checkconsistency = true | false

closesystem = true | false

conservedcurrents

dependency = x | {x, t, ...}

integrabilityconditions = true | false

integratingfactors

jetnotation = jetvariables | jetvariableswithbrackets | jetODE | jetnumbers

moduloPDESYS = true | false

simplifier = anything

split = true | false

typeofsymmetry = pointlike | contact | evolutionary | general

xi_eta = true | false

Given a PDE problem (PDESYS), as an equation, or a set or list of them, the DeterminingPDE command computes the determining PDE system for the symmetries, integrating factors or conserved currents of PDESYS. The output is thus a system of partial differential equations whose solution gives, respectively, the infinitesimals of the symmetry generators leaving invariant PDESYS, the integrating factors or the conserved currents of PDESYS. DeterminingPDE also works with anticommutative variables set using the Physics package.

By default, DeterminingPDE computes the system satisfied by the symmetry infinitesimals of PDESYS and processes PDESYS as given. To request the determining system for the integrating factors or the conserved currents pass the keywords integratingfactors or conservedcurrents. Note that in the case of symmetries, if PDESYS is not closed in the sense of including explicitly all its integrability conditions, the resulting determining system may not include for solutions all the symmetries of PDESYS. To close the system in order to compute all the symmetries use the option closesystem - see details in the Optional arguments section below - see also the related PDEtools Library routine CloseSystem.

Within the framework of computing the so-called group-invariant (GI) solutions of PDESYS, the first step is to compute the determining PDE system, next you attempt solving this system to obtain the infinitesimals, with them 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, Infinitesimals, InvariantTransformation and InvariantSolutions.

If DepVars is not given, DeterminingPDE will consider all the differentiated unknown functions in PDESYS as unknowns of the problem. 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 argument S is optionally used to indicate the functional form of the symmetries - see the examples - if not given, DeterminingPDE will return a determining system whose solution are the infinitesimals of the point symmetries of PDESYS. To override that behavior when not passing S, use the option typeofsymmetry = ... where the right-hand side can be any of pointlike (default), contact, evolutionary or general (see definitions in the Options section).

Among the most commonly used options explained in the Options section, the PDE system returned by DeterminingPDE is automatically reduced taking into account its integrability conditions, a frequently desired but in some cases expensive mathematical computation, unless explicitly requested otherwise by passing the optional argument integrabilityconditions = false, in which case DeterminingPDE will return faster. Also, the output is by default in jetnotation = jetvariables, modulo the given PDESYS, split with respect to powers of independent objects not entering the dependency of the infinitesimals, and simplified in size. That is frequently the most compact and useful form of the determining system. You can override any of these defaults by passing any of the optional arguments integrabilityconditions = false, jetnotation = ..., moduloPDESYS = false, simplifier = none (or the one you prefer).

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, DeterminingPDE, declare, diff_table, casesplit, InfinitesimalGenerator, Infinitesimals, SymmetryTest):

declare(u(x,t));

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

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

PDE1 := U[x,x] - U[t] = 0;

$\mathrm{PDE1}≔u_{x,x}-u_t=0$

DetSys := DeterminingPDE(PDE1);

$\mathrm{DetSys}≔\left\{{\left({\mathrm{\_\ξ}}_t\right)}_{t,t,t}=0\,{\left({\mathrm{\_\η}}_u\right)}_{t,u}=-\frac{{\left({\mathrm{\_\ξ}}_t\right)}_{t,t}}{4}\,{\left({\mathrm{\_\η}}_u\right)}_{u,u}=0\,{\left({\mathrm{\_\η}}_u\right)}_{x,x}={\left({\mathrm{\_\η}}_u\right)}_t\,{\left({\mathrm{\_\ξ}}_t\right)}_u=0\,{\left({\mathrm{\_\ξ}}_t\right)}_x=0\,{\left({\mathrm{\_\ξ}}_x\right)}_t=-2{\left({\mathrm{\_\η}}_u\right)}_{u,x}\,{\left({\mathrm{\_\ξ}}_x\right)}_u=0\,{\left({\mathrm{\_\ξ}}_x\right)}_x=\frac{{\left({\mathrm{\_\ξ}}_t\right)}_t}{2}\right\}$

DeterminingPDE(PDE1, jetnotation = jetnumbers);

$\left\{{\left({\mathrm{\_\ξ}}_2\right)}_{t,t,t}=0\,{\left({\mathrm{\_\η}}_1\right)}_{t,u\left[\right]}=-\frac{{\left({\mathrm{\_\ξ}}_2\right)}_{t,t}}{4}\,{\left({\mathrm{\_\η}}_1\right)}_{x,x}={\left({\mathrm{\_\η}}_1\right)}_t\,{\left({\mathrm{\_\η}}_1\right)}_{u\left[\right],u\left[\right]}=0\,{\left({\mathrm{\_\ξ}}_1\right)}_t=-2{\left({\mathrm{\_\η}}_1\right)}_{x,u\left[\right]}\,{\left({\mathrm{\_\ξ}}_1\right)}_x=\frac{{\left({\mathrm{\_\ξ}}_2\right)}_t}{2}\,{\left({\mathrm{\_\ξ}}_1\right)}_{u\left[\right]}=0\,{\left({\mathrm{\_\ξ}}_2\right)}_x=0\,{\left({\mathrm{\_\ξ}}_2\right)}_{u\left[\right]}=0\right\}$

