Optimal System and Group Invariant Solutions for a Nonlinear Wave Equation

G.Maluleke, D.P.Mason (School of Computational and Applied Mathematics, University of the Witwatersrand, Johannesburg, South Africa )

e-mail: gmalulek@cam.wits.ac.za

We consider a mathematical model proposed by Scott and Stevenson [1984 and 1986] and independently by McKenzie [1984] to explain how the segregation or migration occurs. The mathematical model is one-dimensional and it is in the form of a third order nonlinear partial differential equation describing the two-phase fluid flow of a medium compacting under gravity. The partial differential equation contains two parameters, n and m, which are exponents in power laws relating the permeability of the medium and the viscosity of the solid matrix to the voidage.  We derive the one dimensional optimal system of the partial differential equation by using techniques developed by Olver [1986], Coggeshall and Meyer-ter-vehn [1992] and Chupakhin [1994]. The one dimensional optimal system is then used to classify group invariant solutions of the differential equation.


Section : 12