checkconsistency = true | false

The default value is false; when true, the consistency of PDESYS is checked before proceeding. Note that you could construct the determining system for the symmetries of an inconsistent PDESYS, in which case the solution of the determining system and everything else derived from it would be meaningless.

closesystem = true | false

By default Infinitesimals processes PDESYS as given. If, in PDESYS, all the integrability conditions that can be derived from its equations are explicitly present, then the determining system for the symmetries is assured to have for solution all the symmetries of PDESYS. Otherwise, the determining system for the symmetries may in rare situations also (correctly) include equations that however may restrict the possible values of the symmetry solutions, and hence some of the symmetries be missed. Correspondingly, some infinitesimals may not be present in the output of Infinitesimals. Note also that, for a system to be closed, it is not sufficient to have it in reduced involutive or equivalent form produced by the casesplit and rifsimp commands or the DifferentialAlgebra package. To assure that all the integrability conditions are explicitly present use the option closesystem. See also the Library routine CloseSystem.

degree = nonnegint | set(function = nonnegint)

This option is related to the option typeofsymmetry = polynomial explained below. By default, an upper bound for the degree of the polynomial dependency of the infinitesimals on the jet variables is determined by PDEtools:-Library:-UpperBounds. To override this behavior pass degree = n, where $n$ is a non-negative integer, so that Infinitesimals will search for infinitesimals all of them of degree $n$. You can also specify the degree of each infinitesimal, say, ${\mathrm{\_\ξ}}_x$, ${\mathrm{\_\ξ}}_t$, .., ${\mathrm{\_\η}}_u$, passing, say, degree = {_xi[x] = n, _xi[t] = m, .. , _eta[u] = p}, where $n$, $m$, $p$ are non-negative integers.

dependency = name | nonnegint | range(nonnegint) | set(name) | list(name), or a set or list of any of the previous types

This option is used to specify the dependency of the infinitesimals, typically to restrict it so that their computation is easier.

dependency = name, generates a search for infinitesimals depending only on that name variable.

dependency = n where $n$ is a non-negative integer generates a search for infinitesimals depending on any of the permutations of $n$ jet variables; this is useful in contexts where you need only infinitesimals depending on just some ($n$) variables.

dependency = n..m generates a search for infinitesimals depending on any permutation of $k$ variables where $n<k<m$, so depending on not less than $n$ and no more than $m$ variables.

Let's say the jet variables are $x$, $t$, and $u$: dependency = {[x], [x, t], [t, u]} generates a search for infinitesimals depending on any of the tree lists of variables specified.

dependency = {x, t, u} generates a search for infinitesimals depending on any but only one of the indicated variables, in this case $x,t,u$.

displayfunctionality = true | false

By default, the functionality with which the infinitesimals were searched is displayed together with infinitesimal's labels in the left-hand sides of the equations of the lists returned by Infinitesimals, regardless of the actual dependency found for them (as shown in the dependency of the right-hand sides). To suppress the display of this functionality on the left-hand sides pass displayfunctionality = false.

HINT = <functional form of the infinitesimals>

The functional form of the infinitesimals can be specified as a list of algebraic expressions. There must be as many expressions as the number $n$ of independent variables plus the number $m$ of dependent variables to be correspondingly associated with ${\mathrm{\_\&xi;}}_1$ to ${\mathrm{\_\&xi;}}_n$ and with ${\mathrm{\_\&eta;}}_1$ to  ${\mathrm{\_\&eta;}}_m$ infinitesimals. Alternatively, this list can contain $n+m$ equations, where the left-hand sides contain any functions of name variables and the right-hand sides contains the actual hint being suggested. In both cases, the algebraic expressions (or those in the right-hand- sides) may or not contain functions to be determined so that the problem has solution. If there are no functions, then either the given HINT exactly solves the determining system for the infinitesimals, and hence is returned as the solution, or it doesn't in which case Infinitesimals will returns no result.

jetnotation = jetvariables | jetvariableswithbrackets | jetODE | jetnumbers

The default value is jetvariables so that the infinitesimals are expressed with the dependent variables and its derivatives in jetvariables jet notation. The value jetODE can be used provided that the input PDESYS consists of a single ODE; the infinitesimals are then expressed using the jetODE notation of DEtools,Lie, the package for ODE symmetries developed before the introduction of symmetry commands in PDEtools.

simplifier = anything

The default value is simplify/size. Use this option to specify the simplifier of your choice. To indicate that no simplification is preferred pass the keyword none on the right-hand side.

specialize_Cn = true | false

By default the returned infinitesimals are split into cases by specializing the integration constants $\mathrm{\_Cn}$, all of them but one equal to 0, the remaining one equal to 1. To avoid this splitting and compute a more compact representation of the infinitesimals, parameterized by the $\mathrm{\_Cn}$ entering the solution of the determining system use split = false

specialize_Fn = true | false | set(procedure)

By default the arbitrary functions $\mathrm{Fn}\left(\mathrm{...}\right)$ entering the returned infinitesimals are not specialized. When specialize_Fn = true, these functions are specialized taking all of them but one equal to 0, the remaining one equal to 1, and also equal to each of its arguments. So if an arbitrary function has, for example, three arguments, then there will be four specializations. Alternatively you can specify a set of procedures to be applied to the arguments of the arbitrary functions, each one resulting in a different desired specialization.

typeofsymmetry = pointlike | evolutionary | contact | general | polynomial | functionfield | And(symmetry_type, functionality_type)

