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Consider a PDE problem with two independent variables and one dependent variable, , and consider the list of infinitesimals of a symmetry group.
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In the input above you can also enter the symmetry without labels for the infinitesimals, as in , or use the corresponding infinitesimal generator, which prolonged to order 1 (that is, ready to act on functions depending on x, t, u and partial derivatives of u(x,t) of order 1 at most) is
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The invariants for this symmetry are
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Therefore, applying G to each of the Phi you obtain zero:
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By default Invariants computes differential invariants of order 1, so that they depend on up to 1st order derivatives of (see above) - you can change that using the optional argument order
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It is possible to request the output to be in function notation or jetnumbers notation instead of the default jetvariables jet notation, for that purpose use the optional argument jetnotation = ...
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Alternatively, you can switch back and forth between function notation and jet notation using FromJet and ToJet
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| (9) |
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An example of a symmetry group of dimension greater than one for a problem with two independent and two dependent variables
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So consider the following five lists of infinitesimals, each one associated to a different symmetry transformation of some PDE problem
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There are three expressions simultaneously invariant under each of the five symmetry generators associated to the infinitesimals in L above; these invariants are
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To verify this result, construct first the infinitesimal generators, associated to each of the infinitesimals in L, prolonged to order 1
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Test them on the expressions in Phi (see map to map operators over lists of expressions)
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| (15) |
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