DeterminingPDE(PDE1, integrabilityconditions = false);

$\left\{{\left({\mathrm{\_\η}}_u\right)}_{x,x}-{\left({\mathrm{\_\η}}_u\right)}_t\,-2{\left({\mathrm{\_\ξ}}_t\right)}_{u,x}-2{\left({\mathrm{\_\ξ}}_x\right)}_u\,-2{\left({\mathrm{\_\ξ}}_x\right)}_{u,x}+{\left({\mathrm{\_\η}}_u\right)}_{u,u}\,2{\left({\mathrm{\_\η}}_u\right)}_{u,x}-{\left({\mathrm{\_\ξ}}_x\right)}_{x,x}+{\left({\mathrm{\_\ξ}}_x\right)}_t\,-{\left({\mathrm{\_\ξ}}_t\right)}_{x,x}-2{\left({\mathrm{\_\ξ}}_x\right)}_x+{\left({\mathrm{\_\ξ}}_t\right)}_t\,{\left({\mathrm{\_\ξ}}_t\right)}_{u,u}\,{\left({\mathrm{\_\ξ}}_x\right)}_{u,u}\,{\left({\mathrm{\_\ξ}}_t\right)}_u\,{\left({\mathrm{\_\ξ}}_t\right)}_x\right\}$

DeterminingPDE(PDE1, integrabilityconditions = false, split = false);

$\left\{-{\left({\mathrm{\_\ξ}}_x\right)}_{u,u}{u}_x^3-{\left({\mathrm{\_\ξ}}_t\right)}_{u,u}{u}_x^2{u}_t-2{\left({\mathrm{\_\ξ}}_x\right)}_{u,x}{u}_x^2-2{\left({\mathrm{\_\ξ}}_t\right)}_{u,x}{u}_x{u}_t+{\left({\mathrm{\_\η}}_u\right)}_{u,u}{u}_x^2-2{\left({\mathrm{\_\ξ}}_x\right)}_u{u}_x{u}_t-2{\left({\mathrm{\_\ξ}}_t\right)}_u{u}_x{u}_{x,t}+2u_x{\left({\mathrm{\_\η}}_u\right)}_{u,x}-{\left({\mathrm{\_\ξ}}_x\right)}_{x,x}{u}_x-{\left({\mathrm{\_\ξ}}_t\right)}_{x,x}{u}_t-2{\left({\mathrm{\_\ξ}}_x\right)}_x{u}_t-2{\left({\mathrm{\_\ξ}}_t\right)}_x{u}_{x,t}+{\left({\mathrm{\_\η}}_u\right)}_{x,x}+{\left({\mathrm{\_\ξ}}_x\right)}_t{u}_x+{\left({\mathrm{\_\ξ}}_t\right)}_t{u}_t-{\left({\mathrm{\_\η}}_u\right)}_t\right\}$

show;

$\left\{-\left(\frac{{\partial }^2}{\partial u^2}{\mathrm{\_\ξ}}_x\left(x\,t\,u\right)\right)u_x^3-\left(\frac{{\partial }^2}{\partial u^2}{\mathrm{\_\ξ}}_t\left(x\,t\,u\right)\right)u_x^2{u}_t-2\left(\frac{{\partial }^2}{\partial u\partial x}{\mathrm{\_\ξ}}_x\left(x\,t\,u\right)\right)u_x^2-2\left(\frac{{\partial }^2}{\partial u\partial x}{\mathrm{\_\ξ}}_t\left(x\,t\,u\right)\right)u_x{u}_t+\left(\frac{{\partial }^2}{\partial u^2}{\mathrm{\_\η}}_u\left(x\,t\,u\right)\right)u_x^2-2\left(\frac{\partial }{\partial u}{\mathrm{\_\ξ}}_x\left(x\,t\,u\right)\right)u_x{u}_t-2\left(\frac{\partial }{\partial u}{\mathrm{\_\ξ}}_t\left(x\,t\,u\right)\right)u_x{u}_{x,t}+2u_x\left(\frac{{\partial }^2}{\partial u\partial x}{\mathrm{\_\η}}_u\left(x\,t\,u\right)\right)-\left(\frac{{\partial }^2}{\partial x^2}{\mathrm{\_\ξ}}_x\left(x\,t\,u\right)\right)u_x-\left(\frac{{\partial }^2}{\partial x^2}{\mathrm{\_\ξ}}_t\left(x\,t\,u\right)\right)u_t-2\left(\frac{\partial }{\partial x}{\mathrm{\_\ξ}}_x\left(x\,t\,u\right)\right)u_t-2\left(\frac{\partial }{\partial x}{\mathrm{\_\ξ}}_t\left(x\,t\,u\right)\right)u_{x,t}+\frac{{\partial }^2}{\partial x^2}{\mathrm{\_\η}}_u\left(x\,t\,u\right)+\left(\frac{\partial }{\partial t}{\mathrm{\_\ξ}}_x\left(x\,t\,u\right)\right)u_x+\left(\frac{\partial }{\partial t}{\mathrm{\_\ξ}}_t\left(x\,t\,u\right)\right)u_t-\frac{\partial }{\partial t}{\mathrm{\_\η}}_u\left(x\,t\,u\right)\right\}$

pdsolve(DetSys);

