A singular pertubation theory for reaction-diffusion equations

A pertubation theory is developed for the probability density for the displacement in reaction-diffusion equations of the form ∂p/∂τ = ε{lunate} (∂/∂y) [f(y)∂p/∂y] - (∂/∂y) [v(y)p] - κ(yp. In this equation f(y), v(y) and κ(y) are dimensionless functions of y taken to be O(1), and ε{lunate} is a dimensionless parameter which, in the diffusion-dominated regime satisfies ε{lunate} ≪ 1. We briefly also discuss the case in which v(y) is also proportional to ε{lunate}. Our results are then applied to an exactly solvable example.