In this example we use the package IntroToSymmetry.m to work out the point group of the Blasius equation 

Yxxx+YYxx=0 

discussed in sections 8.10.1 and 14.4.2.

Clear all symbols in the current context.

ClearAll[Evaluate[Context[]<>"*"]]

First read in the package which is located in User Home Folder/Library/Mathematica/Applications/SymmetryAnalysis.

Needs["SymmetryAnalysis`IntroToSymmetry`"]

Enter the input equation as a string. Don't include the == 0 at the end.

blasiusequation=
"D[y[x],x,x,x]+y[x]*D[y[x],x,x]";

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. Be careful to check signs.

rulesarray=
{"D[y[x],x,x,x]->-y[x]*D[y[x],x,x]"};

Enter the list of independent variables.

independentvariables={"x"};

Enter the list of dependent variables.

dependentvariables={"y"};

Enter the list of function names and constant names that need to be preserved when the equation is changed to (x1, y1) variables. In this case there are no such quantities and so we enter a null list.

frozennames={""};

Enter the maximum derivative order.

p=3;

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.

r=3;

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.

xseon=0;

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.

internalrules=1;

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.

Timing[FindDeterminingEquations[
independentvariables,dependentvariables,frozennames,p,r,xseon,blasiusequation,rulesarray,internalrules]]

equationgenericvariables

invarconditiontable

The program expresses the determining equations in terms of z-variables. Here is the correspondence between z-variables and conventional variables.

ztableofrules

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.

Length[zdeterminingequations]

Lets look at the determining equations in a slightly more readable form.

zdeterminingequationscolumn=TraditionalForm[Column[ToExpression[zdeterminingequations],{Left,Below}]];

Notice that the number of determining equations is now down to one. As the order of the Lie-Bäcklund transformation sought increases, the number of determining equations decreases. In this case we are seeking functions that depend on independent derivatives up to third order whereas the equation itself is third order, effectively eliminating any derivatives of third order or higher. Now solve the one  determining equation 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.

Timing[SolveDeterminingEquations[independentvariables,dependentvariables,r,
xseon,zdeterminingequations,order=7]]

Here are the infinitesimals for the independent variables expressed in terms of z-variables.

xsefunctions

and the infinitesimals for the dependent variables.

etafunctions

Express the xse functions in terms of x,y variables.

ToExpression[xsefunctions]/.ztableofrules

Express the eta function in terms of these variables.

ToExpression[etafunctions]/.ztableofrules

Here is a list of the various infinitesimal groups.

infinitesimalgroupsxy=infinitesimalgroups/.{z1->x,z2->y,z3->yx,z4->yxx,z5->yxxx,z6->yxxxx}

Column[infinitesimalgroupsxy]

Note that the third, fourth and fifth groups can be factored in terms of the first two and the Blasius equation.

Column[Factor[infinitesimalgroupsxy]]

Check the infinitesimals in the determining equations.

Simplify[ToExpression[zdeterminingequations]]

In this case, analysis of Lie-Backlund groups leads merely to the two that are equivalent to the point groups of the equation.

rulesarrayexpanded

MaxMemoryUsed[]