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Cell[TextData[StyleBox["In this example we use the package IntroToSymmetry.m \
to work out the point group of the nonlinear heat equation, \n\n\
Ut-(k(U))xUx-k(U)Uxx=0.\n\nwhere k(U)=Exp[U]. Reference: Ibragimov, N. CRC \
Handbook Volume 1, page 110.",
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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["\<\
This next command turns off spurious spelling error warnings.\
\>", "Text"],

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

Cell[TextData[{
 StyleBox["First read in the package which is located in User Home \
Folder/Library/",
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  FontWeight->"Plain",
  FontSlant->"Italic"],
 StyleBox["/Applications/SymmetryAnalysis.",
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Cell[BoxData[
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Cell["Put in the diffusivity function.", "Text"],

Cell[BoxData[
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   RowBox[{"u_", "[", 
    RowBox[{"x_", ",", "t_"}], "]"}], "]"}], ":=", 
  RowBox[{"Exp", "[", 
   RowBox[{"u", "[", 
    RowBox[{"x", ",", "t"}], "]"}], "]"}]}]], "Input"],

Cell[TextData[StyleBox["Enter the input equation as a string. Don't include \
the ==0 at the end.",
 FontWeight->"Plain"]], "Text",
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   ";"}]], "Input"],

Cell[TextData[StyleBox["The function u[x,t] is a solution of the equation and \
this constraint must be applied in the form of a rule to the invariance \
condition. Be careful to check signs.",
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\"", "}"}]}], ";"}]], "Input"],

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

Cell[TextData[StyleBox["Enter the list of function and constant names that \
need to be preserved when the equation is converted to generic y1[x1,x2] \
variables.",
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Cell[TextData[StyleBox["Enter the maximum derivative order of the input \
equation(s).",
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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.",
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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.",
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Cell[TextData[StyleBox["When searching for Lie-Backlund groups it is \
necessary to differentiate the input equation with respect to each of the \
independent variables 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.",
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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",
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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.\"\>"], \
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Cell[BoxData[
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terms of z-variables, the length of zdeterminingequations = \\!\\(9\\), the \
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\\!\\(38473008\\), the time used = \\!\\(15.275725999999999`\\)\"\>",
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length of zdeterminingequations = ``, the byte count of zdeterminingequations \
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Cell[BoxData[
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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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terms of zvariables. Here is the correspondence between z-variables and \
conventional variables.",
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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.",
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E^(2*z3)*Derivative[2, 0, 0][xse2][z1, z2, z3] == 0\"\>", 
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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 \
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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 = \\!\\(20\\)\"\>",
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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 \
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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.\"\
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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 \
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