Exact Solutions Of The Porous Medium Equation Under The Generalized Conditional Symmetry And Invariant Sets

The integrable nonlinear partial differential equations are of current interest in both physics and mathematics. This paper studies one type of the integrable nonlinear partial differential equations: the porous medium equationwhere D(u) is the diffusion coefficient, P(u) and Q(u) are respectively theconvection and source terms. They are smooth functions of the indicated variable. To date, several well-developed methods relating to the symmetry group have been used to construct exact solutions of nonlinear differential equations, these include the classical symmetry group method, the non-classical symmetry group method, the generalized conditional symmetry method, the direct method, the differential constraint method, the sign-invariant and invariant subspace approach, invariant sets method, etc. The motivation of this paper is firstly to use the generalized conditional symmetry method to discuss exact solutions of the porous medium equation whendiffusion coefficient D(u) take two forms of u~m and e~u, variable coefficients take the form of f(x) = 1, g(x)≠0,1, h(x)≠0,1, q(x)≠0,1; secondly, from four cases:scaling invariant set, extended scaling invariant set, rotation invariant set and extended rotation invariant set, to use invariant sets method to discuss exact solutions of the porous medium equation, when variable coefficients take the form off(x) = g(x) = h(x) = q(x). These exact solutions are very helpful to explain and illuminate physical problems.