Oscillation Of Second-order Differential Equations Research Articles

Forced Oscillation of Second-Order Differential Equations Involving Superlinearity in Neutral Term

Forced Oscillation of Second-Order Differential Equations Involving Superlinearity in Neutral Term

On the oscillation of certain second-order linear differential equations

AbstractThis paper consists of three parts: First, letting $b_1(z)$, $b_2(z)$, $p_1(z)$ and $p_2(z)$ be nonzero polynomials such that $p_1(z)$ and $p_2(z)$ have the same degree $k\geq 1$ and distinct leading coefficients $1$ and $\alpha$, respectively, we solve entire solutions of the Tumura–Clunie type differential equation $f^{n}+P(z,\,f)=b_1(z)e^{p_1(z)}+b_2(z)e^{p_2(z)}$, where $n\geq 2$ is an integer, $P(z,\,f)$ is a differential polynomial in $f$ of degree $\leq n-1$ with coefficients having polynomial growth. Second, we study the oscillation of the second-order differential equation $f''-[b_1(z)e^{p_1(z)}+b_2(z)e^{p_2(z)}]f=0$ and prove that $\alpha =[2(m+1)-1]/[2(m+1)]$ for some integer $m\geq 0$ if this equation admits a nontrivial solution such that $\lambda (f)<\infty$. This partially answers a question of Ishizaki. Finally, letting $b_2\not =0$ and $b_3$ be constants and $l$ and $s$ be relatively prime integers such that $l> s\geq 1$, we prove that $l=2$ if the equation $f''-(e^{lz}+b_2e^{sz}+b_3)f=0$ admits two linearly independent solutions $f_1$ and $f_2$ such that $\max \{\lambda (f_1),\,\lambda (f_2)\}<\infty$. In particular, we precisely characterize all solutions such that $\lambda (f)<\infty$ when $l=2$ and $l=4$.

On nested Picard iterative integrators for highly oscillatory second-order differential equations

On nested Picard iterative integrators for highly oscillatory second-order differential equations

Some conditions for the oscillation of second-order differential equations with several mixed delays

In this work, we obtain necessary and sufficient conditions for the oscillation of the solutions to a second-order neutral differential equation with mixed delays. Two examples are provided to show effectiveness and feasibility of main results. Our main tool is the Lebesgue’s Dominated Convergence theorem.

A novel class of explicit two-step Birkhoff-Hermite integrators for highly oscillatory second-order differential equations

This paper focuses on the study of explicit two-step Birkhoff-Hermite (TSBH) integrators for solving nonlinear highly oscillatory second-order ordinary differential equations (ODEs). These new TSBH integrators, involving rth-order derivatives, are of order 2r + 2. Nonlinear stability and convergence of the integrators are analyzed. Two practical TSBH integrators of order four and order six are derived. Numerical results demonstrate the advantage and efficiency of the proposed integrators in comparison with the existing numerical methods reported in the literature.

Oscillation of Second-Order Differential Equations with Multiple and Mixed Delays under a Canonical Operator

In this work, we obtained new sufficient and necessary conditions for the oscillation of second-order differential equations with mixed and multiple delays under a canonical operator. Our methods could be applicable to find the sufficient and necessary conditions for any neutral differential equations. Furthermore, we proved the validity of the obtained results via particular examples. At the end of the paper, we provide the future scope of this study.

New Conditions for the Oscillation of Second-Order Differential Equations with Sublinear Neutral Terms

In continuous applications in electrodynamics, neural networks, quantum mechanics, electromagnetism, and the field of time symmetric, fluid dynamics, neutral differential equations appear when modeling many problems and phenomena. Therefore, it is interesting to study the qualitative behavior of solutions of such equations. In this study, we obtained some new sufficient conditions for oscillations to the solutions of a second-order delay differential equations with sub-linear neutral terms. The results obtained improve and complement the relevant results in the literature. Finally, we show an example to validate the main results, and an open problem is included.

