Nonclassical symmetry solutions and the methods of Bluman–Cole and Clarkson–Kruskal

Nonclassical symmetry solutions of physically relevant partial differential equations are considered via the reduction methods of Bluman and Cole and Clarkson and Kruskal. Consistency conditions will be provided to show that, if satisfied, these two methods are equivalent in the sense that they lead to the same symmetry solutions. The Boussinesq equation and Burgers’ equation are used as illustrative examples. Exact solutions, one of which is new, will be presented for Burgers’ equation obtained from the Bluman and Cole method, yet not obtainable by Clarkson and Kruskal’s method.