This paper introduces a new type of symmetry reductions called extended nonclassical symmetries that can be studied for parameter identification problems described by partial differential equations. Including the data function in the parameter space, we show that specific data and parameter classes that lead to a reduced dimension model can be found. More exactly, since the extended nonclassica...

Reduction operators, i.e., the operators of nonclassical (or conditional) symmetry of a class of variable coefficient nonlinear wave equations with power nonlinearities, are investigated within the framework of a singular reduction operator. A classification of regular reduction operators is performed with respect to generalized extended equivalence groups. Exact solutions of some nonlinear wav...

In this paper we study symmetry reductions and exact solutions of the shallow water wave (SWW) equation uxxxt + αuxuxt + βutuxx − uxt − uxx = 0, (1) where α and β are arbitrary, nonzero, constants, which is derivable using the so-called Boussinesq approximation. Two special cases of this equation, or the equivalent nonlocal equation obtained by setting ux = U , have been discussed in the litera...

The reduction operators, i.e., the operators of nonclassical (conditional) symmetry, of (1 + 1)dimensional second order linear parabolic partial differential equations and all the possible reductions of these equations to ordinary differential ones are exhaustively described. This problem proves to be equivalent, in some sense, to solving the initial equations. The “no-go” result is extended to...

A review of the role of symmetries in solving differential equations is presented. After showing some recent results on the application of classical Lie point symmetries to problems in fluid draining, meteorology, and epidemiology of AIDS, the nonclassical symmetries method is presented. Finally, it is shown that iterations of the nonclassical symmetries method yield new nonlinear equations, wh...

We consider a family of exact solutions to nonlinear reaction–diffusion model, constructed using nonclassical symmetry analysis. In particular limit, the mathematical model approaches well-known Fisher–KPP which means that it is related various applications including cancer progression, wound healing and ecological invasion. The solution mathematically interesting since are rare, often restrict...

Lie group analysis is applied to a mathematical model for thin liquid films, namely nonlinear fourth order partial differential equation in two independent variables. A three-dimensional symmetry algebra found and reductions ordinary equations are obtained by using its one-dimensional subalgebras. Two of these studied the reduction method Jacobi last multiplier method, be linearizable. Furtherm...