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

By R. P. Kuzmina (auth.)

In this publication we think about a Cauchy challenge for a procedure of standard differential equations with a small parameter. The e-book is split into th ree elements in accordance with 3 ways of concerning the small parameter within the procedure. partially 1 we learn the quasiregular Cauchy challenge. Th at is, an issue with the singularity integrated in a bounded functionality j , which depends upon 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 instance, in bankruptcy 2 we reflect on the van der Pol challenge. partly 2 we examine the Tikhonov challenge. this can be, a Cauchy challenge for a approach of standard differential equations the place the coefficients through the derivatives are integer levels of a small parameter.

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Denote the remainder term by u x - Xn(t, c). 14) F( u + Xn(t, c), t, c) - [F(Xn(t, c), t, c)t:;n) , F(u ,t,c) XO(c) - [xO(c)](~n) . UO(c) We have F(O, t, 0) problem 0, UO (0) = 0. 15) has the solution Wl = g(t ,c), W2 = c, and vice versa. 15) has a trivial solution Wl = 0, W2 = 0. 15) turns into the problem of solving the system of ordinary differential equations with small initial perturbations. An estimate of such a solution (and, consequently, an estimate of the remainder term u) can be evaluated by the methods of stability theory.

L) . l ), t , E, ! l)) - Fx(O, t, O, ! ) E} _ BUI + UI dB l dB (u - u). ( t ,ll) Her e ] 32 CHAPT ER 1 Ilxll < < II Zn (t, e, J-L) II + B Ilull + (1 - II Zn(t,c, J-L)1I + 8 B) Wil li < 8. 33) . 31) . 4. 30)). 32) . 5. 32) . 10 for any valu es of e, J-L (0 ~ e ~ C2, 0 < J-L ~ €') . 10). Co nside r t he functions a, b, c. 20) it follows t hat q a = a(t ,e, J-L ) = J II U (q, s, J-L )II· G(O, s,e, J-L ) ds O~q9 0 max SOLUTION EXPANSIONS OF THE QUASIREGULAR CAUCHY. . 33 q < max f II U (q, s , /1)11 · go(s) ds En+1.

VN = V, -v(1 - Nv) + «, (NZv Z + _E_) = 0, l-E _ 1± VK L}. v- 2N(I+I< zN)' = _ 4I + J(2(t) - l/ ~ C < 1, 4J(2N;(;~:J(2 N). 20 ) is a na lytic in u and bounded . 20) it follows t hat 1/J (t ,E) = V 1/J' (t , v) Iv=eK3 (t ) .

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