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.

An Upper Bound for the Singular Parameter in a Stable, Singularly Perturbed System☆

Abstract: This paper considers the problem of stability of a singularly perturbed system and of finding an upper bound for the parameter when the order of the system changes as a result of parameter perturbation. By means of the contraction mapping technique, conditions have been derived for determining explicitly the range of parameter perturbation such that both bounded-input-bounded-output and asymptotic stabilities are insured. In addition, bounds of the state and output of the singularly perturbed system can be found. Two examples are given to illustrate the application and significance of the results.

A boundary value technique for boundary value problems for singularly perturbed fourth-order ordinary differential equations

Abstract: Singularly perturbed two-point boundary value problems (BVPs) for fourth-order ordinary differential equations (ODEs) with a small positive parameter multiplying the highest derivative are considered. A numerical method is suggested in this paper to solve such problems. In this method, the given BVP is transformed into a system of two ODEs subject to suitable boundary conditions. Then, the domain of definition of the differential equation (a closed interval) is divided into two nonoverlapping subintervals, which we call ''inner region'' (boundary layer) and ''outer region''. Then, the DE is solved in these intervals separately. The solutions obtained in these regions are combined to give a solution in the entire interval. To obtain terminal boundary conditions (boundary values inside this interval) we use mostly zero-order asymptotic expansion of the solution of the BVP. First, linear equations are considered and then nonlinear equations. To solve nonlinear equations, Newton's method of quasilinearization is applied. The present method is demonstrated by providing examples. The method is easy to implement.

Exponential asymptotic expansions and approximations of the unstable and stable manifolds of singularly perturbed systems with the Henon map as an example.

Abstract: The subject of this paper is the construction of the exponential asymptotic expansions of the unstable and stable manifolds of the area-preserving Henon map. The approach that is taken enables one to capture the exponentially small effects that result from what is known as the Stokes phenomenon in the analytic theory of equations with irregular singular points. The exponential asymptotic expansions were then used to obtain explicit functional approximations for the stable and unstable manifolds. These approximations are compared with numerical simulations and the agreement is excellent. Several of the main results of the paper have been previously announced in A. Tovbis, M. Tsuchiya, and C. Jaffe [“Chaos-integrability transition in nonlinear dynamical systems: exponential asymptotic approach,” Differential Equations and Applications to Biology and to Industry, edited by M. Martelli, K. Cooke, E. Cumberbatch, B. Tang, and H. Thieme (World Scientific, Singapore, 1996), pp. 495–507, and A. Tovbis, M. Tsuchiya, and C. Jaffe, “Exponential asymptotic expansions and approximations of the unstable and stable manifolds of the Henon map,” preprint, 1994].

Cross-diffusion systems with excluded-volume effects and asymptotic gradient flow structures

Abstract: In this paper, we discuss the analysis of a cross-diffusion PDE system for a mixture of hard spheres, which was derived in Bruna and Chapman (J Chem Phys 137:204116-1–204116-16, 2012a) from a stochastic system of interacting Brownian particles using the method of matched asymptotic expansions. The resulting cross-diffusion system is valid in the limit of small volume fraction of particles. While the system has a gradient flow structure in the symmetric case of all particles having the same size and diffusivity, this is not valid in general. We discuss local stability and global existence for the symmetric case using the gradient flow structure and entropy variable techniques. For the general case, we introduce the concept of an asymptotic gradient flow structure and show how it can be used to study the behavior close to equilibrium. Finally, we illustrate the behavior of the model with various numerical simulations.

Three-dimensional motion of a vortex filament and its relation to the localized induction hierarchy

Abstract: Three-dimensional motion of a slender vortex tube, embedded in an inviscid incompressible fluid, is investigated under the localized induction approximation for the Euler equations. Using the method of matched asymptotic expansions in a small parameter e, the ratio of core radius to curvature radius, the velocity of a vortex filament is derived to O(e3), whereby the influence of elliptical deformation of the core due to the self-induced strain is taken into account. It is found that there is an integrable line in the core whose evolution obeys a summation of the first and third terms of the localized induction hierarchy.