共查询到20条相似文献,搜索用时 31 毫秒
1.
A. A. Abramov S. V. Kurochkin 《Computational Mathematics and Mathematical Physics》2007,47(3):397-406
For the Mathieu equation, we consider finding eigenvalues with a given index (on the basis of oscillation theorems for the relevant difference equations), the stability of solutions to the difference equations, correct definition and calculation of eigenvalues and Mathieu functions with noninteger numbers, correct definition and calculation of the Mathieu characteristic exponent, and the calculation of values of solutions to the Mathieu equation for large arguments. Numerical algorithms are proposed for the problems listed above. 相似文献
2.
《Chaos, solitons, and fractals》2001,12(4):705-720
The method of multiple-scales is used to determine a third-order solution for a cubic nonlinear Mathieu equation. The perturbation solutions are imposed on the so-called solvability conditions. Solvability conditions in the non-resonance case yield the standard Landau equation. Several types of a parametric Landau equation are derived in the neighborhood of five different resonance cases. These parametric Landau equations contain a parametric complex conjugate term or a parametric second-order complex conjugate term or a parametric complex conjugate term as well as a parametric second-order term. Necessary and sufficient conditions for stability are performed in each resonance case. Stability criteria correspond to each parametric Landau equation and are derived by linear perturbation. Stability criteria for the non-trivial steady-state response are discussed. The analysis leads to simultaneous resonance. Transition curves are performed in each case. Numerical calculations are made for some transition curves to illustrate the coupled resonance regions, where the induced stability tongues within the instability tongues are observed. The amplitude of the periodic coefficient of Mathieu equation plays a dual role in the stability criteria for nonlinear Mathieu equation. 相似文献
3.
A. F. Kurin 《Computational Mathematics and Mathematical Physics》2008,48(4):600-617
Mathieu’s equation is solved by an asymptotic averaging method in the fourth approximation for the first to fourth resonance domains and in the third approximation for the zero resonance domain. The general periodic and aperiodic solutions on characteristic curves are found, and the general solution is obtained in instability domains and stability-domain areas adjacent to the characteristic curves. All the solutions are explicitly found in the form of functions of an argument without using the auxiliary parameter employed in Whittaker’s method. Simple formulas depending on two parameters of the equation are derived for the characteristic exponent in instability domains and for the frequency of slow oscillations in stability domains near the characteristic curves. The theory is developed by analyzing the resonances exhibited by Mathieu’s equation. 相似文献
4.
We develop the exact WKB analysis of an M2P1T (merging two simple poles and one simple turning point) Schrödinger equation. In Part II, using a WKB-theoretic transformation to the algebraic Mathieu equation constructed in Part I, we calculate the alien derivative of its Borel transformed WKB solutions at each fixed singular point relevant to the simple poles through the analysis of Borel transformed WKB solutions of the Legendre equations. In the course of the calculation of the alien derivative we make full use of microdifferential operators whose symbols are given by the infinite series that appear in the coefficients of the algebraic Mathieu equation and the Legendre equation. 相似文献
5.
Yan Li Shangchun Fan Zhanshe Guo Jing Li Le Cao Haihan Zhuang 《Communications in Nonlinear Science & Numerical Simulation》2013,18(2):401-410
In this paper, Mathieu equation is applied to analyze the dynamic characteristics of resonant inertial sensors. Unlike previous work, Mathieu equation is not just a differential equation and analyzes the stability of the transition curves, but become an important method in analyzing parametric resonant characteristics and approximate output of resonant inertial sensors. It is demonstrated that the mathematical model of resonant inertial sensors is described by Mathieu equation. The relevant Mathieu equation theory and dynamic characteristics analysis methods were proposed, which include both stability and dynamic linear output. Finally, theoretical and experimental analysis show that the correlation of the theoretical curve and the experimental result coincide so perfectly, which means proposed analysis methods for Mathieu equation could be used to analyze the dynamic output characteristic of resonant inertial sensors. The theoretical analyzing approach of Mathieu equation and experimental results of resonant inertial sensors are obtained, which provide an application area for Mathieu equation and a reference for the robust design for resonant inertial sensors. 相似文献
6.
《Communications in Nonlinear Science & Numerical Simulation》2007,12(1):58-71
The asymptotic solutions and transition curves for the generalized form of the non-homogeneous Mathieu differential equation are investigated in this paper. This type of governing differential equation of motion arises from the dynamic behavior of a pendulum undergoing a butterfly-type end support motion. The strained parameter technique is used to obtain periodic asymptotic solutions. The transition curves for some special cases are presented and their corresponding periodic solutions with the periods of 2π and 4π are evaluated. The stability analyses of those transition curves in the ε–δ plane are carried out, analytically, using the multiple scales method. The numerical simulations for some typical points in the ε–δ plane are performed and the dynamic characteristics of the resulting phase plane trajectories are discussed. 相似文献
7.
