SymmetryTest tests whether a symmetry, given as a list of infinitesimals S or as the corresponding infinitesimal generator differential operator, is a symmetry of a given PDE system PDESYS; if so, S satisfies the determining PDE for PDESYS.

If DepVars is not specified, SymmetryTest will consider all the differentiated unknown functions in PDESYS as unknown of the problems.

$\mathrm{with}\left(\mathrm{PDEtools}\right)\:$

$U≔\mathrm{diff\_table}\left(u\left(x\,y\,z\,t\right)\right)\:$

$\mathrm{declare}\left(U\left[\right]\right)$

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

$\mathrm{pde}\left[1\right]≔U\left[x,x\right]+U\left[y,y\right]+U\left[z,z\right]-U\left[t,t\right]=0$

${\mathrm{pde}}_1≔u_{x,x}+u_{y,y}+u_{z,z}-u_{t,t}=0$

$\mathrm{declare}\left(\left(\mathrm{\_\ξ}\,\mathrm{\_\η}\right)\left(x\,y\,z\,t\,u\right)\right)$

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

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

$S≔\mathrm{Infinitesimals}\left(\mathrm{pde}\left[1\right]\right)$

$S≔\left[{\mathrm{\_\ξ}}_x=0\,{\mathrm{\_\ξ}}_y=1\,{\mathrm{\_\ξ}}_z=0\,{\mathrm{\_\ξ}}_t=0\,{\mathrm{\_\η}}_u=0\right],\left[{\mathrm{\_\ξ}}_x=0\,{\mathrm{\_\ξ}}_y=0\,{\mathrm{\_\ξ}}_z=1\,{\mathrm{\_\ξ}}_t=0\,{\mathrm{\_\η}}_u=0\right],\left[{\mathrm{\_\ξ}}_x=0\,{\mathrm{\_\ξ}}_y=0\,{\mathrm{\_\ξ}}_z=0\,{\mathrm{\_\ξ}}_t=1\,{\mathrm{\_\η}}_u=0\right],\left[{\mathrm{\_\ξ}}_x=1\,{\mathrm{\_\ξ}}_y=0\,{\mathrm{\_\ξ}}_z=0\,{\mathrm{\_\ξ}}_t=0\,{\mathrm{\_\η}}_u=0\right],\left[{\mathrm{\_\ξ}}_x=0\,{\mathrm{\_\ξ}}_y=t\,{\mathrm{\_\ξ}}_z=0\,{\mathrm{\_\ξ}}_t=y\,{\mathrm{\_\η}}_u=0\right],\left[{\mathrm{\_\ξ}}_x=0\,{\mathrm{\_\ξ}}_y=0\,{\mathrm{\_\ξ}}_z=t\,{\mathrm{\_\ξ}}_t=z\,{\mathrm{\_\η}}_u=0\right],\left[{\mathrm{\_\ξ}}_x=t\,{\mathrm{\_\ξ}}_y=0\,{\mathrm{\_\ξ}}_z=0\,{\mathrm{\_\ξ}}_t=x\,{\mathrm{\_\η}}_u=0\right],\left[{\mathrm{\_\ξ}}_x=x\,{\mathrm{\_\ξ}}_y=y\,{\mathrm{\_\ξ}}_z=z\,{\mathrm{\_\ξ}}_t=t\,{\mathrm{\_\η}}_u=0\right],\left[{\mathrm{\_\ξ}}_x=0\,{\mathrm{\_\ξ}}_y=0\,{\mathrm{\_\ξ}}_z=0\,{\mathrm{\_\ξ}}_t=0\,{\mathrm{\_\η}}_u=u\right],\left[{\mathrm{\_\ξ}}_x=0\,{\mathrm{\_\ξ}}_y=z\,{\mathrm{\_\ξ}}_z=-y\,{\mathrm{\_\ξ}}_t=0\,{\mathrm{\_\η}}_u=0\right],\left[{\mathrm{\_\ξ}}_x=z\,{\mathrm{\_\ξ}}_y=0\,{\mathrm{\_\ξ}}_z=-x\,{\mathrm{\_\ξ}}_t=0\,{\mathrm{\_\η}}_u=0\right],\left[{\mathrm{\_\ξ}}_x=y\,{\mathrm{\_\ξ}}_y=-x\,{\mathrm{\_\ξ}}_z=0\,{\mathrm{\_\ξ}}_t=0\,{\mathrm{\_\η}}_u=0\right],\left[{\mathrm{\_\ξ}}_x=xz\,{\mathrm{\_\ξ}}_y=zy\,{\mathrm{\_\ξ}}_z=\frac{z^2}{2}+\frac{t^2}{2}-\frac{x^2}{2}-\frac{y^2}{2}\,{\mathrm{\_\ξ}}_t=zt\,{\mathrm{\_\η}}_u=-uz\right],\left[{\mathrm{\_\ξ}}_x=xy\,{\mathrm{\_\ξ}}_y=\frac{y^2}{2}+\frac{t^2}{2}-\frac{x^2}{2}-\frac{z^2}{2}\,{\mathrm{\_\ξ}}_z=zy\,{\mathrm{\_\ξ}}_t=yt\,{\mathrm{\_\η}}_u=-uy\right],\left[{\mathrm{\_\ξ}}_x=tx\,{\mathrm{\_\ξ}}_y=yt\,{\mathrm{\_\ξ}}_z=zt\,{\mathrm{\_\ξ}}_t=\frac{t^2}{2}+\frac{x^2}{2}+\frac{y^2}{2}+\frac{z^2}{2}\,{\mathrm{\_\η}}_u=-ut\right],\left[{\mathrm{\_\ξ}}_x=-\frac{x^2}{2}-\frac{t^2}{2}+\frac{y^2}{2}+\frac{z^2}{2}\,{\mathrm{\_\ξ}}_y=-xy\,{\mathrm{\_\ξ}}_z=-xz\,{\mathrm{\_\ξ}}_t=-tx\,{\mathrm{\_\η}}_u=ux\right]$