By default Infinitesimals will search for point symmetries. Alternatively you can request to search for symmetries with polynomial or functionfield type of functionality, or of evolutionary, contact or general symmetry types instead of pointlike, where in this context general means infinitesimals depending on derivatives of each unknown of the system up to order $n-1$, where $n$ is the differential order of the unknown as found in the system, and evolutionary is like general but with the $\mathrm{\xi }$ infinitesimals (related to transformations of the independent variables) all equal to zero. The right-hand side can also be of the form And(symmetry_type, functionality_type) where symmetry_type is any of pointlike, evolutionary, contact or general, and functionality_type is one of polynomial or functionfield, the latter related to the FunctionFieldSolutions command. In these cases, You can set the degree of the polynomial or functionfield form using the option degree explained above.

xi_eta = true|false

This option is used to indicate the root names to be used to represent the infinitesimals instead of the default ${\mathrm{\_\&xi;}}_n$ and ${\mathrm{\_\&eta;}}_m$.

Given a PDE problem (PDESYS), as an equation or a set or list of them, the Infinitesimals command computes the infinitesimals of symmetry generators of transformations leaving invariant PDESYS. Infinitesimals also works with anticommutative variables set using the Physics package.

Within the framework of computing the group-invariant (GI) solutions of PDESYS, the first step is to compute the determining PDE system. Next, you attempt to solve this system to obtain the infinitesimals. These are the two steps Infinitesimals does. With these infinitesimals you compute an invariant transformation to reduce the number of independent variables of PDESYS. Finally, you attempt solving the reduced system to change variables back resulting in GI solutions for PDESYS. You can compute any of these steps directly departing from PDESYS, respectively using the commands: DeterminingPDE and pdsolve, or Infinitesimals, InvariantTransformation and InvariantSolutions.

If DepVars is not given, Infinitesimals will consider all the differentiated unknown functions in PDESYS as unknown of the problems. Specifying DepVars however permits not only restricting the unknowns in different ways but also specifying unknowns of the problems which do not appear differentiated in PDESYS.

The infinitesimals returned by Infinitesimals are expressed in jetnotation  = jetvariables, are split into cases by specializing the integration constants $\mathrm{\_Cn}$ in the solution of the determining PDE system, and simplified using the simplifier of DeterminingPDE. You can change these and other defaults, also indicate the functional form of the symmetries, by using the optional arguments explained below.

To avoid having to remember the optional keywords, if you type the keyword misspelled, or just a portion of it, a matching against the correct keywords is performed, and when there is only one match, the input is automatically corrected.

with(PDEtools):

declare(u(x,t));

$u\left(x\&comma;t\right)\mathrm{will\; now\; be\; displayed\; as}u$

U := diff_table(u(x,t)):

PDE := U[x,x] - U[t]=0;

$\mathrm{PDE}≔u_{x,x}-u_t=0$

show;

$\frac{{\partial }^2}{\partial x^2}u\left(x\&comma;t\right)-\frac{\partial }{\partial t}u\left(x\&comma;t\right)=0$

Infinitesimals(PDE);

$\left[{\mathrm{\_\&xi;}}_x\left(x\&comma;t\&comma;u\right)=0\&comma;{\mathrm{\_\&xi;}}_t\left(x\&comma;t\&comma;u\right)=1\&comma;{\mathrm{\_\&eta;}}_u\left(x\&comma;t\&comma;u\right)=0\right],\left[{\mathrm{\_\&xi;}}_x\left(x\&comma;t\&comma;u\right)=1\&comma;{\mathrm{\_\&xi;}}_t\left(x\&comma;t\&comma;u\right)=0\&comma;{\mathrm{\_\&eta;}}_u\left(x\&comma;t\&comma;u\right)=0\right],\left[{\mathrm{\_\&xi;}}_x\left(x\&comma;t\&comma;u\right)=0\&comma;{\mathrm{\_\&xi;}}_t\left(x\&comma;t\&comma;u\right)=0\&comma;{\mathrm{\_\&eta;}}_u\left(x\&comma;t\&comma;u\right)=u\right],\left[{\mathrm{\_\&xi;}}_x\left(x\&comma;t\&comma;u\right)=\frac{x}{2}\&comma;{\mathrm{\_\&xi;}}_t\left(x\&comma;t\&comma;u\right)=t\&comma;{\mathrm{\_\&eta;}}_u\left(x\&comma;t\&comma;u\right)=0\right],\left[{\mathrm{\_\&xi;}}_x\left(x\&comma;t\&comma;u\right)=-2t\&comma;{\mathrm{\_\&xi;}}_t\left(x\&comma;t\&comma;u\right)=0\&comma;{\mathrm{\_\&eta;}}_u\left(x\&comma;t\&comma;u\right)=xu\right],\left[{\mathrm{\_\&xi;}}_x\left(x\&comma;t\&comma;u\right)=0\&comma;{\mathrm{\_\&xi;}}_t\left(x\&comma;t\&comma;u\right)=0\&comma;{\mathrm{\_\&eta;}}_u\left(x\&comma;t\&comma;u\right)={\&ExponentialE;}^{\sqrt{{\mathrm{\_c}}_1}x}{\&ExponentialE;}^{{\mathrm{\_c}}_{1}t}\right],\left[{\mathrm{\_\&xi;}}_x\left(x\&comma;t\&comma;u\right)=0\&comma;{\mathrm{\_\&xi;}}_t\left(x\&comma;t\&comma;u\right)=0\&comma;{\mathrm{\_\&eta;}}_u\left(x\&comma;t\&comma;u\right)=\frac{{\&ExponentialE;}^{{\mathrm{\_c}}_{1}t}}{{\&ExponentialE;}^{\sqrt{{\mathrm{\_c}}_1}x}}\right],\left[{\mathrm{\_\&xi;}}_x\left(x\&comma;t\&comma;u\right)=\frac{tx}{2}\&comma;{\mathrm{\_\&xi;}}_t\left(x\&comma;t\&comma;u\right)=\frac{t^2}{2}\&comma;{\mathrm{\_\&eta;}}_u\left(x\&comma;t\&comma;u\right)=-\frac{\left(x^2+2t\right)u}{8}\right]$