We develop the exact WKB analysis of an M2P1T (merging two simple poles and one simple turning point) Schrödinger equation. Our emphasis is put on the analysis of the singularity structure of its Borel transformed WKB solutions near fixed singular points relevant to the two simple poles contained in the potential of the equation. In Part I, we focus our attention on the construction and analytic properties of a WKB-theoretic transformation that transforms an M2P1T equation to an algebraic Mathieu equation. That transformation plays an important role in Part II ([7]) when we discuss the singularity structure of Borel transformed WKB solutions of an M2P1T equation. 相似文献
8.
A. V. Kashevarov 《Computational Mathematics and Mathematical Physics》2011,51(12):2137-2145
A spherical probe placed in a slowly moving collisional plasma with a large Debye length λD → ∞ is considered. The partial differential equation describing the electron concentration around the probe is reduced to
two ordinary differential equations, namely, to the equation for Coulomb spheroidal functions and Mathieu’s modified equation
with the parameter a of the latter related to the eigenvalue λ of the former by the relation a = λ + 1/4. It is shown that the solutions of Mathieu’s equation are Mathieu functions of half-integer order, which are expressed
as series in terms of spherical Bessel functions and series of products of Bessel functions. These Mathieu functions are numerically
constructed for Mathieu’s modified and usual equations. 相似文献
9.
10.
Michael P. Lamoureux 《Integral Equations and Operator Theory》1997,28(1):45-59
An explicit derivation of a tridiagonal matrix form for the almost Mathieu operator (Harper's equation) is obtained via conjugation with a reflection operator, valid for all rational values of the rotation parameter. The difference between even and odd values of the denominator is highlighted. This tridiagonal form is useful for numerical eigenvalue computations; some Matlab code is included.This research was supported in part by an NSERC individual research grant. 相似文献
11.
本文从可瘪管的运动方程导出有阻尼Mathieu方程,然后用渐近展开法精确到各阶精度求得了稳定界限.指出了用平均变分法所得结果的精度,同时也解释了在实验中观察到的某些现象. 相似文献
12.
A. F. Kurin 《Computational Mathematics and Mathematical Physics》2011,51(8):1325-1338
Four solutions of the Cauchy problem for Mathieu’s equation away from parametric resonance domains are analytically constructed
using an asymptotic averaging method in the fourth approximation. Three solutions occur near fractional parameter values at
which slow combination phases exist. The fourth solution occurs in the absence of slow phases away from parametric resonance
domains and the fractional parameter values. 相似文献
13.
设c为任意正数,记下列级数的和为S: S=sum from n=1 to ∞((2n)/((n~2 c~2)~2 ) (1) 1890年,Mathieu猜测 S<1/c~2 (2) Berg于1952年证实了此猜测。1976年,Diananda又将此结果改进为 相似文献
14.
Hans Volkmer 《Mathematische Nachrichten》1998,192(1):239-253
It is proved that the convergence radii ρn of the eigenvalues of the Mathieu equation satisfy lim inf ρn/n2 > kk′K2 = 2.0418., where the modulus k of the complete elliptic integrals is determined by 2E = K. 相似文献
15.
Wenhua Zhao 《Journal of Pure and Applied Algebra》2010,214(7):1200-272
We first propose a generalization of the image conjecture Zhao (submitted for publication) [31] for the commuting differential operators related with classical orthogonal polynomials. We then show that the non-trivial case of this generalized image conjecture is equivalent to a variation of the Mathieu conjecture Mathieu (1997) [21] from integrals of G-finite functions over reductive Lie groups G to integrals of polynomials over open subsets of Rn with any positive measures. Via this equivalence, the generalized image conjecture can also be viewed as a natural variation of the Duistermaat and van der Kallen theorem Duistermaat and van der Kallen (1998) [14] on Laurent polynomials with no constant terms. To put all the conjectures above in a common setting, we introduce what we call the Mathieu subspaces of associative algebras. We also discuss some examples of Mathieu subspaces from other sources and derive some general results on this newly introduced notion. 相似文献
16.
Cristinel Mortici 《Mathematical and Computer Modelling》2011,53(5-6):909-914
The aim of this paper is to establish new bounds for the Mathieu series. 相似文献
17.
V. A. Bovdi 《代数通讯》2013,41(7):2670-2680
We investigate the classical Zassenhaus conjecture for the unit group of the integral group ring of Mathieu simple group M 23 using the Luthar–Passi method. This work is a continuation of the research that we carried out for Mathieu groups M 11 and M 12. As a consequence, for this group we confirm Kimmerle's conjecture on prime graphs. 相似文献
18.
Norbert Riedel 《Proceedings of the American Mathematical Society》2001,129(6):1681-1687
It is shown that the logarithmic potential associated with the integrated density of states is constant on the spectrum of the almost Mathieu operator in case the irrational frequency is sufficiently well approximable by rationals in terms of a diophantine condition.
19.
20.
V. A. Bovdi A. B. Konovalov S. Siciliano 《Rendiconti del Circolo Matematico di Palermo》1932,56(1):125-136
We consider the Zassenhaus conjecture for the normalized unit group of the integral group ring of the Mathieu sporadic groupM 12. As a consequence, we confirm for this group the Kimmerle’s conjecture on prime graphs. 相似文献