共查询到20条相似文献,搜索用时 15 毫秒
1.
V. B. Levenshtam 《Journal of Mathematical Sciences》2009,163(1):89-110
A complete asymptotic expansion is constructed for solutions of the Cauchy problem for nth order linear ordinary differential
equations with rapidly oscillating coefficients, some of which may be proportional to ω
n/2, where ω is oscillation frequency. A similar problem is solved for a class of systems of n linear first-order ordinary differential
equations with coefficients of the same type. Attention is also given to some classes of first-order nonlinear equations with
rapidly oscillating terms proportional to powers ω
d
. For such equations with d ∈ (1/2, 1], conditions are found that allow for the construction (and strict justification) of the leading asymptotic term
and, in some cases, a complete asymptotic expansion of the solution of the Cauchy problem. 相似文献
2.
The stability problem is considered for certain classes of systems of linear ordinary differential equations with almost periodic coefficients. These systems are characterized by the presence of rapidly oscillating terms with large amplitudes. For each class of equations, a procedure for analyzing the critical stability of solutions is constructed on the basis of the Shtokalo-Kolesov method. A verification scheme is described. The theory proposed is illustrated by using a linearized stability problem for the upper equilibrium of a pendulum with a vibrating suspension point. 相似文献
3.
A. K. Kapikyan V. B. Levenshtam 《Computational Mathematics and Mathematical Physics》2008,48(11):2059-2076
Systems of first-order semilinear partial differential equations with terms that oscillate at a frequency ω ? 1 in a single variable and are proportional to \(\sqrt \omega \) are considered. The Krylov-Bogolyubov-Mitropol’skii averaging method is substantiated for such equations. Based on the two-scale expansion method, an algorithm for constructing complete asymptotics of solutions is proposed and justified. 相似文献
4.
I. N. Yakushina 《Journal of Mathematical Sciences》1995,73(3):408-413
Asymptotic expansions for solutions of n-th order linear differential equation with two turning points are constructed in
Olver's form. Analytic properties of the coefficients of the series obtained are investigated. Bibliography: 7 titles.
Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 186, pp. 172–179, 1990. 相似文献
5.
Jean Mawhin 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》1987,38(2):257-265
The method of upper and lower solutions and convexity arguments are used to prove sharp results for the existence and multiplicity of periodic solutions for first order ordinary differential equations depending upon a parameter.Dedicated to Professor H. W. Knobloch for his sixtieth birthday 相似文献
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7.
Optimal in a certain sense sufficient conditions are given for the existence and uniqueness of ω-periodic solutions of the
nonautonomous ordinary differential equation u
(2m)
=f(t,u,...,u
(m-1)
), where the function f:ℝ×ℝ
m
→ℝ is periodic with respect to the first argument with period ω.
Received: December 21, 1999; in final form: August 12, 2000?Published online: October 2, 2001 相似文献
8.
In this paper we study by Morse theory the existence of multiple periodic solutions of a class of ordinary differential equation with double resonance at infinity between two consecutive eigenvalues and with resonance at origin. 相似文献
9.
Asymptotic and convergent expansions for solutions of third-order linear differential equations with a large parameter 下载免费PDF全文
Chelo Ferreir Jose L. Lopez Ester Perez Sinusia 《Journal of Applied Analysis & Computation》2018,8(3):965-981
In previous papers [6-8,10], we derived convergent and asymptotic expansions of solutions of second order linear differential equations with a large parameter. In those papers we generalized and developed special cases not considered in Olver"s theory [Olver, 1974]. In this paper we go one step forward and consider linear differential equations of the third order: $y"+a\Lambda^2 y"+b\Lambda^3y=f(x)y"+g(x)y$, with $a,b\in\mathbb{C}$ fixed, $f"$ and $g$ continuous, and $\Lambda$ a large positive parameter. We propose two different techniques to handle the problem: (i) a generalization of Olver"s method and (ii) the transformation of the differential problem into a fixed point problem from which we construct an asymptotic sequence of functions that converges to the unique solution of the problem. Moreover, we show that this second technique may also be applied to nonlinear differential equations with a large parameter. As an application of the theory, we obtain new convergent and asymptotic expansions of the Pearcey integral $P(x,y)$ for large $|x|$. 相似文献
10.
John D Dollard Charles N Friedman 《Journal of Mathematical Analysis and Applications》1978,66(2):394-398
We present some conditions which ensure that the solution Y(x) of the ordinary differential equation Y′(x) = A(x) Y(x), Y(x0) = I, where x0 ? x < ∞ and A(x), Y(x) are n × n complex matrix-valued functions with A(x) continuous, has a nonsingular limit as x → ∞. 相似文献
11.
Mathematical Notes - The question of the existence of periodic solutions of linear ordinary differential equations with high-frequency summands in a Banach space is studied. 相似文献
12.
Liqing Zhang 《Numerische Mathematik》1993,66(1):399-409
Summary A two-sided approximation to the periodic orbit of an autonomous ordinary differential equation system is considered. First some results about variational equation systems for periodic solutions are obtained in Sect. 2. Then it is proved that if the periodic orbit is convex and stable, the explicit difference solution approximates the periodic orbit from the outer part and the implicit one from the inner part respectively. Finally a numerical example is given to illustrate our result and to point out that the numerical solution no longer has a one-sided approximation property, if the periodic orbit is not convex.The Work is supported by the National Natural Science Foundation of China 相似文献
13.
Acta Mathematica Hungarica - 相似文献
14.
A. V. Zuev 《Mathematical Notes》2006,79(3-4):518-527
A new version of the method of translation along trajectories, which does not require the uniqueness of the solution of the Cauchy problem, is applied to the proof of the existence theorem for vector-valued periodic solutions of ordinary differential equations of first and second order. This result is applicable to equations and differential inclusions with discontinuous right-hand side. Several applications of the theorems proved in this paper are considered in cases which are not covered by the classical theory of ordinary differential equations with continuous right-hand side and equations with right-hand side satisfying the Carathéodory conditions. 相似文献
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We analyze the asymptotic behavior as x → ∞ of the product integral Πx0xeA(s)ds, where A(s) is a perturbation of a diagonal matrix function by an integrable function on [x0,∞). Our results give information concerning the asymptotic behavior of solutions of certain linear ordinary differential equations, e.g., the second order equation y″ = a(x)y. 相似文献
20.
V. A. Eremenko 《Ukrainian Mathematical Journal》1997,49(8):1279-1285
We consider a scalar linear second-order ordinary differential equation whose coefficient of the second derivative may change its sign when vanishing. For this equation, we obtain sufficient conditions for the existence of a periodic solution in the case of arbitrary periodic inhomogeneity. 相似文献