Explicit criteria for the oscillation of second-order differential equations with several sub-linear neutral coefficients

In this work, we present sufficient conditions for oscillation of all solutions of a second-order functional differential equation. We consider two special cases when gamma >beta and gamma <beta . This new theorem complements and improves a number of results reported in the literature. Finally, we provide examples illustrating our results and state an open problem.

More Effective Criteria for Oscillation of Second-Order Differential Equations with Neutral Arguments

The motivation for this paper is to create new criteria for oscillation of solutions of second-order nonlinear neutral differential equations. In more than one respect, our results improve several related ones in the literature. As proof of the effectiveness of the new criteria, we offer more than one practical example.

On oscillatory second-order nonlinear delay differential equations of neutral type

On oscillatory second-order nonlinear delay differential equations of neutral type

Functionally Fitted Block Method for Solving the General Oscillatory Second-Order Initial Value Problems and Hyperbolic Partial Differential Equations

We present a block hybrid functionally fitted Runge–Kutta–Nyström method (BHFNM) which is dependent on the stepsize and a fixed frequency. Since the method is implemented in a block-by-block fashion, the method does not require starting values and predictors inherent to other predictor-corrector methods. Upon deriving our method, stability is illustrated, and it is used to numerically solve the general second-order initial value problems as well as hyperbolic partial differential equations. In doing so, we demonstrate the method’s relative accuracy and efficiency.

Asymptotic-numerical solvers for highly oscillatory second-order differential equations

In this paper, we propose an approach of combination of asymptotic and numerical techniques to solve highly oscillatory second-order initial value problems. An asymptotic expansion of the solution is derived in inverse of powers of the oscillatory parameter, which develops on two time scales, a slow time t and a fast time τ=ωt. The truncation with the first few terms of the expansion results in a very effective method of discretizing the highly oscillatory differential equation and becomes more accurate when the oscillatory parameter increases. Numerical examples show that our proposed asymptotic-numerical solver is efficient and accurate for highly oscillatory problems.

Asymptotic-numerical approximations for highly oscillatory second-order differential equations by the phase function method

Asymptotic approximations of “phase functions” for linear second-order differential equations, whose solutions are highly oscillatory, can be obtained using Borůvka's theory of linear differential transformations coupled to Liouville–Green (WKB) asymptotics. A numerical method, very effective in case of asymptotically polynomial coefficients, is extended to other cases of rapidly growing coefficients. Zeros of solutions can be computed without prior evaluation of the solutions themselves, but the method can also be applied to Initial- and Boundary-Value problems, as well as to the case of forced oscillations. Numerical examples are given to illustrate the performance of the algorithm. In all cases, the error turns out to be of the order of that made approximating the phase functions.

Adiabatic Filon-type methods for highly oscillatory second-order ordinary differential equations

In this paper, we study efficient numerical integrators for linear and nonlinear systems of highly oscillatory second-order ordinary differential equations. The systems are reformulated as a first-order system, which is then transformed to adiabatic variables. The solution of the transformed system is a smoother function which is more accessible to numerical approximation than the original system. We develop Filon-type methods for linear systems by approximating the integral as a linear combination of function values and derivatives. We then present a special combination of Filon-type methods and waveform relaxation methods for nonlinear systems. Both types of methods can be used with far larger step sizes than those required by traditional schemes and their performance drastically improves as frequency grows, as are illustrated by numerical experiments.

Trigonometric collocation methods based on Lagrange basis polynomials for multi-frequency oscillatory second-order differential equations

In the present work, a kind of trigonometric collocation methods based on Lagrange basis polynomials is developed for effectively solving multi-frequency oscillatory second-order differential equations q″(t)+Mq(t)=f(q(t)). The properties of the obtained methods are investigated. It is shown that the convergent condition of these methods is independent of ‖M‖, which is very crucial for solving oscillatory systems. A fourth-order scheme of the methods is presented. Numerical experiments are implemented to show the remarkable efficiency of the methods proposed in this paper.

