Get Asymptotic Methods for Ordinary Differential Equations PDF

By R. P. Kuzmina (auth.)

ISBN-10: 9048155002

ISBN-13: 9789048155002

ISBN-10: 9401593477

ISBN-13: 9789401593472

In this booklet we give some thought to a Cauchy challenge for a method of standard differential equations with a small parameter. The ebook is split into th ree elements in accordance with 3 ways of concerning the small parameter within the process. partly 1 we learn the quasiregular Cauchy challenge. Th at is, an issue with the singularity incorporated in a bounded functionality j , which relies on time and a small parameter. This challenge is a generalization of the regu­ larly perturbed Cauchy challenge studied through Poincare [35]. a few differential equations that are solved via the averaging technique should be decreased to a quasiregular Cauchy challenge. for instance, in bankruptcy 2 we reflect on the van der Pol challenge. partially 2 we learn the Tikhonov challenge. this can be, a Cauchy challenge for a method of standard differential equations the place the coefficients by means of the derivatives are integer levels of a small parameter.

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ANALYSIS OF THE AUXILIARY PROBLEM Consider the problem d: dv ' =

1 also holds for the function z(k)(t, p). 1 is true for all k = 0, n. 2. INTRODUCTION OF THE AUXILIARY VARIABLE By definition, put n u = z - Zn(t,E,p), Zn (t, E, p) == L z(k)(t, p) Ek . 6) we derive the equations for u: du di=A(t,p) u+G(U ,t,E,p), Ult=o= O. 18) n - L F (k)(t , p) Ek - A(t, p) Zn(t,E,p) - A(t,p) u. 3. 2. 8. 8, n ~ 1, (4. 21) 28 CHAPTER 1 Proof. 22) 1 / Fe(O , t , OE , f(t, /l)) dO E, o ~ IIG(O,t,E,/l)11 CE, i e o; 0 ~ E ~ E, 0< /l ~ E. 4. 20) is true for n = O. n+l IA=19 1 . . 19 n +1 e X 0205 ...

3) and v(t) > V2(t) == p(t) + JT(t) = 2a(t). 4) For t = 0 we have v(O) = Iluoll = a(O) = vt{O) . 10) and satisfying the condition 0 ~ t ~ tl the following inequalities are valid: Ilu(t)1I ~ v(t) ~ vt{t) = p(t) 2a(t) + JT(t) . 10) contains a point t2 such that tl < t2. 10) contains all the points s such that 0 ~ s ~ ii as a(t), b(t) and c(t) are nondecreasing functions; 2) Ilu(t 1)1I = 0 as in the opposite case the solution u(t) could be extended. 10). 5) . This proves the theorem. §7. 11 is similar to that of the Lyapunov theorem for J = N [30] and the Rumyantsev theorem for J < N [38] .

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

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