#### Year

1997

#### Degree Name

Doctor of Philosophy

#### Department

School of Mathematics and Applied Statistics

#### Recommended Citation

Goard, J. M., Symmetry and ad-hoc methods for solving nonlinear partial differential equations, Doctor of Philosophy thesis, School of Mathematics and Applied Statistics, University of Wollongong, 1997. http://ro.uow.edu.au/theses/2060

#### Abstract

In this thesis, we establish the range of applicability of Lie's classical symmetry method, and various of its generalisations, in constucting new exact solutions to topical nonlinear partial differential equations, including reaction-diffusion equations, boundary layer equations and the poorly understood degenerate nonlinear diffusion equations. In addition, some established ad-hoc methods of nonlinear superposition and equation splitting are related to properties of Lie symmetry algebras. These equation solving methods are thereby incorporated in systematic symmetry-finding algorithms.

Lie symmetry analysis of degenerate diffusion equations, in which the diffusivity depends on both concentration and concentration gradient, uncovers an interesting class of integrable equations. Using both linear transform methods and separation of variables, these nonlinear equations are solved subject to initial and boundary conditions on both infinite and finite domains. Solutions on the finite domain evolve towards a discontinuous jump. Also, relevant exact solutions are constructed for nonintegrable models. These solutions include the possibility of strong degeneracy with a step initial condition remaining discontinuous for a finite time.

Every solution of a linear equation with constant coefficients is invariant under some *classical* symmetry. Strictly nonclassical symmetries are rare for nonlinear diffusion equations, a fact which is made evident by proving the equivalence of the nonclassical symmetry determining equations to their classical counterparts. However, new nonclassical solutions are constructed for 2+1-dimensional reaction-diffusion equations, including a case with an Arrhenius reaction term. The higher order symmetry method of generalised conditional symmetries is carried out on a class of degenerate diffusion equations with and without reaction terms.

Recently discovered new solutions to the boundary layer equations are shown to follow from classical symmetry reduction of a larger system of governing partial differential equations (PDEs). This leads to a consideration ofthe classical method of equation splitting and to the more general question of compatibility of differential constraints or side conditions. It is shown that the construction of classically invariant explicit solutions can be obtained by splitting, even when the ordinary differential equation, obtained by symmetry reduction, is intractable.

The simple observation that a nonlinear superposition principle (NLSP) is itself a symmetry of a two-equation system, has led us to relate the method of superposition principles to the structure of the Lie symmetry algebra of the PDE . In so doing, we discover the full class of second order PDEs of two independent variables with a Lie group of NLSPs. This result is then used to solve two related fluid flow problems in scale heterogeneous unsaturated media.