Author : Oleg Mikitchenko
Publisher :
ISBN 13 :
Total Pages : 196 pages
Book Rating : 4.:/5 (474 download)
Book Synopsis Applications of the Resolution of Singularities to Asymptotic Analysis of Differential Equations by : Oleg Mikitchenko
Download or read book Applications of the Resolution of Singularities to Asymptotic Analysis of Differential Equations written by Oleg Mikitchenko and published by . This book was released on 2008 with total page 196 pages. Available in PDF, EPUB and Kindle. Book excerpt: Abstract: The method of resolution of singularities was established in the 17th Century by Newton for finding expansions of solutions of algebraic equations. In this method one uses a polygon in the plane of powers of the variables that appear in the original equation and which is now known as the Newton polygon. Recently, the idea of resolution of singularities was extended by A.D. Bruno into a group of methods, known as Power Geometry, that allows one to compute asymptotics of solutions to ordinary and partial differential equations. In this thesis we show how to solve some problems in asymptotic analysis of differential equations using the resolution of singularities and the power geometry. First, we show how these methods are applied to the linear Airy's equation, doing it in two different ways: by applying the methods directly to the equation and by applying the methods to the autonomous system of ODEs to which the equation can be transformed. This analysis is extended to a larger class of classical second order linear equations with nonautonomous coefficients. Second, we consider a non-linear first order equation for which the origin is an essential singularity, and obtain leading order asymptotic approximations to the solutions in different sectors near the origin, and compare them with numerical solutions. By imposing conditions on the sector boundaries, we obtain approximations of the solutions in the full neighborhood of the origin. Third, we show how the method can be applied to singularly perturbed boundary value problems. We show that the Newton polygon allows us to compute the correct rescaling (or rescalings) of the independent variable as well as to determine the dominant terms of the equation corresponding to this rescaling. We also show how the procedure of matching of inner and outer expansions for such problems can be illustrated by means of the Newton's polygon associated with the equation. We also present a collection of algorithms implemented as a package for the Maple computer algebra system that can be used when one applies power geometry to finding asymptotic expansions of solutions.