$\left\{{\mathrm{\_\η}}_u\left(x\,t\,u\right)=\left(\frac{\left(-x^2-2t\right)\mathrm{c\_\_1}}{8}+\mathrm{c\_\_4}x+\mathrm{c\_\_5}\right)u+\mathrm{c\_\_8}{\ⅇ}^{{\mathrm{\_c}}_{1}t}\mathrm{c\_\_6}{\ⅇ}^{\sqrt{{\mathrm{\_c}}_1}x}+\frac{\mathrm{c\_\_8}{\ⅇ}^{{\mathrm{\_c}}_{1}t}\mathrm{c\_\_7}}{{\ⅇ}^{\sqrt{{\mathrm{\_c}}_1}x}}\,{\mathrm{\_\ξ}}_t\left(x\,t\,u\right)=\frac{1}{2}\mathrm{c\_\_1}{t}^2+\mathrm{c\_\_2}t+\mathrm{c\_\_3}\,{\mathrm{\_\ξ}}_x\left(x\,t\,u\right)=\frac{\left(x\mathrm{c\_\_1}-4\mathrm{c\_\_4}\right)t}{2}+\frac{\mathrm{c\_\_2}x}{2}+\mathrm{c\_\_9}\right\}$

Infinitesimals(PDE1);

$\left[{\mathrm{\_\ξ}}_x\left(x\,t\,u\right)=0\,{\mathrm{\_\ξ}}_t\left(x\,t\,u\right)=1\,{\mathrm{\_\η}}_u\left(x\,t\,u\right)=0\right],\left[{\mathrm{\_\ξ}}_x\left(x\,t\,u\right)=1\,{\mathrm{\_\ξ}}_t\left(x\,t\,u\right)=0\,{\mathrm{\_\η}}_u\left(x\,t\,u\right)=0\right],\left[{\mathrm{\_\ξ}}_x\left(x\,t\,u\right)=0\,{\mathrm{\_\ξ}}_t\left(x\,t\,u\right)=0\,{\mathrm{\_\η}}_u\left(x\,t\,u\right)=u\right],\left[{\mathrm{\_\ξ}}_x\left(x\,t\,u\right)=\frac{x}{2}\,{\mathrm{\_\ξ}}_t\left(x\,t\,u\right)=t\,{\mathrm{\_\η}}_u\left(x\,t\,u\right)=0\right],\left[{\mathrm{\_\ξ}}_x\left(x\,t\,u\right)=-2t\,{\mathrm{\_\ξ}}_t\left(x\,t\,u\right)=0\,{\mathrm{\_\η}}_u\left(x\,t\,u\right)=xu\right],\left[{\mathrm{\_\ξ}}_x\left(x\,t\,u\right)=0\,{\mathrm{\_\ξ}}_t\left(x\,t\,u\right)=0\,{\mathrm{\_\η}}_u\left(x\,t\,u\right)={\ⅇ}^{\sqrt{{\mathrm{\_c}}_1}x}{\ⅇ}^{{\mathrm{\_c}}_{1}t}\right],\left[{\mathrm{\_\ξ}}_x\left(x\,t\,u\right)=0\,{\mathrm{\_\ξ}}_t\left(x\,t\,u\right)=0\,{\mathrm{\_\η}}_u\left(x\,t\,u\right)=\frac{{\ⅇ}^{{\mathrm{\_c}}_{1}t}}{{\ⅇ}^{\sqrt{{\mathrm{\_c}}_1}x}}\right],\left[{\mathrm{\_\ξ}}_x\left(x\,t\,u\right)=\frac{tx}{2}\,{\mathrm{\_\ξ}}_t\left(x\,t\,u\right)=\frac{t^2}{2}\,{\mathrm{\_\η}}_u\left(x\,t\,u\right)=-\frac{\left(x^2+2t\right)u}{8}\right]$

DeterminingPDE(PDE1, integratingfactors);

$\left\{{\left({\mathrm{\_\μ}}_1\right)}_{x,x}=-{\left({\mathrm{\_\μ}}_1\right)}_t\,{\left({\mathrm{\_\μ}}_1\right)}_u=0\,{\left({\mathrm{\_\μ}}_1\right)}_{u_t}=0\,{\left({\mathrm{\_\μ}}_1\right)}_{u_x}=0\right\}$

DeterminingPDE(PDE1, conservedcurrents, dependency = [x, t]);

$\left\{{\left({\mathrm{\_J}}_x\right)}_x+{\left({\mathrm{\_J}}_t\right)}_t\right\}$

declare(u(r, phi, theta, t), (_xi, _eta)((r, phi, theta, t, u)));

$u\left(r\,\mathrm{\phi }\,\mathrm{\theta }\,t\right)\mathrm{will\; now\; be\; displayed\; as}u$

$\mathrm{\_\ξ}\left(r\,\mathrm{\phi }\,\mathrm{\theta }\,t\,u\right)\mathrm{will\; now\; be\; displayed\; as}\mathrm{\_\ξ}$

$\mathrm{\_\η}\left(r\,\mathrm{\phi }\,\mathrm{\theta }\,t\,u\right)\mathrm{will\; now\; be\; displayed\; as}\mathrm{\_\η}$

U := diff_table(u(r,phi,theta,t)):

