Sequential eigenfunction expansion for a problem in combustion theory

A method of sequential eigenfunction expansion is developed for a semi-linear parabolic equation. It allows the time-dependent coefficients of the eigenfunctions to be determined sequentially and iterated to reach convergence. At any stage, only a single ordinary differential equation needs to be considered, in contrast to the Galerkin method which requires the consideration of a system of equations. The method is applied to a central problem in combustion theory to provide a definitive answer to the question of the influence of the initial data in determining whether the solution is sub- or super-critical, in the case of multiple steady-state solutions. It is expected this method will prove useful in dealing with similar problems.