Approximate derivative-dependent functional variable separation for quasi-linear diffusion equations with a weak source

Abstract

By using the approximate derivative-dependent functional variable separation approach, we study the quasi-linear diffusion equations with a weak source u t = (A(u)u x ) x + 𝜖B(u, u x ). A complete classification of these perturbed equations which admit approximate derivative-dependent functional separable solutions is listed. As a consequence, some approximate solutions to the resulting perturbed equations are constructed via examples.

New variable separation solutions for the generalized nonlinear diffusion equations

“…( 1) with perturbation was taken by way of approximate CLBS. [25] The layout of this paper is as follows. In Section 2, we review some basic definitions and fundamental theorems on the CLBS method and the invariant subspace (IS) method.…”

Third-order nonlinear differential operators preserving invariant subspaces of maximal dimension

“…There are various techniques to construct exact solutions of PDEs, which involve the Lie point symmetry, [1] the conditional symmetry, [2] the direct method, [3] the generalized conditional symmetry, [4][5][6] the nonlocal symmetry, [7] the signinvariant method, [8][9][10] the invariant subspace method, [11][12][13][14][15][16][17][18][19][20][21][22] and so on. [23][24][25][26][27] The invariant subspace method, introduced in Refs. [11] and [12], plays an important role in finding exact solutions of nonlinear PDEs.…”