Cell["Length[zdeterminingequations]", "Input",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData["10"], "Output",
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Cell["\<\
Lets look at the determining equations in a slightly more readable form.\
\>", "Text",
 FontWeight->"Plain",
 FontVariations->{"CompatibilityType"->0}],

Cell[BoxData[
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      RowBox[{
       SuperscriptBox["eta1", 
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   },
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 CellChangeTimes->{3.469403386597827*^9, 3.471554994005947*^9, 
  3.47157790919153*^9, 3.483655768725643*^9}],

Cell[TextData[StyleBox["Note that a number of the determining equations are \
redundant.",
 FontWeight->"Plain"]], "Text",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell["\<\
Now solve the determining equations in terms of multivariable polynomials of \
a selected order. 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.\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
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 CellChangeTimes->{{3.4694033789407663`*^9, 3.4694033799245*^9}}],

Cell[BoxData[
 InterpretationBox["\<\"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 = \\!\\(15\\)\"\>",
  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 = ``", 15],
  Editable->False]], "Print",
 CellChangeTimes->{3.4694033866863127`*^9, 3.4715549940804377`*^9, 
  3.471577909241065*^9, 3.483655768761003*^9}],

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",
 CellChangeTimes->{3.4694033866863127`*^9, 3.4715549940804377`*^9, 
  3.471577909241065*^9, 3.483655768762347*^9}],

Cell[BoxData[
 InterpretationBox["\<\"The memory in use = \\!\\(26051568\\), The time = \\!\
\\(1.185949`\\)\"\>",
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Cell[BoxData[
 InterpretationBox["\<\"The number of unknown polynomial coefficients = \
\\!\\(30\\)\"\>",
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  3.471577909241065*^9, 3.483655768764065*^9}],

Cell[BoxData[
 InterpretationBox["\<\"The number of equations for the polynomial \
coefficients = \\!\\(85\\)\"\>",
  StringForm[
  "The number of equations for the polynomial coefficients = ``", 85],
  Editable->False]], "Print",
 CellChangeTimes->{3.4694033866863127`*^9, 3.4715549940804377`*^9, 
  3.471577909241065*^9, 3.4836557687648163`*^9}],

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 \
equation(s). This can take a while.\"\>"], "Print",
 CellChangeTimes->{3.4694033866863127`*^9, 3.4715549940804377`*^9, 
  3.471577909241065*^9, 3.483655768765568*^9}],

Cell[BoxData[
 InterpretationBox["\<\"The memory in use = \\!\\(26071808\\), The time = \\!\
\\(1.188582`\\)\"\>",
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 CellChangeTimes->{3.4694033866863127`*^9, 3.4715549940804377`*^9, 
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Cell[BoxData["\<\"Solve has finished.\"\>"], "Print",
 CellChangeTimes->{3.4694033866863127`*^9, 3.4715549940804377`*^9, 
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Cell[BoxData["\<\"The function SolveDeterminingEquations is finished \
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Cell[BoxData[
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