Multiplicity of solutions of a nonlinear boundary value problem arising in combustion theory : a thesis presented to Massey University of Palmerston North in partial fulfilment of the thesis requirement for the degree of Master of Science in Mathematics

Abstract

The problem of self-heating in spherical and spherically annular domains is addressed in this thesis. In particular, the Frank-Kamenetskii model is used to investigate the multiplicity of steady state solutions in these geometries. The differential equations describing this model depend crucially on a parameter, the "Frank-Kamenetskii" parameter; for spherical geometries it is known that: (a) a unique solution exists for sufficiently small parameter values, (b) there is a value of the parameter such that an infinite number of solutions exist. A convergent infinite series solution is developed for the problem in a spherical domain. The multiplicity of solutions when the problem is posed in spherically annular domains is then explored. It is shown, in contrast to (b), that multiple solutions exist for arbitrarily small parameter values and that no value of the parameter produces infinite multiplicity.