Determining the kinematic properties of an advancing front using a decomposition method

In applications where partial differential equations are used to model populations, there is frequently a critical density threshold below which the population cannot be detected in practice and the corresponding position is often termed the leading edge of the distribution. Historically this position has
been investigated for large time problems, but little attention has been afforded to understanding its short term dynamics. In this work we describe a novel approach, utilizing the Laplace decomposition method, that generates algebraic expressions for the initial kinematic properties of the leading edge in terms
of the initial data and model parameters. The method is demonstrated on two well-studied partial differential equations and two established systems of equations (representing the growth of fungal networks), all of which display travelling fronts. The kinematics of these advancing fronts are determined
using our method and are shown to be in excellent agreement with both exact solutions and numerical approximations of the model equations.