(4)[-1];

$\left[{\mathrm{\_\&xi;}}_x\left(x\&comma;t\&comma;u\right)=\frac{tx}{2}\&comma;{\mathrm{\_\&xi;}}_t\left(x\&comma;t\&comma;u\right)=\frac{t^2}{2}\&comma;{\mathrm{\_\&eta;}}_u\left(x\&comma;t\&comma;u\right)=-\frac{\left(x^2+2t\right)u}{8}\right]$

InfinitesimalGenerator((4)[-1], u(x,t));

$f\&rarr;\frac{1}{2}tx\left(\frac{\partial }{\partial x}f\right)\&plus;\frac{1}{2}{t}^2\left(\frac{\partial }{\partial t}f\right)-\frac{1}{8}\left(x^2\&plus;2t\right)u\left(\frac{\partial }{\partial u}f\right)$

DetSys := DeterminingPDE(PDE);

$\mathrm{DetSys}≔\left\{{\left({\mathrm{\_\&xi;}}_t\right)}_{t,t,t}=0\&comma;{\left({\mathrm{\_\&eta;}}_u\right)}_{t,u}=-\frac{{\left({\mathrm{\_\&xi;}}_t\right)}_{t,t}}{4}\&comma;{\left({\mathrm{\_\&eta;}}_u\right)}_{u,u}=0\&comma;{\left({\mathrm{\_\&eta;}}_u\right)}_{x,x}={\left({\mathrm{\_\&eta;}}_u\right)}_t\&comma;{\left({\mathrm{\_\&xi;}}_t\right)}_u=0\&comma;{\left({\mathrm{\_\&xi;}}_t\right)}_x=0\&comma;{\left({\mathrm{\_\&xi;}}_x\right)}_t=-2{\left({\mathrm{\_\&eta;}}_u\right)}_{u,x}\&comma;{\left({\mathrm{\_\&xi;}}_x\right)}_u=0\&comma;{\left({\mathrm{\_\&xi;}}_x\right)}_x=\frac{{\left({\mathrm{\_\&xi;}}_t\right)}_t}{2}\right\}$

Infinitesimals(PDE, specialize_Cn = false);

$\left[{\mathrm{\_\&xi;}}_x\left(x\&comma;t\&comma;u\right)=\frac{\left(x\mathrm{c\_\_1}-4\mathrm{c\_\_4}\right)t}{2}+\frac{\mathrm{c\_\_2}x}{2}+\mathrm{c\_\_9}\&comma;{\mathrm{\_\&xi;}}_t\left(x\&comma;t\&comma;u\right)=\frac{1}{2}\mathrm{c\_\_1}{t}^2+\mathrm{c\_\_2}t+\mathrm{c\_\_3}\&comma;{\mathrm{\_\&eta;}}_u\left(x\&comma;t\&comma;u\right)=\left(\frac{\left(-x^2-2t\right)\mathrm{c\_\_1}}{8}+\mathrm{c\_\_4}x+\mathrm{c\_\_5}\right)u+\mathrm{c\_\_8}{\&ExponentialE;}^{{\mathrm{\_c}}_{1}t}\mathrm{c\_\_6}{\&ExponentialE;}^{\sqrt{{\mathrm{\_c}}_1}x}+\frac{\mathrm{c\_\_8}{\&ExponentialE;}^{{\mathrm{\_c}}_{1}t}\mathrm{c\_\_7}}{{\&ExponentialE;}^{\sqrt{{\mathrm{\_c}}_1}x}}\right]$

Infinitesimals(PDE, specialize_Cn = false, jetnotation = jetnumbers);

$\left[{\mathrm{\_\&xi;}}_1\left(x\&comma;t\&comma;u\left[\right]\right)=\frac{\left(x\mathrm{c\_\_1}-4\mathrm{c\_\_4}\right)t}{2}+\frac{\mathrm{c\_\_2}x}{2}+\mathrm{c\_\_9}\&comma;{\mathrm{\_\&xi;}}_2\left(x\&comma;t\&comma;u\left[\right]\right)=\frac{1}{2}\mathrm{c\_\_1}{t}^2+\mathrm{c\_\_2}t+\mathrm{c\_\_3}\&comma;{\mathrm{\_\&eta;}}_1\left(x\&comma;t\&comma;u\left[\right]\right)=\left(\frac{\left(-x^2-2t\right)\mathrm{c\_\_1}}{8}+\mathrm{c\_\_4}x+\mathrm{c\_\_5}\right)u\left[\right]+\mathrm{c\_\_8}{\&ExponentialE;}^{{\mathrm{\_c}}_{1}t}\mathrm{c\_\_6}{\&ExponentialE;}^{\sqrt{{\mathrm{\_c}}_1}x}+\frac{\mathrm{c\_\_8}{\&ExponentialE;}^{{\mathrm{\_c}}_{1}t}\mathrm{c\_\_7}}{{\&ExponentialE;}^{\sqrt{{\mathrm{\_c}}_1}x}}\right]$

