Method of matched asymptotic expansions

About: Method of matched asymptotic expansions is a research topic. Over the lifetime, 4233 publications have been published within this topic receiving 73311 citations.

Global asymptotic stability of solutions of cubic stochastic difference equations

Abstract: Global almost sure asymptotic stability of solutions of some nonlinear stochastic difference equations with cubic-type main part in their drift and diffusive part driven by square-integrable martingale differences is proven under appropriate conditions in ℝ1. As an application of this result, the asymptotic stability of stochastic numerical methods, such as partially drift-implicit θ-methods with variable step sizes for ordinary stochastic differential equations driven by standard Wiener processes, is discussed.

Asymptotic modelling for separating boundary layers

Abstract: The method of matched asymptotic expansions is used to explain bow the triple deck structure in a boundary layer can be formed. In the context of a laminar steady flow of an incompressible fluid over a flat plate, a theory is developed to explain the separation over significant wall disturbances. In particular, we show that the triple deck structure is the first perturbation that can both displace the classical boundary layer and cause separation of the flow. Above this exist a serie of perturbations, smaller but “stronger”, that cause a separation of the boundary layer without displacing it. This serie is limited by the smallest perturbation compatible with the hypothesis of the theory, thus leading to a theory in double deck.

An Asymptotic Analysis of a 2-D Model of Dynamically Active Compartments Coupled by Bulk Diffusion

Abstract: A class of coupled cell–bulk ODE–PDE models is formulated and analyzed in a two-dimensional domain, which is relevant to studying quorum-sensing behavior on thin substrates. In this model, spatially segregated dynamically active signaling cells of a common small radius $$\epsilon \ll 1$$ are coupled through a passive bulk diffusion field. For this coupled system, the method of matched asymptotic expansions is used to construct steady-state solutions and to formulate a spectral problem that characterizes the linear stability properties of the steady-state solutions, with the aim of predicting whether temporal oscillations can be triggered by the cell–bulk coupling. Phase diagrams in parameter space where such collective oscillations can occur, as obtained from our linear stability analysis, are illustrated for two specific choices of the intracellular kinetics. In the limit of very large bulk diffusion, it is shown that solutions to the ODE–PDE cell–bulk system can be approximated by a finite-dimensional dynamical system. This limiting system is studied both analytically, using a linear stability analysis and, globally, using numerical bifurcation software. For one illustrative example of the theory, it is shown that when the number of cells exceeds some critical number, i.e., when a quorum is attained, the passive bulk diffusion field can trigger oscillations through a Hopf bifurcation that would otherwise not occur without the coupling. Moreover, for two specific models for the intracellular dynamics, we show that there are rather wide regions in parameter space where these triggered oscillations are synchronous in nature. Unless the bulk diffusivity is asymptotically large, it is shown that a diffusion-sensing behavior is possible whereby more clustered spatial configurations of cells inside the domain lead to larger regions in parameter space where synchronous collective oscillations between the small cells can occur. Finally, the linear stability analysis for these cell–bulk models is shown to be qualitatively rather similar to the linear stability analysis of localized spot patterns for activator–inhibitor reaction–diffusion systems in the limit of long-range inhibition and short-range activation.