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| (1) |
Consider a PDE problem, for example PDESYS, with two independent variables and one dependent variable, , and consider the list of infinitesimals of a symmetry group assumed to be admitted by PDESYS
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In the input above you can also enter the symmetry without infinitesimals' labels, as in . The corresponding infinitesimal generator is
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| (3) |
We say that PDESYS is invariant under the transformations generated by G in that G(PDESYS) = 0 were in this formula G represents the prolongation necessary to act on PDESYS (see InfinitesimalGenerator). The similarity transformation relating the original variables to new variables - say , that reduces by one the number of independent variables of a PDE system invariant under G above is obtained via
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| (4) |
Note these transformation sets are returned with , making explicit that the unknown of the problem you obtain when you change variables does not depend on s.
To express these transformations using jet notation use
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| (5) |
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| (6) |
That this transformation TR reduces the number of independent variables of any PDE system invariant under G above is visible in the fact that it transforms the given infinitesimals (for ) into (for ). To verify this you can use ChangeSymmetry
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| (7) |
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| (8) |
So to this list of infinitesimals corresponds, written in terms of , this infinitesimal generator
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Any PDESYS invariant under G will also be invariant under the operator above, that is, PDESYS will be independent of r after you change variables in it using TR computed with SimilarityTransformation lines above.
If the new variables, here , are not indicated, variables and _phi[k] prefixed by an underscore _ to represent the new variables are introduced
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| (10) |