SymmetryTest((9), PDE);

$\left\{0\right\}$

Infinitesimals(PDE, degree = 0);

$\left[{\mathrm{\_\&xi;}}_x\left(x\&comma;t\&comma;u\right)=0\&comma;{\mathrm{\_\&xi;}}_t\left(x\&comma;t\&comma;u\right)=0\&comma;{\mathrm{\_\&eta;}}_u\left(x\&comma;t\&comma;u\right)=1\right],\left[{\mathrm{\_\&xi;}}_x\left(x\&comma;t\&comma;u\right)=1\&comma;{\mathrm{\_\&xi;}}_t\left(x\&comma;t\&comma;u\right)=0\&comma;{\mathrm{\_\&eta;}}_u\left(x\&comma;t\&comma;u\right)=0\right],\left[{\mathrm{\_\&xi;}}_x\left(x\&comma;t\&comma;u\right)=0\&comma;{\mathrm{\_\&xi;}}_t\left(x\&comma;t\&comma;u\right)=1\&comma;{\mathrm{\_\&eta;}}_u\left(x\&comma;t\&comma;u\right)=0\right]$

Infinitesimals(PDE, degree = {_xi[x] = 1, _eta[u] = 2}, displayfunctionality = false);

$\left[{\mathrm{\_\&xi;}}_x=0\&comma;{\mathrm{\_\&xi;}}_t=0\&comma;{\mathrm{\_\&eta;}}_u=1\right],\left[{\mathrm{\_\&xi;}}_x=0\&comma;{\mathrm{\_\&xi;}}_t=0\&comma;{\mathrm{\_\&eta;}}_u=x\right],\left[{\mathrm{\_\&xi;}}_x=1\&comma;{\mathrm{\_\&xi;}}_t=0\&comma;{\mathrm{\_\&eta;}}_u=0\right],\left[{\mathrm{\_\&xi;}}_x=0\&comma;{\mathrm{\_\&xi;}}_t=1\&comma;{\mathrm{\_\&eta;}}_u=0\right],\left[{\mathrm{\_\&xi;}}_x=0\&comma;{\mathrm{\_\&xi;}}_t=0\&comma;{\mathrm{\_\&eta;}}_u=u\right],\left[{\mathrm{\_\&xi;}}_x=x\&comma;{\mathrm{\_\&xi;}}_t=2t\&comma;{\mathrm{\_\&eta;}}_u=0\right],\left[{\mathrm{\_\&xi;}}_x=0\&comma;{\mathrm{\_\&xi;}}_t=0\&comma;{\mathrm{\_\&eta;}}_u=x^2+2t\right],\left[{\mathrm{\_\&xi;}}_x=-2t\&comma;{\mathrm{\_\&xi;}}_t=0\&comma;{\mathrm{\_\&eta;}}_u=xu\right]$

Infinitesimals(PDE, dependency = {x, t, u});

$\left[{\mathrm{\_\&xi;}}_x\left(t\right)=1\&comma;{\mathrm{\_\&xi;}}_t\left(t\right)=0\&comma;{\mathrm{\_\&eta;}}_u\left(t\right)=0\right],\left[{\mathrm{\_\&xi;}}_x\left(t\right)=0\&comma;{\mathrm{\_\&xi;}}_t\left(t\right)=1\&comma;{\mathrm{\_\&eta;}}_u\left(t\right)=0\right],\left[{\mathrm{\_\&xi;}}_x\left(t\right)=0\&comma;{\mathrm{\_\&xi;}}_t\left(t\right)=0\&comma;{\mathrm{\_\&eta;}}_u\left(t\right)=1\right],\left[{\mathrm{\_\&xi;}}_x\left(u\right)=0\&comma;{\mathrm{\_\&xi;}}_t\left(u\right)=0\&comma;{\mathrm{\_\&eta;}}_u\left(u\right)=u\right],\left[{\mathrm{\_\&xi;}}_x\left(x\right)=0\&comma;{\mathrm{\_\&xi;}}_t\left(x\right)=0\&comma;{\mathrm{\_\&eta;}}_u\left(x\right)=x\right]$

Infinitesimals(PDE, dependency = {[x,t] ,u});

