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

fxx+fyy-(c^2/4)f=0.

This equation comes up in connection with a model of a laminar jet diffusion flame with convection. See the 1993 paper by S. Mahalingam entitled "Self-similar diffusion flame including effects of streamwise diffusion", Combustion Science and Technology, 89: 363-373.

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.

inputequation=
"D[f[x,y],x,x]+D[f[x,y],y,y]-(c^2/4)*f[x,y]";

The function f[x, y] 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[f[x,y],y,y]->-D[f[x,y],x,x]+(c^2/4)*f[x,y]"};

Enter the list of independent variables.

independentvariables={"x","y"};

Enter the list of dependent variables.

dependentvariables={"f"};

Enter the list of function and/or constant names that need to be preserved when the equation is converted to generic y1[x1,x2] variables.

frozennames={"c"};

Enter the maximum derivative order of the input equation(s).

p=2;

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=0;

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=1;

When searching for Lie-Backlund groups it is necessary to differentiate the input equation with respect to each independent variable 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=0;

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,inputequation,rulesarray,internalrules]]

equationgenericvariables

invarconditiontable

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

ztableofrules

Here are the determining equations expressed in terms of z-variables.The equations in the table can be distinguished by the == 0 at the end of each item.

Column[zdeterminingequations]

How many determining equations are there?

Length[zdeterminingequations]

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.

Timing[SolveDeterminingEquations[
independentvariables,dependentvariables,r,xseon,zdeterminingequations,order=4]]

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

xsefunctions

and the infinitesimals for the dependent variables.

etafunctions

Here is a list of the various infinitesimal groups.

infinitesimalgroupsxy=infinitesimalgroups/.{z1->x,z2->y,z3->f,z4->fx,z5->fy};

Column[infinitesimalgroupsxy]

Check the solution in the determining equations.

Simplify[ToExpression[zdeterminingequations]]

Note that the infinitesimal polynomials truncate in spite of the linearity of the equation. In other words the order of the solution does not continue to increase as the order of the trial polynomial is increased. The reason for this is that the presence of f[x,y] in the last term of the equation rather than a derivative of f[x,y] makes it impossible for a polynomial of any order to satisfy the equation.  Nevertheless the equation must admit the group {{0,0},{g[x,y]}} where g[x,y] is any solution of the equation. Now generate the commutator table.

MakeCommutatorTable[
independentvariables,dependentvariables,infinitesimalgroupsxy]

Put the commutator table in readable form.

MatrixForm[commutatortable]

MaxMemoryUsed[ ]