New variable separation solutions for the generalized nonlinear diffusion equations**Project supported by the National Natural Science Foundation of China (Grant Nos. 11371293, 11401458, and 11501438), the National Natural Science Foundation of China, Tian Yuan Special Foundation (Grant No. 11426169), and the Natural Science Basic Research Plan in Shaanxi Province of China (Grant No. 2015JQ1014).

Abstract

The functionally generalized variable separation of the generalized nonlinear diffusion equations ut = A(u,ux)uxx + B(u,ux) is studied by using the conditional Lie–Bäcklund symmetry method. The variant forms of the considered equations, which admit the corresponding conditional Lie–Bäcklund symmetries, are characterized. To construct functionally generalized separable solutions, several concrete examples defined on the exponential and trigonometric invariant subspaces are provided.