$\left[{\mathrm{\_\&xi;}}_x\left(u\right)=1\&comma;{\mathrm{\_\&xi;}}_t\left(u\right)=0\&comma;{\mathrm{\_\&eta;}}_u\left(u\right)=0\right],\left[{\mathrm{\_\&xi;}}_x\left(u\right)=0\&comma;{\mathrm{\_\&xi;}}_t\left(u\right)=1\&comma;{\mathrm{\_\&eta;}}_u\left(u\right)=0\right],\left[{\mathrm{\_\&xi;}}_x\left(u\right)=0\&comma;{\mathrm{\_\&xi;}}_t\left(u\right)=0\&comma;{\mathrm{\_\&eta;}}_u\left(u\right)=1\right],\left[{\mathrm{\_\&xi;}}_x\left(u\right)=0\&comma;{\mathrm{\_\&xi;}}_t\left(u\right)=0\&comma;{\mathrm{\_\&eta;}}_u\left(u\right)=u\right],\left[{\mathrm{\_\&xi;}}_x\left(x\&comma;t\right)=x\&comma;{\mathrm{\_\&xi;}}_t\left(x\&comma;t\right)=2t\&comma;{\mathrm{\_\&eta;}}_u\left(x\&comma;t\right)=0\right],\left[{\mathrm{\_\&xi;}}_x\left(x\&comma;t\right)=0\&comma;{\mathrm{\_\&xi;}}_t\left(x\&comma;t\right)=0\&comma;{\mathrm{\_\&eta;}}_u\left(x\&comma;t\right)={\&ExponentialE;}^{\sqrt{{\mathrm{\_c}}_1}x}{\&ExponentialE;}^{{\mathrm{\_c}}_{1}t}\right],\left[{\mathrm{\_\&xi;}}_x\left(x\&comma;t\right)=0\&comma;{\mathrm{\_\&xi;}}_t\left(x\&comma;t\right)=0\&comma;{\mathrm{\_\&eta;}}_u\left(x\&comma;t\right)=\frac{{\&ExponentialE;}^{{\mathrm{\_c}}_{1}t}}{{\&ExponentialE;}^{\sqrt{{\mathrm{\_c}}_1}x}}\right]$

Infinitesimals(PDE, HINT = [_xi[x] = f(x)*g(t), _xi[t] = f(x) + g(t), _eta[u] = f(x)*g(t)]);

$\left[{\mathrm{\_\&xi;}}_x=0\&comma;{\mathrm{\_\&xi;}}_t=1\&comma;{\mathrm{\_\&eta;}}_u=0\right]$

declare((u,v)(x,t), (xi,eta)(x,t,u,v));

$u\left(x\&comma;t\right)\mathrm{will\; now\; be\; displayed\; as}u$

$v\left(x\&comma;t\right)\mathrm{will\; now\; be\; displayed\; as}v$

$\mathrm{\xi }\left(x\&comma;t\&comma;u\&comma;v\right)\mathrm{will\; now\; be\; displayed\; as}\mathrm{\xi }$

$\mathrm{\eta }\left(x\&comma;t\&comma;u\&comma;v\right)\mathrm{will\; now\; be\; displayed\; as}\mathrm{\eta }$

DepVars := [u,v](x,t);

$\mathrm{DepVars}≔\left[u\&comma;v\right]$

U, V := diff_table(u(x,t)), diff_table(v(x,t)):

e1 := U[t] + 1/2*U[x]^2 + 2*V[] - 2*x = 1/2*(-V[x]^2 + 2*V[x,x] * V[])/V[]^2;

$\mathrm{e1}≔u_t+\frac{{u_x}^2}{2}+2v-2x=\frac{2v{v}_{x,x}-{v_x}^2}{2v^2}$

e2 := V[t] + diff(V[]*U[x], x)=0;

$\mathrm{e2}≔v_t+v_x{u}_x+v{u}_{x,x}=0$

PDESYS := [e1, e2]:

G := Infinitesimals(PDESYS);

$G≔\left[{\mathrm{\_\&xi;}}_x\left(x\&comma;t\&comma;u\&comma;v\right)=0\&comma;{\mathrm{\_\&xi;}}_t\left(x\&comma;t\&comma;u\&comma;v\right)=1\&comma;{\mathrm{\_\&eta;}}_u\left(x\&comma;t\&comma;u\&comma;v\right)=0\&comma;{\mathrm{\_\&eta;}}_v\left(x\&comma;t\&comma;u\&comma;v\right)=0\right],\left[{\mathrm{\_\&xi;}}_x\left(x\&comma;t\&comma;u\&comma;v\right)=0\&comma;{\mathrm{\_\&xi;}}_t\left(x\&comma;t\&comma;u\&comma;v\right)=0\&comma;{\mathrm{\_\&eta;}}_u\left(x\&comma;t\&comma;u\&comma;v\right)=1\&comma;{\mathrm{\_\&eta;}}_v\left(x\&comma;t\&comma;u\&comma;v\right)=0\right],\left[{\mathrm{\_\&xi;}}_x\left(x\&comma;t\&comma;u\&comma;v\right)=1\&comma;{\mathrm{\_\&xi;}}_t\left(x\&comma;t\&comma;u\&comma;v\right)=0\&comma;{\mathrm{\_\&eta;}}_u\left(x\&comma;t\&comma;u\&comma;v\right)=2t\&comma;{\mathrm{\_\&eta;}}_v\left(x\&comma;t\&comma;u\&comma;v\right)=0\right],\left[{\mathrm{\_\&xi;}}_x\left(x\&comma;t\&comma;u\&comma;v\right)=t\&comma;{\mathrm{\_\&xi;}}_t\left(x\&comma;t\&comma;u\&comma;v\right)=0\&comma;{\mathrm{\_\&eta;}}_u\left(x\&comma;t\&comma;u\&comma;v\right)=t^2+x\&comma;{\mathrm{\_\&eta;}}_v\left(x\&comma;t\&comma;u\&comma;v\right)=0\right],\left[{\mathrm{\_\&xi;}}_x\left(x\&comma;t\&comma;u\&comma;v\right)=\frac{x}{2}+\frac{3t^2}{2}\&comma;{\mathrm{\_\&xi;}}_t\left(x\&comma;t\&comma;u\&comma;v\right)=t\&comma;{\mathrm{\_\&eta;}}_u\left(x\&comma;t\&comma;u\&comma;v\right)=t\left(t^2+3x\right)\&comma;{\mathrm{\_\&eta;}}_v\left(x\&comma;t\&comma;u\&comma;v\right)=-v\right]$

map(SymmetryTest, [G], PDESYS);