PDE2 := 1/r^2*(2*r*U[r]+r^2*U[r,r]) + 1/r^2/sin(theta)*(cos(theta)*U[theta] + sin(theta)*U[theta,theta]) + 1/r^2/sin(theta)^2*U[phi,phi] = U[t,t];

$\mathrm{PDE2}≔\frac{r^2{u}_{r,r}+2r{u}_r}{r^2}+\frac{\cos \left(\mathrm{\theta }\right)u_{\mathrm{\theta }}+\sin \left(\mathrm{\theta }\right)u_{\mathrm{\theta },\mathrm{\theta }}}{r^2\sin \left(\mathrm{\theta }\right)}+\frac{u_{\mathrm{\phi },\mathrm{\phi }}}{r^2{\sin \left(\mathrm{\theta }\right)}^2}=u_{t,t}$

Infinitesimals(PDE2);

declare(y(x), prime = x);

$y\left(x\right)\mathrm{will\; now\; be\; displayed\; as}y$

$\mathrm{derivatives\; with\; respect\; to}x\mathrm{of\; functions\; of\; one\; variable\; will\; now\; be\; displayed\; with\; \text{'}}$

ODE2 := diff(y(x),x,x) = Phi(y(x));

$\mathrm{ODE2}≔\mathrm{y\text{'}\text{'}}=\mathrm{\Phi }\left(y\right)$

DS := [_xi = f(x,y,_y1), _eta = 0];

$\mathrm{DS}≔\left[\mathrm{\_\ξ}=f\left(x\,y\,\mathrm{\_y1}\right)\,\mathrm{\_\η}=0\right]$

DeterminingPDE(ODE2, y(x), DS, 'jetnotation' = 'jetODE');

$\left\{f_{x,x}=\frac{-f_{\mathrm{\_y1},\mathrm{\_y1}}\mathrm{\_y1}{\mathrm{\Phi }\left(y\right)}^2-2f_{\mathrm{\_y1},y}{\mathrm{\_y1}}^2\mathrm{\Phi }\left(y\right)-{\mathrm{\Phi }}_y{\mathrm{\_y1}}^2{f}_{\mathrm{\_y1}}-f_{y,y}{\mathrm{\_y1}}^3-2f_{\mathrm{\_y1}}{\mathrm{\Phi }\left(y\right)}^2-2f_{\mathrm{\_y1},x}\mathrm{\_y1}\mathrm{\Phi }\left(y\right)-3f_y\mathrm{\_y1}\mathrm{\Phi }\left(y\right)-2f_{x,y}{\mathrm{\_y1}}^2-2f_x\mathrm{\Phi }\left(y\right)}{\mathrm{\_y1}}\right\}$

ODE2 := diff(z(x,y),x) = y/x*z(x,y);

$\mathrm{ODE2}≔z_x=\frac{yz\left(x\,y\right)}{x}$

S2 := [0, _F1(y,z*x^(-y)), x^y*z*x^(-y)*_F1(y,z*x^(-y))*ln(x)+x^y*_F2(y,z*x^(-y))];

$\mathrm{S2}≔\left[0\,\mathrm{f\_\_1}\left(y\,z{x}^{-y}\right)\,x^{y}z{x}^{-y}\mathrm{f\_\_1}\left(y\,z{x}^{-y}\right)\ln \left(x\right)+x^y\mathrm{f\_\_2}\left(y\,z{x}^{-y}\right)\right]$

DeterminingPDE(ODE2, S2);

$\left\{0\right\}$

SymmetryTest(S2, ODE2);

$\left\{0\right\}$

ODE3 := diff(y(x), x,x) = x*y(x)*diff(y(x), x);

$\mathrm{ODE3}≔\mathrm{y\text{'}\text{'}}=xy\mathrm{y\text{'}}$

DeterminingPDE(ODE3, typeofsymmetry = evolutionary, notation = jetODE);

$\left\{{\left({\mathrm{\_\η}}_y\right)}_{x,x}=-{\left({\mathrm{\_\η}}_y\right)}_{\mathrm{\_y1},\mathrm{\_y1}}{x}^2{y}^2{\mathrm{\_y1}}^2-2{\left({\mathrm{\_\η}}_y\right)}_{\mathrm{\_y1},y}{\mathrm{\_y1}}^{2}xy-{\left({\mathrm{\_\η}}_y\right)}_{\mathrm{\_y1}}{\mathrm{\_y1}}^{2}x-2xy\mathrm{\_y1}{\left({\mathrm{\_\η}}_y\right)}_{\mathrm{\_y1},x}+{\mathrm{\_\η}}_y\left(x\,y\,\mathrm{\_y1}\right)x\mathrm{\_y1}-{\left({\mathrm{\_\η}}_y\right)}_{y,y}{\mathrm{\_y1}}^2-y{\left({\mathrm{\_\η}}_y\right)}_{\mathrm{\_y1}}\mathrm{\_y1}+{\left({\mathrm{\_\η}}_y\right)}_{x}xy-2\mathrm{\_y1}{\left({\mathrm{\_\η}}_y\right)}_{x,y}\right\}$

declare(u(x,y));

$u\left(x\,y\right)\mathrm{will\; now\; be\; displayed\; as}u$

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

DE := [U[x] - U[y] = 0, U[y,y] = 0];

