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Notebook[{
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\nfor one more choice of k(U). Reference: Ibragimov, \
N. CRC Handbook Volume 1, page 110.",
  FontSize->14,
  FontWeight->"Plain"]], "Text",
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Cell["\<\
This next command turns off spurious spelling error warnings.\
\>", \
"Text"],

Cell[BoxData[
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Cell[TextData[StyleBox["First read in the package which is located in a \
folder called SymmetryAnalysis inside the Mathematica folder.",
  FontWeight->"Plain"]], "Text",
  ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData[
    \(Needs["\<SymmetryAnalysis`IntroToSymmetry`\>"]\)], "Input"],

Cell[TextData[StyleBox["Define the diffusivity function.",
  FontWeight->"Plain"]], "Text",
  ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData[
    \(k[u_[x_, t_]] := u[x, t]^sigma\)], "Input"],

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

Cell[BoxData[
    \(\(inputequation = \n\
"\<D[u[x,t],t]-D[k[u[x,t]],x]*D[u[x,t],x]-k[u[x,t]]*D[u[x,t],x,x]\>";\)\)], \
"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.",
  FontWeight->"Plain"]], "Text",
  ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData[
    \(\(rulesarray = \
\[IndentingNewLine]{"\<D[u[x,t],t]->D[k[u[x,t]],x]*D[u[x,t],x]+k[u[x,t]]*D[u[\
x,t],x,x]\>"};\)\)], "Input"],

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

Cell[BoxData[
    \(\(independentvariables = {"\<x\>", "\<t\>"};\)\)], "Input"],

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

Cell[BoxData[
    \(\(dependentvariables = {"\<u\>"};\)\)], "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.",
  FontWeight->"Plain"]], "Text",
  ImageRegion->{{0, 1}, {0, 1}}],

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

Cell[BoxData[
    \(\(porder = 2;\)\)], "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[
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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 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.",
  FontWeight->"Plain"]], "Text",
  ImageRegion->{{0, 1}, {0, 1}}],

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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.",
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Cell[BoxData[
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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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x1,x2],x1,x1]"\)], "Output"]
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Cell[BoxData[
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Cell[TextData[StyleBox["Remove redundancies to try to simplify the \
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                RowBox[{
                  RowBox[{\(-2\), " ", "sigma", " ", 
                    RowBox[{
                      SuperscriptBox["eta1", 
                        TagBox[\((1, 0, 0)\),
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                  Plus[ -1, sigma]], 
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              Times[ 
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Cell[TextData[StyleBox["Remove the factor z3^sigma that multiplies every \
term.",
  FontWeight->"Plain"]], "Text",
  ImageRegion->{{0, 1}, {0, 1}}],

Cell[CellGroupData[{

Cell[BoxData[
    \(imax = Length[zdeterminingequationsreduced4]\)], "Input"],

Cell[BoxData[
    \(10\)], "Output"]
}, Open  ]],

Cell[BoxData[
    \(\(zdeterminingequationsreduced4strings = 
        Table[ToString[
            InputForm[zdeterminingequationsreduced4[\([i]\)]]], {i, 1, 
            imax}];\)\)], "Input"],

Cell["\<\
zdeterminingequationsnoequalszero=
StringReplace[zdeterminingequationsreduced4strings,
{\"== 0\"->\" \"}];\
\>", "Input",
  ImageRegion->{{0, 1}, {0, 1}}],

Cell["\<\
Now convert each determining equation to an expression and multiply \
through by (x^2*y^3). The built-in function Together cancels terms in the \
numerator and denominator.\
\>", "Text"],

Cell["\<\
zdeterminingequationsnoequalszeroexpression=
Together[(z3^2)*(z3^(-sigma))*ToExpression[
zdeterminingequationsnoequalszero]];\
\>", "Input",
  ImageRegion->{{0, 1}, {0, 1}}],