$\left[\left\{0\right\}\&comma;\left\{0\right\}\&comma;\left\{0\right\}\&comma;\left\{0\right\}\&comma;\left\{0\right\}\right]$

S := Infinitesimals(PDESYS, typeofsymmetry = And(evolutionary, polynomial), displayfunctionality = false);

$S≔\left[{\mathrm{\_\&xi;}}_x=0\&comma;{\mathrm{\_\&xi;}}_t=0\&comma;{\mathrm{\_\&eta;}}_u=1\&comma;{\mathrm{\_\&eta;}}_v=0\right],\left[{\mathrm{\_\&xi;}}_x=0\&comma;{\mathrm{\_\&xi;}}_t=0\&comma;{\mathrm{\_\&eta;}}_u=u_t\&comma;{\mathrm{\_\&eta;}}_v=v_t\right],\left[{\mathrm{\_\&xi;}}_x=0\&comma;{\mathrm{\_\&xi;}}_t=0\&comma;{\mathrm{\_\&eta;}}_u=-2t+u_x\&comma;{\mathrm{\_\&eta;}}_v=v_x\right],\left[{\mathrm{\_\&xi;}}_x=0\&comma;{\mathrm{\_\&xi;}}_t=0\&comma;{\mathrm{\_\&eta;}}_u=-t^2+t{u}_x-x\&comma;{\mathrm{\_\&eta;}}_v=t{v}_x\right]$

map(SymmetryTest, [S], PDESYS);

$\left[\left\{0\right\}\&comma;\left\{0\right\}\&comma;\left\{0\right\}\&comma;\left\{0\right\}\right]$

IG := InfinitesimalGenerator(G[5], DepVars, expanded, prolongation = 1);

$\mathrm{IG}≔f\&rarr;\left(\frac{1}{2}x\&plus;\frac{3}{2}{t}^2\right)\left(\frac{\partial }{\partial x}f\right)\&plus;t\left(\frac{\partial }{\partial t}f\right)\&plus;t\left(t^2\&plus;3x\right)\left(\frac{\partial }{\partial u}f\right)-v\left(\frac{\partial }{\partial v}f\right)\&plus;\left(3t-\frac{1}{2}{u}_x\right)\left(\frac{\partial }{\partial u_x}f\right)-\frac{3}{2}{v}_x\left(\frac{\partial }{\partial v_x}f\right)\&plus;\left(3t^2-3t{u}_x\&plus;3x-u_t\right)\left(\frac{\partial }{\partial u_t}f\right)\&plus;\left(-3t{v}_x-2v_t\right)\left(\frac{\partial }{\partial v_t}f\right)$

Invariants(G[5], DepVars);

$tv,\frac{-t^2+x}{\sqrt{t}},\frac{2}{3}{t}^3-2tx+u,v_x{t}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.},-2t^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}+u_x\sqrt{t},2t^3{v}_x+t^2{v}_t,-2t\left(t^2-t{u}_x+x-\frac{1}{2}{u}_t\right)$

map(IG, [(25)]);

$\left[0\&comma;\frac{\frac{x}{2}+\frac{3t^2}{2}}{\sqrt{t}}+t\left(-2\sqrt{t}-\frac{-t^2+x}{2t^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\right)\&comma;-2\left(\frac{x}{2}+\frac{3t^2}{2}\right)t+t\left(2t^2-2x\right)+t\left(t^2+3x\right)\&comma;0\&comma;t\left(-3\sqrt{t}+\frac{u_x}{2\sqrt{t}}\right)+\left(3t-\frac{u_x}{2}\right)\sqrt{t}\&comma;t\left(6t^2{v}_x+2t{v}_t\right)-3t^3{v}_x+\left(-3t{v}_x-2v_t\right)t^2\&comma;-2\left(\frac{x}{2}+\frac{3t^2}{2}\right)t+t\left(-2t^2+2t{u}_x-2x+u_t-2t\left(2t-u_x\right)\right)+2\left(3t-\frac{u_x}{2}\right)t^2+\left(3t^2-3t{u}_x+3x-u_t\right)t\right]$

simplify((26));

$\left[0\&comma;0\&comma;0\&comma;0\&comma;0\&comma;0\&comma;0\right]$

NewVars := [f, g](r, s);

$\mathrm{NewVars}≔\left[f\left(r\&comma;s\right)\&comma;g\left(r\&comma;s\right)\right]$

SymmetryTransformation(G[5], DepVars, NewVars);

