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

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

By R. P. Kuzmina (auth.)

In this booklet we think about a Cauchy challenge for a approach of normal differential equations with a small parameter. The booklet is split into th ree elements in keeping with 3 ways of concerning the small parameter within the process. partially 1 we research the quasiregular Cauchy challenge. Th at is, an issue with the singularity integrated in a bounded functionality j , which is dependent upon time and a small parameter. This challenge is a generalization of the regu­ larly perturbed Cauchy challenge studied by means of Poincare [35]. a few differential equations that are solved by way of the averaging process might be diminished to a quasiregular Cauchy challenge. for example, in bankruptcy 2 we think about the van der Pol challenge. partly 2 we research the Tikhonov challenge. this can be, a Cauchy challenge for a method of standard differential equations the place the coefficients via the derivatives are integer levels of a small parameter.

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8) and the inequalities IgEI ~ te + lalE 2 ~ C < 1. 3. 9) where a is a constant independent of x, t, E. L) = O. 10) SOLUTION EXPA NSIO NS OF THE QUASIREG ULAR CAUCHY. . 10) takes the form 00 z = (e h - 1) e + L eh(e h - l )k-l c k, h == t + aJ-L sin ( ~ ) . 10) for t he pro blem (8. 9) is eq ual to x = (e9 - 1) e + 00 L g==t+ac sin(~). 8. 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.

2. 5) were a is a constant independent of x, t, c . Take f(t, c) = cos(t/c) and consider the problem with two small parameters ~: = [1 + a cos ( ;) ] (z + c) 2 , Z ( 0, c, It) = O. 5) is equal to 00 x= L gk -lck , g == t + ae sin (~) . 8. 7) . The var iational equation takes t he form d( - =0. dt The Cauchy mat rix is equal to U(t , 5 , It) = 1. 3) for U is not valid . 7) . 3 , so we do not consider it here . 5) exists, is unique, and satisfies the ineq ua lity 46 CHAPTER 1 for 0 ~ t ~ T, 0

Let J be an integer , 1 ~ J ~ N; x be a vector of J components of the vector x , D be a set in vector space RN+2 3 (x, t,c). By D* denote the set D* {(x, t , c): Ilxll ~ 8, t ~ 0, ~ e ~ E}.

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