$\mathrm{DE}≔\left[u_x-u_y=0\,u_{y,y}=0\right]$

detsys := DeterminingPDE(DE);

$\mathrm{detsys}≔\left\{{\left({\mathrm{\_\η}}_u\right)}_{u,u,u}=0\,{\left({\mathrm{\_\η}}_u\right)}_{u,u,y}=0\,{\left({\mathrm{\_\η}}_u\right)}_{y,y}=0\,{\left({\mathrm{\_\ξ}}_y\right)}_{u,u}=0\,{\left({\mathrm{\_\ξ}}_y\right)}_{u,y}=\frac{{\left({\mathrm{\_\η}}_u\right)}_{u,u}}{2}\,{\left({\mathrm{\_\ξ}}_y\right)}_{y,y}=2{\left({\mathrm{\_\η}}_u\right)}_{u,y}\,{\left({\mathrm{\_\η}}_u\right)}_x={\left({\mathrm{\_\η}}_u\right)}_y\,{\left({\mathrm{\_\ξ}}_x\right)}_u=0\,{\left({\mathrm{\_\ξ}}_x\right)}_y=0\,{\left({\mathrm{\_\ξ}}_y\right)}_x={\left({\mathrm{\_\ξ}}_y\right)}_y-{\left({\mathrm{\_\ξ}}_x\right)}_x\right\}$

pdsolve(detsys);

$\left\{{\mathrm{\_\η}}_u\left(x\,y\,u\right)=\frac{\mathrm{c\_\_3}{u}^2}{2}+\left(\left(x+y\right)\mathrm{c\_\_1}+\mathrm{c\_\_4}\right)u+\left(x+y\right)\mathrm{c\_\_2}+\mathrm{c\_\_5}\,{\mathrm{\_\ξ}}_x\left(x\,y\,u\right)=\mathrm{f\_\_1}\left(x\right)\,{\mathrm{\_\ξ}}_y\left(x\,y\,u\right)=-\mathrm{f\_\_1}\left(x\right)+\mathrm{c\_\_1}{x}^2+\frac{\left(4\mathrm{c\_\_1}y+\mathrm{c\_\_3}u+2\mathrm{c\_\_7}\right)x}{2}+\mathrm{c\_\_1}{y}^2+\frac{\left(\mathrm{c\_\_3}u+2\mathrm{c\_\_7}\right)y}{2}+\mathrm{c\_\_6}u+\mathrm{c\_\_8}\right\}$

closed_DE := PDEtools:-Library:-CloseSystem(DE, [u(x, y)]);

$\mathrm{closed\_DE}≔\left\{u_x-u_y\,u_{x,x}\,u_{x,y}\,u_{y,y}\right\}$

DeterminingPDE(closed_DE);

$\left\{{\left({\mathrm{\_\η}}_u\right)}_{u,u,u}=0\,{\left({\mathrm{\_\η}}_u\right)}_{u,u,y}=0\,{\left({\mathrm{\_\η}}_u\right)}_{y,y}=0\,{\left({\mathrm{\_\ξ}}_y\right)}_{u,u}=-{\left({\mathrm{\_\ξ}}_x\right)}_{u,u}\,{\left({\mathrm{\_\ξ}}_y\right)}_{u,y}=-{\left({\mathrm{\_\ξ}}_x\right)}_{u,y}+\frac{{\left({\mathrm{\_\η}}_u\right)}_{u,u}}{2}\,{\left({\mathrm{\_\ξ}}_y\right)}_{y,y}=2{\left({\mathrm{\_\η}}_u\right)}_{u,y}-{\left({\mathrm{\_\ξ}}_x\right)}_{y,y}\,{\left({\mathrm{\_\η}}_u\right)}_x={\left({\mathrm{\_\η}}_u\right)}_y\,{\left({\mathrm{\_\ξ}}_y\right)}_x={\left({\mathrm{\_\ξ}}_x\right)}_y+{\left({\mathrm{\_\ξ}}_y\right)}_y-{\left({\mathrm{\_\ξ}}_x\right)}_x\right\}$

DeterminingPDE(DE, closesystem);

$\left\{{\left({\mathrm{\_\η}}_u\right)}_{u,u,u}=0\,{\left({\mathrm{\_\η}}_u\right)}_{u,u,y}=0\,{\left({\mathrm{\_\η}}_u\right)}_{y,y}=0\,{\left({\mathrm{\_\ξ}}_y\right)}_{u,u}=-{\left({\mathrm{\_\ξ}}_x\right)}_{u,u}\,{\left({\mathrm{\_\ξ}}_y\right)}_{u,y}=-{\left({\mathrm{\_\ξ}}_x\right)}_{u,y}+\frac{{\left({\mathrm{\_\η}}_u\right)}_{u,u}}{2}\,{\left({\mathrm{\_\ξ}}_y\right)}_{y,y}=2{\left({\mathrm{\_\η}}_u\right)}_{u,y}-{\left({\mathrm{\_\ξ}}_x\right)}_{y,y}\,{\left({\mathrm{\_\η}}_u\right)}_x={\left({\mathrm{\_\η}}_u\right)}_y\,{\left({\mathrm{\_\ξ}}_y\right)}_x={\left({\mathrm{\_\ξ}}_x\right)}_y+{\left({\mathrm{\_\ξ}}_y\right)}_y-{\left({\mathrm{\_\ξ}}_x\right)}_x\right\}$

pdsolve((31));

