Octavian G. Mustafa
Conditions are given for a class of nonlinear ordinary differential equations x''(t)+a(t)w(x)=0, t>=1, which includes the linear equation to possess solutions x(t) with prescribed oblique asymptote that have an oscillatory pseudo-wronskian x'(t)-x(t)/t.
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The Riccati equation method is used for study the oscillatory and non oscillatory behavior of solutions of systems of two first order linear two by two dimensional matrix differential equations. An integral and an interval oscillatory criteria are obtained. Two non oscillatory criteria are obtained as well. On an example one of the obtained oscillatory criteria is compared with some well known results.
The Riccati equation method is used to establish three new oscillatory criteria for the second order linear ordinary differential equations in the marginal, sub extremal and extremal cases.We show that the first of these criteria implies the J. Deng's oscillatory criterion. An extremal oscillatory condition for the Mathieu's equation is obtained. The obtained results are compared with some known oscillatory criteria.
The Riccati equation method is used to establish an oscillatory and a non oscillatory criteria for nonhomogeneous linear systems of two first-order ordinary differential equations. It is shown that the obtained oscillatory criterion is a generalization of J. S. W. Wong's oscillatory criterion.
The Riccati equation method is used to establish Kamenev type conditions for the existence of oscillatory solutions to third order linear ordinary differential equations. Three oscillatory theorems are proved, which generalize the Lazer's oscillation criterion.
By revisiting an asymptotic integration theory of nonlinear ordinary differential equations due to J.K. Hale and N. Onuchic [Contributions Differential Equations 2 (1963), 61--75], we improve and generalize several recent results in the literature. As an application, we study the existence of bounded positive solutions to a large class of semi-linear elliptic partial differential equations via the subsolution-supersolution approach.
A system of linear differential equations with oscillatory decreasing coefficients is considered. The coefficients has the form $t^{-α}a(t)$,~$α>0$, where $a(t)$ is trigonometric polynomial with an arbitrary set of frequencies. The asymptotic behavior of the solutions of this system as $t\to\infty$ is studied. We construct an invertible (for sufficiently large $t$) change of variables that takes the original system to a system not containing oscillatory coefficients in its principal part. The study of the asymptotic behavior of the solutions of the transformed system is a simpler problem. As a…
Asymptotic regimes in oscillatory systems with damped non-resonant perturbations
本文研究平面上非线性振荡系统在满足非共振条件且随时间衰减的扰动下的长期渐近行为，结合平均化方法与Lyapunov函数构造，证明受扰系统既可保持极限系统行为，也可能产生新的渐近稳定状态。
Real and imaginary part of the limit 2N->infinity of the integral int_{x=1..2N} exp(i*pi*x)*x^(1/x) dx are evaluated to 20 digits with brute force methods after multiple partial integration, or combining a standard Simpson integration over the first halve wave with series acceleration techniques for the alternating series co-phased to each of its points. The integrand is of the logarithmic kind; its branch cut limits the performance of integration techniques that rely on smooth higher order derivatives.
We prove the stability under integration and under Fourier transform of a concrete class of functions containing all globally subanalytic functions and their complex exponentials. This paper extends the investigation started in [J.-M. Lion, J.-P. Rolin: "Volumes, feuilles de Rolle de feuilletages analytiques et théorème de Wilkie" Ann. Fac. Sci. Toulouse Math. (6) 7 (1998), no. 1, 93-112] and [R. Cluckers, D. J. Miller: "Stability under integration of sums of products of real globally subanalytic functions and their logarithms" Duke Math. J. 156 (2011), no. 2, 311-348] to an enriched framework…
Some properties of global solution of scalar Riccati equation are studied. On the basis of these properties using the Whiburn's and Leighton - Nehary's theorems some oscillatory and criteria are proved for second order linear systems of ordinary differential equations.
The Riccati equations reducible to first-order linear equations by an appropriate change the dependent variable are singled out. All these equations are integrable by quadrature. A wide class of linear ordinary differential equations reducible to algebraic equations is found. It depends on two arbitrary functions. The method for solving all these equations is given. The new class contains the constant coefficient equations and Euler's equations as particular cases.
We discuss the occurrence of oscillatory solutions which decay to 0 as $s\to+\infty$ for a class of perturbed second order ordinary differential equations. As opposed to other results in the recent literature, the perturbation is as small as desired in terms of its improper integrals and it is independent of the coefficients of the non-oscillatory unperturbed equation. This class of equations reveals thus a new pathology in the theory of perturbed oscillations.