$\left\{r=\frac{t^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\&ExponentialE;}^{2\mathrm{\_\&epsilon;}}-\sqrt{t{\&ExponentialE;}^{\mathrm{\_\&epsilon;}}}{t}^2+\sqrt{t{\&ExponentialE;}^{\mathrm{\_\&epsilon;}}}x}{\sqrt{t}}\&comma;s=t{\&ExponentialE;}^{\mathrm{\_\&epsilon;}}\&comma;f\left(r\&comma;s\right)=2{\&ExponentialE;}^{\mathrm{\_\&epsilon;}}\left(\sqrt{t}x-t^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)\sqrt{t{\&ExponentialE;}^{\mathrm{\_\&epsilon;}}}+\frac{4t^3{\&ExponentialE;}^{3\mathrm{\_\&epsilon;}}}{3}+\frac{2t^3}{3}-2tx+u\&comma;g\left(r\&comma;s\right)={\&ExponentialE;}^{-\mathrm{\_\&epsilon;}}v\right\}$

SimilarityTransformation(G[5], DepVars, NewVars);

U := diff_table(u(x,t)):

PDE := U[x,x] - U[t,t]=0;

$\mathrm{PDE}≔u_{x,x}-u_{t,t}=0$

Infinitesimals(PDE);

$\left[{\mathrm{\_\&xi;}}_x\left(x\&comma;t\&comma;u\right)=\mathrm{f\_\_1}\left(t+x\right)+\mathrm{f\_\_2}\left(t-x\right)\&comma;{\mathrm{\_\&xi;}}_t\left(x\&comma;t\&comma;u\right)=-\mathrm{f\_\_2}\left(t-x\right)+\mathrm{f\_\_1}\left(t+x\right)+1\&comma;{\mathrm{\_\&eta;}}_u\left(x\&comma;t\&comma;u\right)=\mathrm{f\_\_3}\left(t+x\right)+\mathrm{f\_\_4}\left(t-x\right)\right],\left[{\mathrm{\_\&xi;}}_x\left(x\&comma;t\&comma;u\right)=\mathrm{f\_\_5}\left(t+x\right)+\mathrm{f\_\_6}\left(t-x\right)\&comma;{\mathrm{\_\&xi;}}_t\left(x\&comma;t\&comma;u\right)=-\mathrm{f\_\_6}\left(t-x\right)+\mathrm{f\_\_5}\left(t+x\right)\&comma;{\mathrm{\_\&eta;}}_u\left(x\&comma;t\&comma;u\right)=u+\mathrm{f\_\_7}\left(t+x\right)+\mathrm{f\_\_8}\left(t-x\right)\right]$

Infinitesimals(PDE, specialize_Fn);

$\left[{\mathrm{\_\&xi;}}_x\left(x\&comma;t\&comma;u\right)=1\&comma;{\mathrm{\_\&xi;}}_t\left(x\&comma;t\&comma;u\right)=0\&comma;{\mathrm{\_\&eta;}}_u\left(x\&comma;t\&comma;u\right)=0\right],\left[{\mathrm{\_\&xi;}}_x\left(x\&comma;t\&comma;u\right)=1\&comma;{\mathrm{\_\&xi;}}_t\left(x\&comma;t\&comma;u\right)=2\&comma;{\mathrm{\_\&eta;}}_u\left(x\&comma;t\&comma;u\right)=0\right],\left[{\mathrm{\_\&xi;}}_x\left(x\&comma;t\&comma;u\right)=0\&comma;{\mathrm{\_\&xi;}}_t\left(x\&comma;t\&comma;u\right)=1\&comma;{\mathrm{\_\&eta;}}_u\left(x\&comma;t\&comma;u\right)=1\right],\left[{\mathrm{\_\&xi;}}_x\left(x\&comma;t\&comma;u\right)=1\&comma;{\mathrm{\_\&xi;}}_t\left(x\&comma;t\&comma;u\right)=1\&comma;{\mathrm{\_\&eta;}}_u\left(x\&comma;t\&comma;u\right)=u\right],\left[{\mathrm{\_\&xi;}}_x\left(x\&comma;t\&comma;u\right)=1\&comma;{\mathrm{\_\&xi;}}_t\left(x\&comma;t\&comma;u\right)=\mathrm{-1}\&comma;{\mathrm{\_\&eta;}}_u\left(x\&comma;t\&comma;u\right)=u\right],\left[{\mathrm{\_\&xi;}}_x\left(x\&comma;t\&comma;u\right)=0\&comma;{\mathrm{\_\&xi;}}_t\left(x\&comma;t\&comma;u\right)=1\&comma;{\mathrm{\_\&eta;}}_u\left(x\&comma;t\&comma;u\right)=t+x\right],\left[{\mathrm{\_\&xi;}}_x\left(x\&comma;t\&comma;u\right)=0\&comma;{\mathrm{\_\&xi;}}_t\left(x\&comma;t\&comma;u\right)=1\&comma;{\mathrm{\_\&eta;}}_u\left(x\&comma;t\&comma;u\right)=t-x\right],\left[{\mathrm{\_\&xi;}}_x\left(x\&comma;t\&comma;u\right)=0\&comma;{\mathrm{\_\&xi;}}_t\left(x\&comma;t\&comma;u\right)=0\&comma;{\mathrm{\_\&eta;}}_u\left(x\&comma;t\&comma;u\right)=u+1\right],\left[{\mathrm{\_\&xi;}}_x\left(x\&comma;t\&comma;u\right)=0\&comma;{\mathrm{\_\&xi;}}_t\left(x\&comma;t\&comma;u\right)=0\&comma;{\mathrm{\_\&eta;}}_u\left(x\&comma;t\&comma;u\right)=u+t+x\right],\left[{\mathrm{\_\&xi;}}_x\left(x\&comma;t\&comma;u\right)=0\&comma;{\mathrm{\_\&xi;}}_t\left(x\&comma;t\&comma;u\right)=0\&comma;{\mathrm{\_\&eta;}}_u\left(x\&comma;t\&comma;u\right)=u+t-x\right],\left[{\mathrm{\_\&xi;}}_x\left(x\&comma;t\&comma;u\right)=t+x\&comma;{\mathrm{\_\&xi;}}_t\left(x\&comma;t\&comma;u\right)=1+t+x\&comma;{\mathrm{\_\&eta;}}_u\left(x\&comma;t\&comma;u\right)=0\right],\left[{\mathrm{\_\&xi;}}_x\left(x\&comma;t\&comma;u\right)=t-x\&comma;{\mathrm{\_\&xi;}}_t\left(x\&comma;t\&comma;u\right)=-t+x+1\&comma;{\mathrm{\_\&eta;}}_u\left(x\&comma;t\&comma;u\right)=0\right],\left[{\mathrm{\_\&xi;}}_x\left(x\&comma;t\&comma;u\right)=t+x\&comma;{\mathrm{\_\&xi;}}_t\left(x\&comma;t\&comma;u\right)=t+x\&comma;{\mathrm{\_\&eta;}}_u\left(x\&comma;t\&comma;u\right)=u\right],\left[{\mathrm{\_\&xi;}}_x\left(x\&comma;t\&comma;u\right)=t-x\&comma;{\mathrm{\_\&xi;}}_t\left(x\&comma;t\&comma;u\right)=-t+x\&comma;{\mathrm{\_\&eta;}}_u\left(x\&comma;t\&comma;u\right)=u\right]$

