(* Content-type: application/mathematica *)

(*** Wolfram Notebook File ***)
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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 a coupled pair of PDEs, the boundary layer stream function \
equation with a buoyancy induced forcing term, \n\n",
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 StyleBox["\[CapitalPsi]",
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 StyleBox["yy-",
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 StyleBox["\[Beta](",
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 StyleBox["ginf/rhoinf)(T-Tinf)-nu*",
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 StyleBox["\[CapitalPsi]",
  FontFamily->"Symbol",
  FontSize->14,
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 StyleBox["yyy=0, \n\nand a convection-diffusion equation for the \
temperature, \n\n(alpha)Tyy-",
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 StyleBox["\[CapitalPsi]",
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 StyleBox["yTx+",
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 StyleBox["\[CapitalPsi]",
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  FontVariations->{"CompatibilityType"->0}],
 StyleBox["xTy=0.",
  FontSize->14,
  FontVariations->{"CompatibilityType"->0}]
}], "Text",
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Cell[TextData[StyleBox["The following command turns off spurious spelling \
error messages.",
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Cell[TextData[{
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Folder/Library/",
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Cell[BoxData[
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Cell[TextData[StyleBox["Enter the list of independent variables.",
 FontWeight->"Plain"]], "Text",
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Cell[BoxData[
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   RowBox[{"{", 
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Cell["Enter the list of dependent variables.", "Text"],

Cell[BoxData[
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Cell[TextData[StyleBox["Enter the input equations as strings. Don't include \
the ==0 at the end. The ambient reference temperature is tinf.",
 FontWeight->"Plain"]], "Text",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData[
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  "equation1", "=", 
   "\"\<D[\[CapitalPsi][x,y],y,y,y]+\n\
(1/nu)*beta*(ginf/rhoinf)*(t[x,y]-tinf)+\n(1/nu)*D[\[CapitalPsi][x,y],x]*D[\
\[CapitalPsi][x,y],y,y]-\n\
(1/nu)*D[\[CapitalPsi][x,y],y]*D[\[CapitalPsi][x,y],x,y]\>\""}], 
  ";"}]], "Input"],

Cell["\<\
equation2=
\"D[t[x,y],y,y]-
(1/alpha)*D[\[CapitalPsi][x,y],y]*D[t[x,y],x]+
(1/alpha)*D[\[CapitalPsi][x,y],x]*D[t[x,y],y]\";\
\>", "Input",
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Cell[BoxData[
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   RowBox[{"{", 
    RowBox[{
    "\"\<D[\[CapitalPsi][x,y],y,y,y]->-\n\
(1/nu)*beta*(ginf/rhoinf)*(t[x,y]-tinf)-\n(1/nu)*D[\[CapitalPsi][x,y],x]*D[\
\[CapitalPsi][x,y],y,y]+\n\
(1/nu)*D[\[CapitalPsi][x,y],y]*D[\[CapitalPsi][x,y],x,y]\>\"", ",", 
     "\[IndentingNewLine]", 
     "\"\<D[t[x,y],y,y]->\n(1/alpha)*D[\[CapitalPsi][x,y],y]*D[t[x,y],x]-\n\
(1/alpha)*D[\[CapitalPsi][x,y],x]*D[t[x,y],y]\>\""}], "}"}]}], ";"}]], "Input"],

Cell[TextData[StyleBox["The input equations are expressed internally in terms \
of generic variables, (x1,x2,x3,..., y1,y2,y3,...) using the function \
StringReplace. When this is done it is important that function names and \
constant names that may appear in the equation are not modified. Enter these \
protected names in string format.",
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Cell[BoxData[
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Cell[TextData[StyleBox["Enter the maximum derivative order of the input \
equation set.",
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Cell[BoxData[
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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",
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Cell[BoxData[
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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[
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Cell[TextData[StyleBox["When searching for Lie-Backlund groups it is \
necessary to differentiate the input equation to produce derivatives of order \
p+r and append these higher order differential consequences 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.",
 
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Cell[TextData[StyleBox["Now work out the determining equations of the Lie \
point group that leaves equation1 invariant. The output is available as a \
table of strings called zdeterminingequations. Notice that rulesarray \
contains both equations and that therefore both equations in the set are \
applied to the invariance condition for each equation.",
 FontWeight->"Plain"]], "Text",
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 InterpretationBox["\<\"The function FindDetermining Equations has begun, the \
memory in use = \\!\\(27120680\\), the time used = \\!\\(1.946431`\\)\"\>",
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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 \
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the generation of the extended derivatives of the infinitesimals.\"\>"], \
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terms of z-variables, the length of zdeterminingequations = \\!\\(73\\), the \
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\\(27921536\\), the time used = \\!\\(2.5723949999999998`\\)\"\>",
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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 \
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 \
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Cell[TextData[StyleBox["Here are the determining equations in terms of \
z-variables. Set them equal to a new table called zdeterminingequations1. \
Remove the semicolon at the end if you want to look at the equations.",
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Cell[TextData[StyleBox["Now work out the determining equations of the point \
group which leaves equation2 invariant.",
 FontWeight->"Plain"]], "Text",
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The invariance condition has been created with all rules applied. The final \
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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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table infinitesimaltable once for each possible combination of y derivative \
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FindDeterminingEquations.Virtually all other steps are quite fast including \
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\\!\\(27144976\\), the time used = \\!\\(2.7426969999999997`\\)\"\>",
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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 \
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Cell[TextData[StyleBox["Here are the determining equations in terms of \
z-variables. Set them equal to a new table called zdeterminingequations2. \
Remove the semicolon at the end if you want to look at the equations.",
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Cell[BoxData[
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Cell[TextData[StyleBox["Now concatenate these two sets of determining \
equations together to form the table zdeterminingequations3 that contains the \
determining equations for the entire equation set.",
 FontWeight->"Plain"]], "Text",
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Cell["\<\
zdeterminingequations3=Join[
zdeterminingequations1,zdeterminingequations2];\
\>", "Input",
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Cell[TextData[StyleBox["How many determining equations are we dealing with?",
 FontWeight->"Plain"]], "Text",
 ImageRegion->{{0, 1}, {0, 1}}],

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required for each multivariate polynomial used for the infinitesimals. This \
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zvariables. The time needed to solve the determining equations increases as \
powertable increases. powertablelength = \\!\\(35\\)\"\>",
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multivariate polynomial used for the infinitesimals. This number is \
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The time needed to solve the determining equations increases as powertable \
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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.\"\
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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 \
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Cell[TextData[StyleBox["Notice that the infinite dimensional translation \
group on the wall normal coordinate still comes out, ie, this symmetry is not \
lost when the temperature equation is coupled to the boundary layer equation. \
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Cell[TextData[StyleBox["Now generate the commutator table of the Lie algebra. \
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