Arbitrary-Order Trigonometric Fourier Collocation Methods for Multi-Frequency Oscillatory Systems

We rigorously study a novel type of trigonometric Fourier collocation methods for solving multi-frequency oscillatory second-order ordinary differential equations (ODEs) $$q^{\prime \prime }(t)+Mq(t)=f(q(t))$$q?(t)+Mq(t)=f(q(t)) with a principal frequency matrix $$M\in \mathbb {R}^{d\times d}$$M?Rd×d. If $$M$$M is symmetric and positive semi-definite and $$f(q) = -\nabla U(q)$$f(q)=-?U(q) for a smooth function $$U(q)$$U(q), then this is a multi-frequency oscillatory Hamiltonian system with the Hamiltonian $$H(q,p)=p^{T}p/2+q^{T}Mq/2+U(q),$$H(q,p)=pTp/2+qTMq/2+U(q), where $$p = q'$$p=q?. The solution of this system is a nonlinear multi-frequency oscillator. The new trigonometric Fourier collocation method takes advantage of the special structure brought by the linear term $$Mq$$Mq, and its construction incorporates the idea of collocation methods, the variation-of-constants formula and the local Fourier expansion of the system. The properties of the new methods are analysed. The analysis in the paper demonstrates an important feature, namely that the trigonometric Fourier collocation methods can be of an arbitrary order and when $$M\rightarrow 0$$M?0, each trigonometric Fourier collocation method creates a particular Runge---Kutta---Nystrom-type Fourier collocation method, which is symplectic under some conditions. This allows us to obtain arbitrary high-order symplectic methods to deal with a special and important class of systems of second-order ODEs in an efficient way. The results of numerical experiments are quite promising and show that the trigonometric Fourier collocation methods are significantly more efficient in comparison with alternative approaches that have previously appeared in the literature.

Forced oscillation of second-order differential equations with mixed nonlinearities

Abstract We study oscillatory behavior of a class of second-order forced differential equations with mixed nonlinearities. Some new oscillation theorems are presented that improve and complement those related results given in the literature. An example is provided to illustrate the main results. MSC:34K11.

Improved Filon-type asymptotic methods for highly oscillatory differential equations with multiple time scales

In this paper we consider multi-frequency highly oscillatory second-order differential equations x″(t)+Mx(t)=f(t,x(t),x′(t)) where high-frequency oscillations are generated by the linear part Mx(t), and M is positive semi-definite (not necessarily nonsingular). It is known that Filon-type methods are effective approach to numerically solving highly oscillatory problems. Unfortunately, however, existing Filon-type asymptotic methods fail to apply to the highly oscillatory second-order differential equations when M is singular. We study and propose an efficient improvement on the existing Filon-type asymptotic methods, so that the improved Filon-type asymptotic methods can be able to numerically solving this class of multi-frequency highly oscillatory systems with a singular matrix M. The improved Filon-type asymptotic methods are designed by combining Filon-type methods with the asymptotic methods based on the variation-of-constants formula. We also present one efficient and practical improved Filon-type asymptotic method which can be performed at lower cost. Accompanying numerical results show the remarkable efficiency.

Explicit multi-frequency symmetric extended RKN integrators for solving multi-frequency and multidimensional oscillatory reversible systems

This paper studies explicit multi-frequency symmetric extended Runge---Kutta---Nystrom (ERKN) integrators tailored to numerically computing the multi-frequency and multidimensional oscillatory reversible second-order differential equations $$q''(t)+Mq(t)=f\big (q(t)\big )$$q??(t)+Mq(t)=f(q(t)). We establish the symmetry conditions in a simplified way for multi-frequency ERKN integrators. Five explicit multi-frequency symmetric ERKN integrators are derived based on the simplified symmetry conditions. The arbitrary high-order explicit multi-frequency symmetric ERKN integrators can be achieved by the application of the symmetric composition. The stability and phase properties of the new integrators are discussed. Five numerical experiments are carried out and the numerical results demonstrate the remarkable numerical behavior of the new explicit multi-frequency symmetric integrators when applied to the multi-frequency and multidimensional oscillatory reversible second-order differential equations.

Oscillation of second-order differential equations with a sublinear neutral term

This paper is concerned with oscillation of a certain class of second-order differential equations with a sublinear neutral term. Two oscillation criteria and two illustrative examples are included. In particular, the results obtained improve those reported in the literature.