Bistability and oscillatory behaviours of cyclic feedback loops
本文分析循环反馈环 ODE 在仿射与 Hill 等函数下的稳定性，给出双稳态和振荡行为产生的参数条件，并统一涵盖已有基因调控网络研究。
Some new decay estimates for $(2+1)$-dimensional degenerate oscillatory integral operators
针对具有三次齐次相位的(2+1)维退化振荡积分算子，该研究将L^2→L^2衰减率改进至3/8，并利用分数次积分方法建立尖锐的L^2→L^6衰减估计。
Some more general "inheritance conditions" have been found for a given set of symmetry generators $\{\mathbf{Z}_{\bar{l}}\}$ acting on some set of coupled ordinary differential equations, once the "first integration method" has been applied upon some Abelian sub-algebra of generators $\{\mathbf{Z}_{i}\}$ for the system of equations to be reduced. We have proven the following theorem: all the generators of some solvable sub-algebra can be used via the "first integration method" in order to reduce (the order) and/or (possibly) integrate the system of equations. The specific order in which the st…
In this paper we consider the linear ordinary equation of the second order $$ L x(t)\equiv \ddot{x}(t) +a(t)\dot{x}(t)+b(t)x(t)=f(t), \eqno{(1)} $$ and the corresponding homogeneous equation $$ \ddot{x}(t) +a(t)\dot{x}(t)+b(t)x(t)=0. \eqno{(2)} $$ Note that $[α,β]$ is called a nonoscillation interval if every nontrivial solution has at most one zero on this interval. Many investigations which seem to have no connection such as differential inequalities, the Polia-Mammana decomposition (i.e. representation of the operator $L$ in the form of products of the first order differential operators), u…
Symmetry Approach to Integration of Ordinary Differential Equations with Retarded Argument
本文综述了利用李群对称性研究带延迟变元常微分方程的方法，完成一阶和二阶单延迟方程的李群分类，并通过诺特型定理给出二阶延迟方程的首积分，最后提出若干开放问题。
Recently, a new fractional derivative called the conformable fractional derivative is given which is based on the basic limit definition of the derivative in [1]. Then, the fractional versions of chain rules, exponential functions, Gronwall's inequality, integration by parts, Taylor power series expansions is developed in [2]. In this paper, we give conformable fractional differential transform method and its application to conformable fractional differential equations.
Deterministic Integral and Ordinary Differential Equations over Irregular Paths
本文将关于不规则路径的确定性积分定义为标准线积分的极限，描述了Hölder指数在(0,1]范围内积分存在的路径类，并利用该积分理论研究依赖时间与路径的常微分方程解的存在性、唯一性与连续性，补充了Hölder指数小于1/3时的Young-Lyons-Gubinelli积分理论。
Recently, it was observed that solutions of a large class of highly oscillatory second order linear ordinary differential equations can be approximated using nonoscillatory phase functions. In particular, under mild assumptions on the coefficients and wavenumber $λ$ of the equation, there exists a function whose Fourier transform decays as $\exp(-μ|ξ|)$ and which represents solutions of the differential equation with accuracy on the order of $λ^{-1} \exp(-μλ)$. In this article, we establish an improved existence theorem for nonoscillatory phase functions. Among other things, we show that solut…
Integration of an arbitrary linear ODE
本文提出multex积分算子，用于显式求解任意线性常微分方程，弥补了传统方法在处理变量系数方程时的不足。
Integration of first-order ODEs by Jacobi fields
本文引入一类非Lie点对称的新向量场实现一阶常微分方程积分，基于二维Riemann流形与可积性关系提供积分程序，并揭示积分因子与Schrödinger型方程的联系。
A six-stage third order additive method for stiff ordinary differential equations
本文构造了一种用于求解加性分裂自治刚性常微分方程组的六阶段三阶加性方法，该方法对隐式部分L稳定且允许使用任意雅可比矩阵近似，并通过局部误差和稳定性控制实现自动步长选择。
In this article, we consider a class of functions on $\mathbb{R}^d$, called positive homogeneous functions, which interact well with certain continuous one-parameter groups of (generally anisotropic) dilations. Generalizing the Euclidean norm, positive homogeneous functions appear naturally in the study of convolution powers of complex-valued functions on $\mathbb{Z}^d$. As the spherical measure is a Radon measure on the unit sphere which is invariant under the symmetry group of the Euclidean norm, to each positive homogeneous function $P$, we construct a Radon measure $σ_P$ on $S=\{η\in \math…
A non-homogeneous method of third order for additive stiff systems of ordinary differential equations
针对加性分裂自治刚性常微分方程组，构造了一种三阶L-稳定方法，其第四阶段显式且允许任意雅可比矩阵近似，通过局部误差和稳定性控制实现自动步长选择，数值实验验证了算法的可靠性和效率。
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