Download Asymptotic Methods for Ordinary Differential Equations by R. P. Kuzmina (auth.) PDF

By R. P. Kuzmina (auth.)

In this booklet we ponder a Cauchy challenge for a approach of standard differential equations with a small parameter. The booklet is split into th ree components in accordance with 3 ways of concerning the small parameter within the method. partly 1 we examine the quasiregular Cauchy challenge. Th at is, an issue with the singularity incorporated in a bounded functionality j , which will depend on time and a small parameter. This challenge is a generalization of the regu­ larly perturbed Cauchy challenge studied by way of Poincare [35]. a few differential equations that are solved by way of the averaging process should be lowered to a quasiregular Cauchy challenge. for example, in bankruptcy 2 we ponder the van der Pol challenge. partially 2 we learn the Tikhonov challenge. this can be, a Cauchy challenge for a method of normal differential equations the place the coefficients via the derivatives are integer levels of a small parameter.

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Extra resources for Asymptotic Methods for Ordinary Differential Equations

Example text

11) 0 are continuous with respect to t. q a(t,E) = ~;~t - IIU(q, O,E)' UO(E) + - ! U(q, S,E) . G(O, S,E) dsll, 0 q b(t, E) max O

39) have positive exponents, and the right hand sides are small for small values of lEI. 10 , we have: 1. for 0 ~ t ~ t*(c: ) , 0 ~ e ~ e. L)11 ~ C al (t, s) ; 2. L)11 ~ Cal(t, C:) ; 3. 1) exists , is unique a nd sat isfies t he inequ ality Il x(t ,c:) - X n(t ,c:)11 ~ C al (t,c:). 5-2. 8. §s. 1. 1. L ~"t . 1. 2. 2. L ), k = 0,1, . 1) . 2. 3. 9 in case 0 = 1, t: = 1. 2), t hey can be given t his form by a n appropriate scaling of x, e. SOL UTION EXPANSIONS OF THE QUASIREGULAR CAUCHY. . l , f) dZl .

11) . The variational eq uation t akes t he form ~; = [1 + a cos (~ )] (. 4) for U a re not valid . 11). 9) exists , is unique , a nd sat isfies t he inequ ali t y for 0 ~ t ~ T , 0 2. : 1), n t nt C* e + e (e -1), < C* e IIX(t,e)11 < IIX(t ,e) - Xn(t,e)11 1 n> 1 for 0 ::; t ::; T - Xlne, 0 < e ::; e. 9) exists for 0 ::; t < t; (e), e > 0, where t; is the smallest positive solution of the equation t .

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