Topic
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.
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TL;DR: A high-order uniformly convergent method to solve singularly perturbed delay parabolic convection diffusion problems exhibiting a regular boundary layer is introduced and it is shown that the method is -uniformly convergent of second-order accurate in time and in the spatial direction.
Abstract: In this article, we aim to introduce a high-order uniformly convergent method to solve singularly perturbed delay parabolic convection diffusion problems exhibiting a regular boundary layer. The domain is discretized by a uniform mesh in the time direction and a piecewise-uniform Shishkin mesh for the spatial direction. We use the Crank–Nicolson method for the time derivative and we develop a fourth-order compact difference method to solve the set of ordinary differential equations at each time level. The stability analysis and the truncation error are discussed. Parameter-uniform error estimates are derived and it is shown that the method is e-uniformly convergent of second-order accurate in time, and in the spatial direction it is of second-order outside region of boundary layer, and of almost fourth-order inside the layer region. Numerical examples are presented to verify the theoretical results and to confirm the efficiency and high accuracy of the proposed method.
19 citations
TL;DR: In this article, it was shown that there are no two-peaked solutions of singularly perturbed elliptic equations in a strictly convex domain, and that these conditions are related to the geometry of the domain.
Abstract: We obtain necessary conditions for the existence of two-peaked solutions of singularly perturbed elliptic equations. These conditions are related to the geometry of the domain. In particular, we prove there are no two-peaked solutions in a strictly convex domain.
19 citations
TL;DR: The Euler-Poincare equations describing geodesics on Lie groups with invariant metrics fall into the class of quasihomogeneous systems with homogeneous quadratic right-hand members as mentioned in this paper.
Abstract: An example is a system with homogeneous quadratic right-hand members: in it, gl = ... = gn = i. Among others, the Euler-Poincare equations describing geodesics on Lie groups with invariant metrics fall into this class. A popular example from dynamics is Kirchoff's problem on the motion of a rigid body in an unbounded volume of an ideal liquid. Quasihomogeneous systems are also exemplified by the equations of the problem of many gravitating bodies and by the Euler-Poisson equations describing the rotation of a heavy rigid body about a fixed point. These remarks show that it is expedient to consider quasihomogeneous systems from the viewpoint of applications.
19 citations
19 citations
01 Jun 1968
19 citations