1. Introduction

The focus here is on showing the equivalence of different approaches to finding the nonclassical determining equations for the partial differential equation (PDE),

In particular we address the problem posed by Hashemi and Nucci [4] who considered equations of the form (1.1): “We hope that an independent researcher will take up the task of comparing the two methods [of Bîlã and Niesen and Nucci] since we conjecture that Bîlã and Niesen’s method, and its extension, as given in [2], are equivalent to Nucci’s method.”

We consider the symmetry generator

and start by considering the case when the infinitesimal T(x, t, u) ≠ 0..

2. T ≠ 0

With T ≠ 0, the corresponding invariant surface condition (ISC) is given by

In the traditional approach, nonclassical symmetries of (1.1) are defined by

where Γ⁽²⁾ is the second prolongation of Γ, namely, where

Hence for nonclassical symmetries, we seek the invariance of the governing PDE subject to the PDE itself and the ISC (and its differential consequences). We note however that if f(x, t, u) is an arbitrary function, then the prolongation formula implies [f(x, t, u)Γ]⁽ⁿ⁾|_{Xux+Tut=U} = f(x, t, u)Γ⁽ⁿ⁾|_{Xux+Tut=U}. That is, by imposing the ISC we have [f(x, t, u)Γ]⁽ⁿ⁾|_{Xux+Tu_t=U} = f(x, t, u)Γ⁽ⁿ⁾. Hence if Γ is a nonclassical symmetry then f(x, t, u)Γ is also a nonclassical symmetry yielding the same invariant surface condition. This allows us to normalise any one of the nonzero coefficients of the vector field by setting it equal to one when finding nonclassical symmetries. Hence in the following, WLOG we set T = 1.

Applying (2.2) with T = 1 we get the condition

subject to u_t = K ∩ {Xu_x + u_t = U}.

We can further expand this as

subject to U − Xu_x = K, where we have used u_t = U − Xu_x and the definition of U_[t].

Bîlã and Niesen [1] use the approach

where the ISC u_t = ϕ/T′ − ξ/T′u_x has been substituted into the governing equation before taking the second prolongation. Treating ϕ and ξ as arbitrary functions of x, t, u means that (2.7) is equivalent to finding the determining equations for classical symmetries of the ordinary differential equation ϕ/T′ − ξ/T′u_x − K(x, t, u, u_x, u_xx) = 0. Then substituting ϕ = U, ξ = X, T′ = 1 leads to the determining equations for nonclassical symmetries of (1.1).

Hence Bîlã and Niesen essentially use the approach

This gives

subject to U − Xu_x = K, or

subject to U − Xu_x = K. The condition simplifies to

subject to U − Xu_x = K. As (2.6) and (2.11) are the same conditions they lead to the same nonclassical determining equations.

3. T= 0

When the infinitesimal symmetry T = 0 in (1.2), then as explained in the previous section, WLOG we can set X = 1. In the traditional approach we find the nonclassical determining equations using

where Γ0(2) is the second prolongation of Γ0=∂∂x+U(x,t,u)∂∂u, namely,

With T = 0, X = 1 we have U_[x] = D_xU, U_[t] = D_tU, U_[xx] = D_x(U_[x]).

Hence applying (3.1) we get the condition U_[t] − K_x −UK_u −U_[x]K_ux −U_[xx]K_uxx = 0, subject to u_t = K ∩ u_x = U.

We can further expand this as

subject to u_t = K, where we have used u_x = U and the definition of U_[x] and U_[xx].

In [2], Bruzón and Gandarias extend the method of Bîlã and Niesen to the case T = 0. They use the approach

where the ISC u_x = U′(x, t, u)/X′(x, t, u) has been substituted into the governing equation before taking the second prolongation. Treating U′ and X′ as arbitrary functions of x, t, u means that (3.4) is equivalent to finding the determining equations for classical symmetries of u_t = K(x, t, u, U′/X′,D_x(U′/X′)). Then substituting U′ = U, X′ = 1, leads to the determining equations for nonclassical symmetries of (1.1).

Hence Bruzón and Gandarias essentially use the approach

Letting z = U_x +U_uu_x(= u_xx), this gives

subject to u_t = K, or

subject to u_t = K. As (3.3) and (3.7) are the same conditions they lead to the same nonclassical determining equations.