$\left\{{\mathrm{\_\η}}_u\left(x\,y\,u\right)=\frac{\mathrm{c\_\_3}{u}^2}{2}+\left(\left(x+y\right)\mathrm{c\_\_1}+\mathrm{c\_\_4}\right)u+\left(x+y\right)\mathrm{c\_\_2}+\mathrm{c\_\_5}\,{\mathrm{\_\ξ}}_x\left(x\,y\,u\right)={\mathrm{\_\ξ}}_x\left(x\,y\,u\right)\,{\mathrm{\_\ξ}}_y\left(x\,y\,u\right)={\left(x+y\right)}^2\mathrm{c\_\_1}-{\mathrm{\_\ξ}}_x\left(x\,y\,u\right)+\frac{\left(\left(x+y\right)\mathrm{c\_\_3}+2\mathrm{c\_\_6}\right)u}{2}+\left(x+y\right)\mathrm{c\_\_7}+\mathrm{c\_\_8}\right\}$

Infinitesimals(DE, closesystem, dis = false);

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

$\left[{\mathrm{\_\ξ}}_x=\mathrm{f\_\_1}\left(x\,y\,u\right)\,{\mathrm{\_\ξ}}_y=-\mathrm{f\_\_1}\left(x\,y\,u\right)\,{\mathrm{\_\η}}_u=1\right],\left[{\mathrm{\_\ξ}}_x=\mathrm{f\_\_2}\left(x\,y\,u\right)\,{\mathrm{\_\ξ}}_y=-\mathrm{f\_\_2}\left(x\,y\,u\right)+1\,{\mathrm{\_\η}}_u=0\right],\left[{\mathrm{\_\ξ}}_x=\mathrm{f\_\_3}\left(x\,y\,u\right)\,{\mathrm{\_\ξ}}_y=-\mathrm{f\_\_3}\left(x\,y\,u\right)\,{\mathrm{\_\η}}_u=u\right],\left[{\mathrm{\_\ξ}}_x=\mathrm{f\_\_4}\left(x\,y\,u\right)\,{\mathrm{\_\ξ}}_y=-\mathrm{f\_\_4}\left(x\,y\,u\right)+x+y\,{\mathrm{\_\η}}_u=0\right],\left[{\mathrm{\_\ξ}}_x=\mathrm{f\_\_5}\left(x\,y\,u\right)\,{\mathrm{\_\ξ}}_y=-\mathrm{f\_\_5}\left(x\,y\,u\right)+u\,{\mathrm{\_\η}}_u=0\right],\left[{\mathrm{\_\ξ}}_x=\mathrm{f\_\_6}\left(x\,y\,u\right)\,{\mathrm{\_\ξ}}_y=-\mathrm{f\_\_6}\left(x\,y\,u\right)\,{\mathrm{\_\η}}_u=x+y\right],\left[{\mathrm{\_\ξ}}_x=\mathrm{f\_\_7}\left(x\,y\,u\right)\,{\mathrm{\_\ξ}}_y=-\mathrm{f\_\_7}\left(x\,y\,u\right)+\frac{\left(x+y\right)u}{2}\,{\mathrm{\_\η}}_u=\frac{u^2}{2}\right],\left[{\mathrm{\_\ξ}}_x=\mathrm{f\_\_8}\left(x\,y\,u\right)\,{\mathrm{\_\ξ}}_y=-\mathrm{f\_\_8}\left(x\,y\,u\right)+x^2+2xy+y^2\,{\mathrm{\_\η}}_u=\left(x+y\right)u\right]$

with(Physics);

$\left[\mathrm{`*`}\,\mathrm{`.`}\,\mathrm{Annihilation}\,\mathrm{AntiCommutator}\,\mathrm{Antisymmetrize}\,\mathrm{Assume}\,\mathrm{Bra}\,\mathrm{Bracket}\,\mathrm{Check}\,\mathrm{Christoffel}\,\mathrm{Coefficients}\,\mathrm{Commutator}\,\mathrm{CompactDisplay}\,\mathrm{Coordinates}\,\mathrm{Creation}\,\mathrm{D\_}\,\mathrm{Dagger}\,\mathrm{Decompose}\,\mathrm{Define}\,\mathrm{D\γ}\,\mathrm{DiracConjugate}\,\mathrm{Einstein}\,\mathrm{EnergyMomentum}\,\mathrm{Expand}\,\mathrm{ExteriorDerivative}\,\mathrm{Factor}\,\mathrm{FeynmanDiagrams}\,\mathrm{FeynmanIntegral}\,\mathrm{Fundiff}\,\mathrm{Geodesics}\,\mathrm{GrassmannParity}\,\mathrm{Gtaylor}\,\mathrm{Intc}\,\mathrm{Inverse}\,\mathrm{Ket}\,\mathrm{KillingVectors}\,\mathrm{KroneckerDelta}\,\mathrm{LagrangeEquations}\,\mathrm{LeviCivita}\,\mathrm{Library}\,\mathrm{LieBracket}\,\mathrm{LieDerivative}\,\mathrm{Normal}\,\mathrm{NumericalRelativity}\,\mathrm{Parameters}\,\mathrm{PerformOnAnticommutativeSystem}\,\mathrm{Projector}\,\mathrm{Psigma}\,\mathrm{Redefine}\,\mathrm{Ricci}\,\mathrm{Riemann}\,\mathrm{Setup}\,\mathrm{Simplify}\,\mathrm{SortProducts}\,\mathrm{SpaceTimeVector}\,\mathrm{StandardModel}\,\mathrm{Substitute}\,\mathrm{SubstituteTensor}\,\mathrm{SubstituteTensorIndices}\,\mathrm{SumOverRepeatedIndices}\,\mathrm{Symmetrize}\,\mathrm{TensorArray}\,\mathrm{Tetrads}\,\mathrm{ThreePlusOne}\,\mathrm{ToContravariant}\,\mathrm{ToCovariant}\,\mathrm{ToFieldComponents}\,\mathrm{ToSuperfields}\,\mathrm{Trace}\,\mathrm{TransformCoordinates}\,\mathrm{Vectors}\,\mathrm{Weyl}\,\mathrm{`^`}\,\mathrm{dAlembertian}\,\mathrm{d\_}\,\mathrm{diff}\,\mathrm{g\_}\,\mathrm{gamma\_}\right]$

