In this example we use the package IntroToSymmetry.m to work out the point group of the boundary layer stream function equation, 

ΨyΨxy-ΨxΨyy-UeUex-nu*Ψyyy=0.

This next command turns off spurious spelling error warnings.

Off[General::spell]

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. Notice we are treating Ue as an arbitrary function of x.

inputequation=
"D[Ψ[x,y],y,y,y]+(1/nu)*Ue[x]*D[Ue[x],x]+(1/nu)*D[Ψ[x,y],x]*D[Ψ[x,y],y,y]-(1/nu)*D[Ψ[x,y],y]*D[Ψ[x,y],x,y]";

The function Ψ[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. Note the additional rule that makes Ue independent of y.

rulesarray=
{"D[Ψ[x,y],y,y,y]->-(1/nu)*Ue[x]*D[Ue[x],x]-(1/nu)*D[Ψ[x,y],x]*D[Ψ[x,y],y,y]+(1/nu)*D[Ψ[x,y],y]*D[Ψ[x,y],x,y]"};

Enter the list of independent variables.

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

Enter the list of dependent variables.

dependentvariables={"Ψ"};

Enter the list of function and/or constant names that must be preserved when the equation is expressed in terms of generic variables y1[x1, x2].

frozennames={"Ue","nu"};

Enter the maximum derivative order of the equation set.

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=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. The Mathematica function Timing outputs the time required for the FindDeterminingEquations function to execute.

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

equationgenericvariables

Let's look at the partial derivatives that appear in the invariance condition.

invarconditiontable

Here are the determining equations in terms of z-variables.

zdeterminingequations;

Length[zdeterminingequations]

Here is the correspondence between z-variables and conventional variables.

ztableofrules

Now solve the determining equations 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=4]]

Here are the infinitesimal transformation functions for the independent variables.

xsefunctions

and for the dependent variables.

etafunctions

infinitesimalgroupsxy=infinitesimalgroups/.{z1->x,z2->y,z3->Ψ}

Check the groups; xse's are on the left, eta's are on the right.

Column[infinitesimalgroupsxy]

Check the infinitesimals in the determining equations.

Simplify[ToExpression[zdeterminingequations]]

Note the higher powers of x that appear in the transformation of the wall normal coordinate. This suggests the existence of an arbitrary function of x in the transformation of y. Drop these terms for the moment.

infinitesimalgroupsxy1=Drop[infinitesimalgroupsxy,{2,6}]

Construct the commutator table.

MakeCommutatorTable[independentvariables,dependentvariables,infinitesimalgroupsxy1]

MatrixForm[commutatortable]

*********************************************************

First let's check to see if these infinitesimals satisfy the invariance condition by forming all the terms that appear in the invariance condition. Here we will make use of the function PthInfinitesimal that generates the infinitesimal transformation of the pth derivative.

ToExpression[xsefunctions]/.{z1->x1,z2->x2,z3->y1[x1,x2]}

ToExpression[etafunctions]/.{z1->x1,z2->x2,z3->y1[x1,x2]}

n=Length[independentvariables]

m=Length[dependentvariables]

plocal is order of the desired derivative.

xse1noderiv=a10

eta1x=PthInfinitesimal[
n,m,yindex=1,plocal=1,r,xseon,j1=1]

eta1y=PthInfinitesimal[
n,m,yindex=1,plocal=1,r,xseon,j1=2]

eta1xy=PthInfinitesimal[
n,m,yindex=1,plocal=2,r,xseon,j1=1,j2=2]

eta1yy=PthInfinitesimal[
n,m,yindex=1,plocal=2,r,xseon,j1=2,j2=2]

eta1yyy=PthInfinitesimal[
n,m,yindex=1,plocal=3,r,xseon,j1=2,j2=2,j3=2]

invariancecondcoefficients={
xse1noderiv,0,0,eta1x,eta1y,0,eta1xy,eta1yy,0,0,0,eta1yyy}

invarianceconditionsum=
Expand[
invarconditiontable.invariancecondcoefficients]/.
{D[y1[x1,x2],x2,x2,x2]->
-(1/nu)*Ue[x1]*D[Ue[x1],x1]-
(1/nu)*D[y1[x1,x2],x1]*D[y1[x1,x2],x2,x2]+
(1/nu)*D[y1[x1,x2],x2]*D[y1[x1,x2],x1,x2]}

Simplify[%]

This final expression is equal to zero if and only if Ue[x] satisfies Ue[x]Ue''=(Ue')^2. Thus invariance of the equation holds if Ue=a*Exp[b*x].  Let's show this.

%/.{Ue[x1]->aa*Exp[bb*x1],Ue'[x1]->aa*bb*Exp[bb*x1],Ue''[x1]->aa*bb*bb*Exp[bb*x1],Ue'''[x1]->aa*bb*bb*bb*Exp[bb*x1]}

The implication of this result is that self-similar behavior occurs for an exponential distribution of the free stream velocity over a flat plate.

MaxMemoryUsed[ ]