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.

On the summability of formal solutions for doubly singular nonlinear partial differential equations

Abstract: We study Gevrey asymptotic properties of solutions of singularly perturbed singular nonlinear partial differential equations of irregular type in the complex domain. We construct actual holomorphic solutions of these problems with the help of the Borel---Laplace transforms. Using the Malgrange---Sibuya theorem, we show that these holomorphic solutions have a common formal power series asymptotic expansion of Gevrey order 1 in the perturbation parameter.

Numerical and asymptotic aspects of parabolic cylinder functions

Abstract: Several uniform asymptotics expansions of the Weber parabolic cylinder functions are considered, one group in terms of elementary functions, another group in terms of Airy functions. Starting point for the discussion are asymptotic expansions given earlier by F.W.J. Olver. Some of his results are modified to improve the asymptotic properties and to enlarge the intervals for using the expansions in numerical algorithms. Olver's results are obtained from the differential equation of the parabolic cylinder functions; we mention how modified expansions can be obtained from integral representations. Numerical tests are given for three expansions in terms of elementary functions. In this paper only real values of the parameters will be considered.

Asymptotic expansions of eigenvalues and eigenfunctions of an elliptic operator in a domain with many "light" concentrated masses situated on the boundary. Two-dimensional case

Abstract: We consider vibrations of a membrane which contains many "light" concentrated masses on the boundary. We study the asymptotic behaviour of the frequencies of eigenvibrations of the membrane as the small parameter (which char- acterizes the diameter and density of the concentrated masses) tends to zero. We construct asymptotic expansions of eigenelements of the corresponding problems and carefully justify these expansions.

Lifting-line theory for an unsteady wing as a singular perturbation problem

Abstract: A linearized theory, which treats unsteady motions of a wing of large aspect ratio at variable forward speed in an inviscid incompressible fluid, is developed, using the method of matched asymptotic expansions. The wing geometry and motions are specified; and the time-dependent lift and moment are obtained. Long-time asymptotic behaviour of an initial-value harmonic motion is presented, as are the short-time solutions of a wing starting from rest, with constant acceleration and with impulsive acceleration to constant speed. Some attention is given to flapping flight. Results are presented in quadrature form for a general class of unsteady wing motions.