Cell["The number of determining equations.", "Text"],

Cell[CellGroupData[{

Cell["\<\
imax=Length[
zdeterminingequationsnoequalszeroexpression]\
\>", "Input",
  ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData[
    \(10\)], "Output"]
}, Open  ]],

Cell["\<\
Convert each determining equation back to a string and reinstate \
the == 0.\
\>", "Text"],

Cell[CellGroupData[{

Cell["\<\
zdeterminingequationsmod1=Table[
ToString[InputForm[
zdeterminingequationsnoequalszeroexpression[[i]]]]<>
\"== 0\",
{i,1,imax}]\
\>", "Input",
  ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData[
    \({"sigma*eta1[z1, z2, z3] - sigma^2*eta1[z1, z2, z3] - \
sigma*z3*Derivative[0, 0, 1][eta1][z1, z2, z3] - z3^2*Derivative[0, 0, \
2][eta1][z1, z2, z3] - sigma*z3*Derivative[0, 1, 0][xse2][z1, z2, z3] + \
2*sigma*z3*Derivative[1, 0, 0][xse1][z1, z2, z3]== 0", 
      "-(z3*(sigma*eta1[z1, z2, z3] + z3*Derivative[0, 1, 0][xse2][z1, z2, \
z3] - 2*z3*Derivative[1, 0, 0][xse1][z1, z2, z3]))== 0", 
      "True*z3^(2 - sigma)== 0", "True*z3^(2 - sigma)== 0", 
      "True*z3^(2 - sigma)== 0", "True*z3^(2 - sigma)== 0", 
      "True*z3^(2 - sigma)== 0", "True*z3^(2 - sigma)== 0", 
      "-(z3*(2*sigma*Derivative[1, 0, 0][eta1][z1, z2, z3] + \
2*z3*Derivative[1, 0, 1][eta1][z1, z2, z3] - z3*Derivative[2, 0, 0][xse1][z1, \
z2, z3]))== 0", "True*z3^(2 - sigma)== 0"}\)], "Output"]
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sigma*z3*Derivative[0, 0, 1][eta1][z1, z2, z3] - z3^2*Derivative[0, 0, \
2][eta1][z1, z2, z3] - sigma*z3*Derivative[0, 1, 0][xse2][z1, z2, z3] + \
2*sigma*z3*Derivative[1, 0, 0][xse1][z1, z2, z3]== 0\"\>"},
            {"\<\"-(z3*(sigma*eta1[z1, z2, z3] + z3*Derivative[0, 1, \
0][xse2][z1, z2, z3] - 2*z3*Derivative[1, 0, 0][xse1][z1, z2, z3]))== \
0\"\>"},
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            {"\<\"-(z3*(2*sigma*Derivative[1, 0, 0][eta1][z1, z2, z3] + \
2*z3*Derivative[1, 0, 1][eta1][z1, z2, z3] - z3*Derivative[2, 0, 0][xse1][z1, \
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2][eta1][z1, z2, z3] - sigma*z3*Derivative[0, 1, 0][xse2][z1, z2, z3] + \
2*sigma*z3*Derivative[1, 0, 0][xse1][z1, z2, z3]== 0", 
          "-(z3*(sigma*eta1[z1, z2, z3] + z3*Derivative[0, 1, 0][xse2][z1, \
z2, z3] - 2*z3*Derivative[1, 0, 0][xse1][z1, z2, z3]))== 0", 
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2*z3*Derivative[1, 0, 1][eta1][z1, z2, z3] - z3*Derivative[2, 0, 0][xse1][z1, \
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Cell["\<\
Now let's put together the final set of determining equations with \
redundancies removed.\
\>", "Text"],