Setup(anticommutativepre = {theta, Q});

$\mathrm{*\; Partial\; match\; of\; \text{'}}\mathrm{anticommutativepre}\mathrm{\text{'}\; against\; keyword\; \text{'}}\mathrm{anticommutativeprefix}\text{'}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\left[\mathrm{anticommutativeprefix}=\left\{Q\,\mathrm{\theta }\right\}\right]$

PDEtools:-declare(Q(x, y, theta[1], theta[2]));

$Q\left(x\,y\,{\mathrm{\theta }}_1\,{\mathrm{\theta }}_2\right)\mathrm{will\; now\; be\; displayed\; as}Q$

q := PDEtools:-diff_table(Q(x, y, theta[1], theta[2])):

pde[1] := q[x, y, theta[1]] + q[x, y, theta[2]] - q[y, theta[1], theta[2]] = 0;

${\mathrm{pde}}_1≔Q_{x,y,{\mathrm{\theta }}_1}+Q_{x,y,{\mathrm{\theta }}_2}-Q_{y,{\mathrm{\theta }}_1,{\mathrm{\theta }}_2}=0$

pde[2] := q[theta[1]] = 0;

${\mathrm{pde}}_2≔Q_{{\mathrm{\theta }}_1}=0$

Setup(anticommutativepre = {Xi, Rho}, additionally);

$\mathrm{*\; Partial\; match\; of\; \text{'}}\mathrm{anticommutativepre}\mathrm{\text{'}\; against\; keyword\; \text{'}}\mathrm{anticommutativeprefix}\text{'}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\left[\mathrm{anticommutativeprefix}=\left\{Q\,\mathrm{{\rm P}}\,\mathrm{\Xi }\,\mathrm{\theta }\right\}\right]$

S := [xi[1], xi[2], Xi[1], Xi[2], Eta](x, y, theta[1], theta[2]);

$S≔\left[{\mathrm{\xi }}_1\left(x\,y\,{\mathrm{\theta }}_1\,{\mathrm{\theta }}_2\right)\,{\mathrm{\xi }}_2\left(x\,y\,{\mathrm{\theta }}_1\,{\mathrm{\theta }}_2\right)\,{\mathrm{\Xi }}_1\left(x\,y\,{\mathrm{\theta }}_1\,{\mathrm{\theta }}_2\right)\,{\mathrm{\Xi }}_2\left(x\,y\,{\mathrm{\theta }}_1\,{\mathrm{\theta }}_2\right)\,\mathrm{{\rm H}}\left(x\,y\,{\mathrm{\theta }}_1\,{\mathrm{\theta }}_2\right)\right]$

PDEtools:-declare(S);

$\mathrm{{\rm H}}\left(x\,y\,{\mathrm{\theta }}_1\,{\mathrm{\theta }}_2\right)\mathrm{will\; now\; be\; displayed\; as}\mathrm{{\rm H}}$

$\mathrm{\Xi }\left(x\,y\,{\mathrm{\theta }}_1\,{\mathrm{\theta }}_2\right)\mathrm{will\; now\; be\; displayed\; as}\mathrm{\Xi }$

$\mathrm{\xi }\left(x\,y\,{\mathrm{\theta }}_1\,{\mathrm{\theta }}_2\right)\mathrm{will\; now\; be\; displayed\; as}\mathrm{\xi }$

InfinitesimalGenerator(S, Q(x,y,theta[1], theta[2]));

$f\mapsto {\mathrm{\xi }}_1\left(x\,y\,{\mathrm{\theta }}_1\,{\mathrm{\theta }}_2\right)\left(\frac{\ⅆf}{\ⅆx}\right)+{\mathrm{\xi }}_2\left(x\,y\,{\mathrm{\theta }}_1\,{\mathrm{\theta }}_2\right)\left(\frac{\ⅆf}{\ⅆy}\right)+{\mathrm{\Xi }}_1\left(x\,y\,{\mathrm{\theta }}_1\,{\mathrm{\theta }}_2\right)\left(\frac{\ⅆf}{\ⅆ{\mathrm{\theta }}_1}\right)+{\mathrm{\Xi }}_2\left(x\,y\,{\mathrm{\theta }}_1\,{\mathrm{\theta }}_2\right)\left(\frac{\ⅆf}{\ⅆ{\mathrm{\theta }}_2}\right)+\mathrm{{\rm H}}\left(x\,y\,{\mathrm{\theta }}_1\,{\mathrm{\theta }}_2\right)\left(\frac{\ⅆf}{\ⅆQ}\right)$

