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Extra info for Applied math. Part 1: Probability
Secondly, we consider the situation for nonlinear PDEs. 72) where L is a linear differential operator. 72) has a nontrivial local symmetry X = η µ (x, u, ∂u, . . , ∂ s u) ∂ . 73) if and only if η satisfies the local symmetry determining equations = 0. 75) in terms of some linear differential operator R. 76) R = r(x) + ri (x)Di + · · · + ri1 ···ik (x)Di1 . . 77) with for some functions r(x), ri (x), . . , ri1 ···ik (x). 78) and R is a matrix differential operator with matrix elements α i1 ···ik α αi Rα (x)Di1 .
M. 23b). 24) with X = ξ i (x, u) ∂ ∂ ∂ ∂ + η µ (x, u) µ = ξ i (x∗ , u∗ ) + η µ (x∗ , u∗ ) . 24), one then has u = g(eεX x, Θ(eεX x); −ε) = g(f (x, u; ε), Θ(f (x, u; ε)); −ε). 4. 21). 25) implicitly defines a mapping of the family of surfaces uµ = Θµ (x) into a one-parameter family of surfaces uµ = φµ (x; ε). In order to effectively generalize one-parameter Lie groups of point or contact transformations to one-parameter higher-order transformations, it is important to consider the mapping of surfaces from the point of view of transformations acting directly on the space of functions u = u(x) instead of transformations acting on (x, u)-space (or (x, u, ∂u)-space in the case of contact transformations).
S, σ = 1, . . , N. , transformations that preserve the differential structure of the equations in the PDE system but may change the form of the constitutive functions and/or parameters. In particular, the consideration of equivalence transformations is useful in analyses that involve classifications with respect to constitutive functions and/or parameters, such as local symmetry and local conservation law analysis. , only for forms of constitutive functions and/or parameters that are not related by an equivalence transformation.