Given the Solution to a problem, as an equation or a set (list) of equations, or given a set (list) of these solutions, and given the infinitesimal generators of a symmetry S, or a list of them, SymmetrySolutions computes other solutions obtained by exploring the given symmetries. This command complements Infinitesimals and InvariantSolutions, which respectively compute the symmetry infinitesimals and the group (symmetry) invariant solutions of a PDE system.

The dependent variables of the problem do not need to be indicated: SymmetrySolutions will automatically take the left-hand-sides of the given solutions as dependent variables.

What SymmetrySolutions does can be decomposed into two steps that can be reproduced interactively:

compute the finite form of the transformation associated to all the given symmetries (can be performed using SymmetryTransformation);

apply this transformation to each of the given solutions (can be performed using dchange).

You can optionally specify a simplifier to be applied to the result using the optional argument simplifier = ...

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.

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

$\mathrm{pde}\left[1\right]≔\mathrm{diff}\left(u\left(x\,y\,t\right)\,x\,x\right)+\mathrm{diff}\left(u\left(x\,y\,t\right)\,y\,y\right)=0$

${\mathrm{pde}}_1≔\frac{{\partial }^2}{\partial x^2}u\left(x\,y\,t\right)+\frac{{\partial }^2}{\partial y^2}u\left(x\,y\,t\right)=0$

$\mathrm{sol}\left[1\right]≔\mathrm{pdsolve}\left(\mathrm{pde}\left[1\right]\,\mathrm{HINT}=\mathrm{`*`}\,\mathrm{build}\right)$

${\mathrm{sol}}_1≔u\left(x\,y\,t\right)=\mathrm{f\_\_4}\left(t\right)\mathrm{c\_\_1}{\ⅇ}^{\sqrt{{\mathrm{\_c}}_1}x}\mathrm{c\_\_3}\sin \left(\sqrt{{\mathrm{\_c}}_1}y\right)+\mathrm{f\_\_4}\left(t\right)\mathrm{c\_\_1}{\ⅇ}^{\sqrt{{\mathrm{\_c}}_1}x}\mathrm{c\_\_4}\cos \left(\sqrt{{\mathrm{\_c}}_1}y\right)+\frac{\mathrm{f\_\_4}\left(t\right)\mathrm{c\_\_2}\mathrm{c\_\_3}\sin \left(\sqrt{{\mathrm{\_c}}_1}y\right)}{{\ⅇ}^{\sqrt{{\mathrm{\_c}}_1}x}}+\frac{\mathrm{f\_\_4}\left(t\right)\mathrm{c\_\_2}\mathrm{c\_\_4}\cos \left(\sqrt{{\mathrm{\_c}}_1}y\right)}{{\ⅇ}^{\sqrt{{\mathrm{\_c}}_1}x}}$

$S\left[1\right]≔\left[\mathrm{Infinitesimals}\left(\mathrm{pde}\left[1\right]\,\mathrm{typeofsymmetry}=\mathrm{polynomial}\,\mathrm{displayfunctionality}=\mathrm{false}\right)\left[-2..-1\right]\right]$

$S_1≔\left[\left[{\mathrm{\_\ξ}}_x=\frac{x^2}{2}-\frac{y^2}{2}\,{\mathrm{\_\ξ}}_y=xy\,{\mathrm{\_\ξ}}_t=0\,{\mathrm{\_\η}}_u=0\right]\,\left[{\mathrm{\_\ξ}}_x=xy\,{\mathrm{\_\ξ}}_y=-\frac{x^2}{2}+\frac{y^2}{2}\,{\mathrm{\_\ξ}}_t=0\,{\mathrm{\_\η}}_u=0\right]\right]$

$\mathrm{newsol}\left[1\right]≔\mathrm{SymmetrySolutions}\left(\mathrm{sol}\left[1\right]\,S\left[1\right]\right)$

${\mathrm{newsol}}_1≔\left\{u\left(x\,y\,t\right)=\mathrm{f\_\_4}\left(t\right)\left({\ⅇ}^{\frac{4\left(\left(x^2+y^2\right)\mathrm{\_\ε}+2x\right)\sqrt{{\mathrm{\_c}}_1}}{4+\left(x^2+y^2\right){\mathrm{\_\ε}}^2+4\mathrm{\_\ε}x}}\mathrm{c\_\_1}+\mathrm{c\_\_2}\right)\left(\cos \left(\frac{4\sqrt{{\mathrm{\_c}}_1}y}{4+\left(x^2+y^2\right){\mathrm{\_\ε}}^2+4\mathrm{\_\ε}x}\right)\mathrm{c\_\_4}+\sin \left(\frac{4\sqrt{{\mathrm{\_c}}_1}y}{4+\left(x^2+y^2\right){\mathrm{\_\ε}}^2+4\mathrm{\_\ε}x}\right)\mathrm{c\_\_3}\right){\ⅇ}^{-\frac{2\left(\left(x^2+y^2\right)\mathrm{\_\ε}+2x\right)\sqrt{{\mathrm{\_c}}_1}}{4+\left(x^2+y^2\right){\mathrm{\_\ε}}^2+4\mathrm{\_\ε}x}}\right\},\left\{u\left(x\,y\,t\right)=\mathrm{f\_\_4}\left(t\right)\left(\cos \left(\frac{2\sqrt{{\mathrm{\_c}}_1}\left(\left(x^2+y^2\right)\mathrm{\_\ε}+2y\right)}{4+\left(x^2+y^2\right){\mathrm{\_\ε}}^2+4y\mathrm{\_\ε}}\right)\mathrm{c\_\_4}+\sin \left(\frac{2\sqrt{{\mathrm{\_c}}_1}\left(\left(x^2+y^2\right)\mathrm{\_\ε}+2y\right)}{4+\left(x^2+y^2\right){\mathrm{\_\ε}}^2+4y\mathrm{\_\ε}}\right)\mathrm{c\_\_3}\right)\left({\ⅇ}^{\frac{8\sqrt{{\mathrm{\_c}}_1}x}{4+\left(x^2+y^2\right){\mathrm{\_\ε}}^2+4y\mathrm{\_\ε}}}\mathrm{c\_\_1}+\mathrm{c\_\_2}\right){\ⅇ}^{-\frac{4\sqrt{{\mathrm{\_c}}_1}x}{4+\left(x^2+y^2\right){\mathrm{\_\ε}}^2+4y\mathrm{\_\ε}}}\right\}$

$\mathrm{map}\left(\mathrm{pdetest}\,\left[\mathrm{newsol}\left[1\right]\right]\,\mathrm{pde}\left[1\right]\right)$

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