Partially nonclassical method and conformal invariance

Abstract

A generalization of the classical and nonclassical methods for finding similarity reductions of a partial differential equation (PDE), termed as the “partially nonclassical method”, is developed. Applying the partially nonclassical method to the flat steady-state boundary layer (BL) equations shows that the partially nonclassical method enables one to obtain similarity reductions obtainable neither classical nor nonclassical methods. Based on the observation, that applying the classical Lie group method to a PDE yields transformations which do not leave the PDE invariant but modify it by a conformal factor, a unified representation of the classical, nonclassical and partially nonclassical methods for finding similarity reductions of PDEs is given. That representation may lead to further generalizations of the nonclassical method.A generalization of the classical and nonclassical methods for finding similarity reductions of a partial differential equation (PDE), termed as the “partially nonclassical method”, is developed. Applying the partially nonclassical method to the flat steady-state boundary layer (BL) equations shows that the partially nonclassical method enables one to obtain similarity reductions obtainable neither classical nor nonclassical methods. Based on the observation, that applying the classical Lie group method to a PDE yields transformations which do not leave the PDE invariant but modify it by a conformal factor, a unified representation of the classical, nonclassical and partially nonclassical methods for finding similarity reductions of PDEs is given. That representation may lead to further generalizations of the nonclassical method.