$\mathrm{SymmetryTest}\left(S\left[1\right]\,\mathrm{pde}\left[1\right]\right)$

$\left\{0\right\}$

$\mathrm{map}\left(\mathrm{SymmetryTest}\,\left[S\right]\,\mathrm{pde}\left[1\right]\right)$

$\left[\left\{0\right\}\,\left\{0\right\}\,\left\{0\right\}\,\left\{0\right\}\,\left\{0\right\}\,\left\{0\right\}\,\left\{0\right\}\,\left\{0\right\}\,\left\{0\right\}\,\left\{0\right\}\,\left\{0\right\}\,\left\{0\right\}\,\left\{0\right\}\,\left\{0\right\}\,\left\{0\right\}\,\left\{0\right\}\right]$

$\mathrm{declare}\left(u\left(x\,y\right)\right)$

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

$\mathrm{SGE}≔\mathrm{diff}\left(u\left(x\,y\right)\,x\,y\right)=\sin \left(u\left(x\,y\right)\right)$

$\mathrm{SGE}≔u_{x,y}=\sin \left(u\right)$

$S≔\left[\mathrm{\_\ξ}\left[1\right]=0\,\mathrm{\_\ξ}\left[2\right]=0\,\mathrm{\_\η}\left[1\right]=u\left[1,1,1\right]+\frac{1}{2}{u\left[1\right]}^3\right]$

$S≔\left[{\mathrm{\_\ξ}}_1=0\,{\mathrm{\_\ξ}}_2=0\,{\mathrm{\_\η}}_1=u_{1,1,1}+\frac{u_1^3}{2}\right]$

$\mathrm{FromJet}\left(S\,u\left(x\,y\right)\right)$

$\left[{\mathrm{\_\ξ}}_1=0\,{\mathrm{\_\ξ}}_2=0\,{\mathrm{\_\η}}_1=u_{x,x,x}+\frac{{u_x}^3}{2}\right]$

$\mathrm{SymmetryTest}\left(S\,\mathrm{SGE}\right)$

$\left\{0\right\}$