DeterminingPDE([pde[1], pde[2]], S);

$\left\{-{\mathrm{{\rm H}}}_{{\mathrm{\theta }}_1}=0\,-{\mathrm{{\rm H}}}_{x,y,{\mathrm{\theta }}_2}=0\,-{\left({\mathrm{\xi }}_2\right)}_{{\mathrm{\theta }}_1}=0\,-{\left({\mathrm{\xi }}_2\right)}_{{\mathrm{\theta }}_2}=0\,-{\left({\mathrm{\xi }}_1\right)}_{{\mathrm{\theta }}_1,{\mathrm{\theta }}_2}{\mathrm{\theta }}_1-{\left({\mathrm{\xi }}_1\right)}_{{\mathrm{\theta }}_2}=0\,-{\left({\mathrm{\xi }}_1\right)}_{{\mathrm{\theta }}_1}+{\left({\mathrm{\xi }}_1\right)}_{{\mathrm{\theta }}_1,{\mathrm{\theta }}_2}{\mathrm{\theta }}_2=0\,-{\mathrm{\theta }}_1{\left({\mathrm{\xi }}_1\right)}_{y,{\mathrm{\theta }}_1}-{\mathrm{\theta }}_2{\left({\mathrm{\xi }}_1\right)}_{y,{\mathrm{\theta }}_2}+{\left({\mathrm{\xi }}_1\right)}_y=0\,-{\mathrm{\theta }}_1{\left({\mathrm{\xi }}_2\right)}_{x,{\mathrm{\theta }}_1}-{\mathrm{\theta }}_2{\left({\mathrm{\xi }}_2\right)}_{x,{\mathrm{\theta }}_2}+{\left({\mathrm{\xi }}_2\right)}_x=0\,-{\mathrm{\theta }}_1{\left({\mathrm{\xi }}_1\right)}_{x,x,{\mathrm{\theta }}_1}-{\mathrm{\theta }}_2{\left({\mathrm{\xi }}_1\right)}_{x,x,{\mathrm{\theta }}_2}+{\left({\mathrm{\xi }}_1\right)}_{x,x}=0\,{\left({\mathrm{\Xi }}_1\right)}_x-{\left({\mathrm{\Xi }}_1\right)}_{x,{\mathrm{\theta }}_2}{\mathrm{\theta }}_2-{\left({\mathrm{\Xi }}_1\right)}_{x,{\mathrm{\theta }}_1}{\mathrm{\theta }}_1=0\,{\left({\mathrm{\Xi }}_2\right)}_x-{\left({\mathrm{\Xi }}_2\right)}_{x,{\mathrm{\theta }}_2}{\mathrm{\theta }}_2-{\left({\mathrm{\Xi }}_2\right)}_{x,{\mathrm{\theta }}_1}{\mathrm{\theta }}_1=0\,{\mathrm{{\rm H}}}_{{\mathrm{\theta }}_1,{\mathrm{\theta }}_2}=0\,{\left({\mathrm{\Xi }}_1\right)}_{{\mathrm{\theta }}_1,{\mathrm{\theta }}_2}=0\,{\left({\mathrm{\Xi }}_2\right)}_{{\mathrm{\theta }}_1,{\mathrm{\theta }}_2}=0\,{\left({\mathrm{\xi }}_1\right)}_{{\mathrm{\theta }}_1,{\mathrm{\theta }}_2}=0\,{\left({\mathrm{\xi }}_2\right)}_{{\mathrm{\theta }}_1,{\mathrm{\theta }}_2}=0\,{\left({\mathrm{\Xi }}_2\right)}_{x,{\mathrm{\theta }}_2}=0\,{\left({\mathrm{\Xi }}_2\right)}_{y,{\mathrm{\theta }}_2}=0\,{\left({\mathrm{\Xi }}_1\right)}_{{\mathrm{\theta }}_1}=-{\mathrm{\theta }}_1{\left({\mathrm{\xi }}_1\right)}_{x,{\mathrm{\theta }}_1}-{\mathrm{\theta }}_2{\left({\mathrm{\xi }}_1\right)}_{x,{\mathrm{\theta }}_2}+{\left({\mathrm{\xi }}_1\right)}_x\,{\left({\mathrm{\Xi }}_1\right)}_{{\mathrm{\theta }}_2}={\mathrm{\theta }}_1{\left({\mathrm{\xi }}_1\right)}_{x,{\mathrm{\theta }}_1}+{\mathrm{\theta }}_2{\left({\mathrm{\xi }}_1\right)}_{x,{\mathrm{\theta }}_2}-{\left({\mathrm{\xi }}_1\right)}_x+{\left({\mathrm{\Xi }}_2\right)}_{{\mathrm{\theta }}_2}\,{\left({\mathrm{\Xi }}_2\right)}_{{\mathrm{\theta }}_1}=0\right\}$

pdsolve((43));