Self-similar perturbation near a corner: matching versus multiscale expansions for a model problem
نویسنده
چکیده
In this paper we consider the Laplace-Dirichlet equation in a polygonal domain perturbed at the small scale ε near a vertex. We assume that this perturbation is self-similar, that is, derives from the same pattern for all relevant values of ε. We construct and validate asymptotic expansions of the solution in powers of ε via two different techniques, namely the method of multiscale expansions and the method of matched asymptotic expansions. Then we show how the terms of each expansion can be split into a finite number of sub-terms in order to reconstruct the other expansion. Compared with the fairly general approach of Maz’ya, Nazarov and Plamenevskiı̆ relying on multiscale expansions, the novelty of our paper is the rigorous validation of the method of matched asymptotic expansions, and its comparison with the multiscale method. The consideration of a model problem allows to simplify the exposition of these rather complicated two techniques. CONTENTS
منابع مشابه
Freezing in a Finite Slab Using Extensive Perturbation Expansions Method
In this paper Mathematica is used to solve the moving boundary problem of freezing in a finite slab for higher order perturbations. Mathematica is a new system which makes it possible to do algebra with computer. More specifically, it enables researchers to find the location of the ice at any time for as high order of perturbation as one whishes. Using of Mathematica and outer solution and an i...
متن کاملA NOVEL HOMOTOPY PERTURBATION METHOD: KOUROSH´S METHOD FOR A THERMAL BOUNDARY LAYER IN A SATURATED POROUS MEDIUM
this paper a novel homotopy perturbation method has been presented for forced convection boundary layer problems in a porous medium. Noting the infinite condition, a homotopy form which is similar to the singular perturbation form has been considered. The inner and outer solutions have been achieved and the coincidence of the results has been investigated with a proper matching method. The resu...
متن کاملTransient Natural Convection Flow on an Isothermal Vertical Wall at High Prandtl Numbers: Second-Order Approximation
The method of matched asymptotic expansions, which has been used in previous studies of steady natural convection flow, is extended here to transient natural convection flow at high Prandtl number (Pr). Second-order expansion solutions, valid for large Prandtl numbers, are presented for the transient natural convection flow near a vertical surface which undergoes a step change in temperature. T...
متن کاملEigenvalues of Self-Similar Solutions of the Dafermos Regularization of a System of Conservation Laws via Geometric Singular Perturbation Theory
The Dafermos regularization of a system of n conservation laws in one space dimension admits smooth self-similar solutions of the form u= u(X/T ). In particular, there are such solutions near a Riemann solution consisting of n possibly large Lax shocks. In Lin and Schecter (2004, SIAM. J. Math. Anal. 35, 884–921), eigenvalues and eigenfunctions of the linearized Dafermos operator at such a solu...
متن کاملThe Initial Value Problem for the Binormal Flow with Rough Data
In this article we consider the initial value problem of the binormal flow with initial data given by curves that are regular except at one point where they have a corner. We prove that under suitable conditions on the initial data a unique regular solution exists for strictly positive and strictly negative times. Moreover, this solution satisfies a weak version of the equation for all times an...
متن کامل