Infinitesimals(PDE, specialize_Fn = {`*`, `+`});

$\left[{\mathrm{\_\&xi;}}_x\left(x\&comma;t\&comma;u\right)=0\&comma;{\mathrm{\_\&xi;}}_t\left(x\&comma;t\&comma;u\right)=1\&comma;{\mathrm{\_\&eta;}}_u\left(x\&comma;t\&comma;u\right)=t+x\right],\left[{\mathrm{\_\&xi;}}_x\left(x\&comma;t\&comma;u\right)=0\&comma;{\mathrm{\_\&xi;}}_t\left(x\&comma;t\&comma;u\right)=1\&comma;{\mathrm{\_\&eta;}}_u\left(x\&comma;t\&comma;u\right)=t-x\right],\left[{\mathrm{\_\&xi;}}_x\left(x\&comma;t\&comma;u\right)=0\&comma;{\mathrm{\_\&xi;}}_t\left(x\&comma;t\&comma;u\right)=0\&comma;{\mathrm{\_\&eta;}}_u\left(x\&comma;t\&comma;u\right)=u+t+x\right],\left[{\mathrm{\_\&xi;}}_x\left(x\&comma;t\&comma;u\right)=0\&comma;{\mathrm{\_\&xi;}}_t\left(x\&comma;t\&comma;u\right)=0\&comma;{\mathrm{\_\&eta;}}_u\left(x\&comma;t\&comma;u\right)=u+t-x\right],\left[{\mathrm{\_\&xi;}}_x\left(x\&comma;t\&comma;u\right)=t+x\&comma;{\mathrm{\_\&xi;}}_t\left(x\&comma;t\&comma;u\right)=1+t+x\&comma;{\mathrm{\_\&eta;}}_u\left(x\&comma;t\&comma;u\right)=0\right],\left[{\mathrm{\_\&xi;}}_x\left(x\&comma;t\&comma;u\right)=t-x\&comma;{\mathrm{\_\&xi;}}_t\left(x\&comma;t\&comma;u\right)=-t+x+1\&comma;{\mathrm{\_\&eta;}}_u\left(x\&comma;t\&comma;u\right)=0\right],\left[{\mathrm{\_\&xi;}}_x\left(x\&comma;t\&comma;u\right)=t+x\&comma;{\mathrm{\_\&xi;}}_t\left(x\&comma;t\&comma;u\right)=t+x\&comma;{\mathrm{\_\&eta;}}_u\left(x\&comma;t\&comma;u\right)=u\right],\left[{\mathrm{\_\&xi;}}_x\left(x\&comma;t\&comma;u\right)=t-x\&comma;{\mathrm{\_\&xi;}}_t\left(x\&comma;t\&comma;u\right)=-t+x\&comma;{\mathrm{\_\&eta;}}_u\left(x\&comma;t\&comma;u\right)=u\right]$

PDE := U[x,x]^2*t-U[t]*u^2*x = 0;

$\mathrm{PDE}≔-u^{2}x{u}_t+t{u_{x,x}}^2=0$

Infinitesimals(PDE, display = false);

$\mathrm{*\; Partial\; match\; of\; \text{'}display\text{'}\; against\; keyword\; \text{'}displayfunctionality\text{'}}$

$\left[{\mathrm{\_\&xi;}}_x=0\&comma;{\mathrm{\_\&xi;}}_t=\frac{1}{t}\&comma;{\mathrm{\_\&eta;}}_u=0\right],\left[{\mathrm{\_\&xi;}}_x=0\&comma;{\mathrm{\_\&xi;}}_t=t\&comma;{\mathrm{\_\&eta;}}_u=2u\right],\left[{\mathrm{\_\&xi;}}_x=x\&comma;{\mathrm{\_\&xi;}}_t=0\&comma;{\mathrm{\_\&eta;}}_u=-5u\right]$