Cell[BoxData[
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sigma^2*eta1[z1, z2, z3] - sigma*z3*Derivative[0, 0, 1][eta1][z1, z2, z3] - \
z3^2*Derivative[0, 0, 2][eta1][z1, z2, z3] - sigma*z3*Derivative[0, 1, \
0][xse2][z1, z2, z3] + 2*sigma*z3*Derivative[1, 0, 0][xse1][z1, z2, z3]== \
0\>", "\<-(z3*(sigma*eta1[z1, z2, z3] + z3*Derivative[0, 1, 0][xse2][z1, z2, \
z3] - 2*z3*Derivative[1, 0, 0][xse1][z1, z2, z3]))== 0\>", \
"\<-(z3*(2*sigma*Derivative[1, 0, 0][eta1][z1, z2, z3] + 2*z3*Derivative[1, \
0, 1][eta1][z1, z2, z3] - z3*Derivative[2, 0, 0][xse1][z1, z2, z3]))== 0\>", \
\[IndentingNewLine]"\<D[xse2[z1,z2,z3],z1]== 0\>", \
\[IndentingNewLine]"\<D[xse2[z1,z2,z3],z1,z1]== 0\>", \
\[IndentingNewLine]"\<D[xse2[z1,z2,z3],z1,z2]== 0\>", \
\[IndentingNewLine]"\<D[xse2[z1,z2,z3],z1,z3]== 0\>", \
\[IndentingNewLine]"\<D[xse2[z1,z2,z3],z3]== 0\>", \
\[IndentingNewLine]"\<D[xse2[z1,z2,z3],z3,z3]== 0\>", \
\[IndentingNewLine]"\<D[xse2[z1,z2,z3],z2,z3]== 0\>", \
\[IndentingNewLine]"\<D[xse1[z1,z2,z3],z3]== 0\>", \
\[IndentingNewLine]"\<D[xse1[z1,z2,z3],z2,z3]== 0\>", \
\[IndentingNewLine]"\<D[xse1[z1,z2,z3],z3,z3]== 0\>", \
\[IndentingNewLine]"\<D[xse1[z1,z2,z3],z1,z3]== 0\>", \
\[IndentingNewLine]"\<D[eta1[z1,z2,z3],z2]== 0\>", \
\[IndentingNewLine]"\<D[eta1[z1,z2,z3],z1,z1]== 0\>", \
\[IndentingNewLine]"\<D[eta1[z1,z2,z3],z1,z2]== 0\>", \
\[IndentingNewLine]"\<D[eta1[z1,z2,z3],z2,z3]== 0\>", \
\[IndentingNewLine]"\<D[xse1[z1,z2,z3],z2]== 0\>", \
\[IndentingNewLine]"\<D[xse1[z1,z2,z3],z1,z2]== 0\>", \
\[IndentingNewLine]"\<D[xse1[z1,z2,z3],z2,z3]== 0\>", \
\[IndentingNewLine]"\<D[xse1[z1,z2,z3],z2,z2]== 0\>"};\)\)], "Input"],

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"],

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Cell[BoxData[
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Cell[BoxData[
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coefficients = \\!\\(110\\)\"\>",
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Cell[BoxData[
    \("SolveDeterminingEquations has finished executing. You can look at the \
output in the tables xsefunctions and etafunctions. Each entry in these \
tables is an infinitesimal function in string format expressed in terms of \
z-variables and the group parameters. The output can also be viewed with the \
group parameters stripped away by looking at the table infinitesimalgroups. \
In either case you may wish to convert the z-variables to conventional \
variables using the table ztableofrules.   Keep in mind that this function \
only finds solutions of the determining equations that are of algebraic form. \
The determining equations may admit solutions that involve transcendental \
functions and/or integrals. Note that arbitrary functions may appear in the \
infinitesimals and that these can be detected by running the package function \
SolveDeterminingEquations for several polynomial orders. If terms of ever \
increasing order appear, then an arbitrary function is indicated."\)], "Print"],

Cell[BoxData[
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Cell[BoxData[
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in the table commutatortable. To present the output in the most readable form \
you may want view it as a matrix using MatrixForm[commutatortable]. \
Occasionally the entries in the commutatortable will have terms that cancel. \
To get rid of these terms use the function Simplify before viewing the \
table."\)], "Print"]
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*)




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End of Mathematica Notebook file.
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