The method of generalised conditional symmetries and its various implementations

Abstract

We relate Kaptsov's method of B-determining equations for nding invariant solutions of PDEs to the nonclassical method and to the method of generalised conditional symmetries. An extension of Kaptsov's method is then used to nd new solutions of degenerate diusion equations. Disciplines Physical Sciences and Mathematics Publication DetailsThis article was originally published as Goard, JM, The method of generalised conditional symmetries and its various implementations, European Journal of Applied Mathematics, 11, 2000, 215-224 We relate Kaptsov's method of B-determining equations for finding invariant solutions of PDEs to the nonclassical method and to the method of generalised conditional symmetries. An extension of Kaptsov's method is then used to find new solutions of degenerate diffusion equations.

Separation of variables and exact solutions to nonlinear diffusion equations with x-dependent convection and absorption

“…Equation (30) with (39) has the product separable solution (7) (2) When c = 0, then φ − c 4 φ ln|φ| = λφ, c 1 ψ + c 2 ψ + c 3 ψ + c 4 ψ ln|ψ| = λψ.…”

“…In the case that Q(x, u) and P (x, u) are independent of x, a number of methods relating to symmetry group have been developed successfully to find exact solutions and symmetry reductions of various special forms of (1). These methods are the Lie-point symmetry method [7,17], the nonclassical method [5,8,10,13,36,37], the direct method [9,12,21], the generalized conditional symmetry (GCS) method [11,15,16,[22][23][24][25][26][27][28][29][30][31][32][33][34][35]39], the sign-invariant and invariantsubspace methods [18][19][20] and Painlevé analysis [6,14]. Many exciting results have been derived.…”

Symmetries and solutions to the thin film equations

“…It has been known that the generalized conditional symmetry approach is very efficient in seeking symmetry reductions and exact solutions of nonlinear diffusion equations with convection and source terms [21][22][23][24][25][32][33][34][35][36][37][38]. In this section, we show that Eq.…”

“…Further extensions to scaling and rotation groups were introduced by Qu and Estevez [31]. The generalized conditional symmetries for second-order nonlinear parabolic equations have been studied by several authors, see [21][22][23][24][25][32][33][34][35][36][37][38].…”

Conditional Lie Bäcklund Symmetries and Sign‐Invariants to Quasi‐Linear Diffusion Equations

“…In the case of n = 1, Equation (1) has been studied extensively from many different points of view and a huge number of results on symmetries, exact solutions and qualitative analysis including existence, uniqueness, blow up, extinction, large time behavior and geometrical properties have been obtained (see [4][5][6][7] and references therein). It has been known that the approaches of sign-invariant [8][9][10][11] and conditional Lie Bäcklund symmetry [13][14][15][16][17][18][19][20][21][22][23][24][25][26] have been successfully applied to obtain solutions and to explore the properties of Equation (1) with n = 1. So it is of great interest to develop both methods to study (1) with n = 1.…”