By means of the modified CK’s direct method, we give out the relationship between variable coefficients of the fifth-order KdV equation and the corresponding constant coefficient ones. At the same time, we have studied the generalized fifth-order KdV equation with constants coefficients using the Lie symmetry group methods. By applying the nonclassical symmetry method we found that the analyzed...

in this paper, heir-equations method is applied to investigate nonclassical symmetries and new solutions of the black-scholes equation. nonlinear self-adjointness is proved and infinite number of conservation laws are computed by a new conservation laws theorem.

In this paper we study a shallow water equation derivable using the Boussinesq approximation, which includes as two special cases, one equation discussed by Ablowitz et al [Stud. Appl. Math., 53 (1974) 249–315] and one by Hirota and Satsuma [J. Phys. Soc. Japan, 40 (1976) 611–612]. A catalogue of classical and nonclassical symmetry reductions, and a Painlevé analysis, are given. Of particular i...

The notion of singular reduction operators, i.e., of singular operators of nonclassical (conditional) symmetry, of partial differential equations in two independent variables is introduced. All possible reductions of these equations to first-order ODEs are are exhaustively described. As examples, properties of singular reduction operators of (1+ 1)-dimensional evolution and wave equations are s...

The symmetry group method is important and widely used in the reduction and construction of explicit solutions of PDEs (partial differential equations). As shown in [1 – 4], the Lie symmetry group method can be used to find explicit solutions of PDEs using symmetry reduction and construction. This method is known as the classical method. By a classical symmetry group of a system of PDEs we mean...

In this paper, by using the nonclassical method, several new symmetries and solutions are obtained, which are unobtainable by Lie classical symmetries.