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By Lisle I.G.

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43) The auxiliary system A is ˆ is The most general transformation τˆ in the equivalence group Q v = x = t = u = a = λ (αv + βx ) + κ0 ρ λ (γv + δx ) + κ1 ρ λ2 t + κ2 ρ αu + β , αδ − βγ = ±1 γu + δ ρ−1 (γu + δ)2 a , λ = 0, ρ > 0. 44) This group acts on diffusivity functions by D (u ) = ρ αu + β , D 2 (γu + δ) γu + δ αδ − βγ = ±1. 44) has eight independent parameters, only four (indeNote that although Q pendent) parameters affect the diffusivity D(u). 5. 3. 46) These operators were found by Akhatov, et al.

We follow the convention that an object (such as a transformation) on augmented space is ‘hatted’ (ˆ τ ); its projection to a corresponding object on base space is unhatted (τ ). 1. 20) α, ρ = 0 maps a nonlinear diffusion equation ut = (D(u)ux )x to a diffusion equation ut = (D (u )ux )x , where D (u ) = ρ2 D(αu + β). 20) and a = ρ−2 a . In this section we give a theory for equivalence transformations of the type described by Ovsiannikov [52]. In the next section we give the corresponding infinitesimal form of the theory.

F s ), where u = (u1 , u2 , . . , um ) are dependent variables, x = (x1 , x2 , . . , xn ) are independent variables, and (x, u) ≡ w. (Note that m + n = ν). ), we obtain the system 1 1 E(φ) f x, u, u, . . , u, φ(x, u), φ(x, u), . . ’s for u as a function of x. 7) represents a class C of differential equations, namely C = {E(φ) | φ solves A}. 9) We call the functions φ(x, u) solving A the arbitrary elements characterizing the class C. The system A we call the auxiliary system of the class; E will be called the primary system.

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Equivalence Transformations for Classes of Differential Equations by Lisle I.G.

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