On two classes of autonomous third-order nonlinear ordinary differential equations

Two autonomous, nonlinear, third-order ordinary differential equations whose dynamics can be represented by second-order nonlinear ordinary differential equations for the first-order derivative of the solution are studied analytically and numerically. The analytical study includes both the obtention of closed-form solutions and the use of an artificial parameter method that provides approximations to both the solution and the frequency of oscillations. It is shown that both the analytical solution and the accuracy of the artificial parameter method depend greatly on the sign of the nonlinearities and the initial value of the first-order derivative.     

A Volterra integral formulation for determining the periodic solutions of some autonomous, nonlinear, third-order ordinary differential equations

A Volterra integral formulation based on the introduction of a term proportional to the velocity times the square of the (unknown) frequency of oscillation, a new independent variable equal to the original one times the (unknown) frequency of oscillation, the method of variation of parameters and series expansions of both the solution and the frequency of oscillation, is used to determine the periodic solutions to three nonlinear, autonomous, third-order, ordinary differential equations. It is shown that the first term of the series expansion of the frequency of oscillation coincides with that determined from a first-order harmonic balance procedure, whereas the two-term approximation to the frequency of oscillation is shown to be more accurate than that of a parameter perturbation procedure for the second equation considered in this paper. For the third equation, it is shown that the two-term approximation presented in this paper is more accurate than the corresponding one for one of the parameter perturbation methods, and for initial velocities less than one, for the other parameter perturbation approach.     

某些非线性三阶常微分方程的几个存在性结论

考察了某些非线性三阶常微分方程的存在性．主要结论的条件涉及非线性项在无穷远处的增长速度．改进和推广了某些现有的结论．     

First integrals and reduction of a class of nonlinear higher order ordinary differential equations

We propose a method for constructing first integrals of higher order ordinary differential equations. In particular third, fourth and fifth order equations of the form are considered. The relation of the proposed method to local and nonlocal symmetries are discussed.     

Remarks on Barbashin-Ezeilo Problem on third-order nonlinear differential equations

In this paper, we use the frequency domain criteria to initiate a general result on the Barbashin-Ezeilo Problem on third-order nonlinear differential equation. Earlier ideas of Burkin [I.M. Burkin, Orbital stability of second-kind limiting cycles for dynamical systems with cylindrical phase space, Differential Equations 29 (1993) 1262-1264], Leonov [G.A. Leonov, A frequency criterion for the existence of limit cycles of dynamical systems with cylindrical phase spaces, Differential Equations 23 (1987) 1375-1378] on the problem are being improved upon.     

On elliptic solutions of nonlinear ordinary differential equations

The problem of constructing and classifying exact elliptic solutions of autonomous nonlinear ordinary differential equations is studied. An algorithm for finding elliptic solutions in explicit form is presented.     

Comparative study on order-reduced methods for linear third-order ordinary differential equations

The linear third-order ordinary differential equation (ODE) can be transformed into a system of two second-order ODEs by introducing a variable replacement, which is different from the common order-reduced approach. We choose the functions p(x) and q(x) in the variable replacement to get different cases of the special order-reduced system for the linear third-order ODE. We analyze the numerical behavior and algebraic properties of the systems of linear equations resulting from the sinc discretizations of these special second-order ODE systems. Then the block-diagonal preconditioner is used to accelerate the convergence of the Krylov subspace iteration methods for solving the discretized system of linear equation. Numerical results show that these order-reduced methods are effective for solving the linear third-order ODEs.     

General nonlinear self-adjoint eigenvalue problem for systems of ordinary differential equations

The general nonlinear self-adjoint eigenvalue problem for systems of ordinary differential equations is considered. A method is proposed for reducing the problem to one for a Hamiltonian system. Results for Hamiltonian systems previously obtained by the authors are extended to this system.     

Some oscillation criteria for second order nonlinear functional ordinary differential equations

The main objective of this article is to study the oscillatory behavior of the solutions of the following nonlinear functional differential equations (a(t)x'(t))' δ1p(t)x'(t) δ2q(t)f(x(g(t))) = 0,for 0 ≤ t0 ≤ t, where δ1 = ±1 and δ2 = ±1. The functions p,q,g : [t0, ∞) → R, f :R → R are continuous, a(t) ＞ 0, p(t) ≥ 0,q(t) ≥ 0 for t ≥ t0, limt→∞ g(t) = ∞, and q is not identically zero on any subinterval of [t0, ∞). Moreover, the functions q(t),g(t), and a(t) are continuously differentiable.     

Nonoscillation theorems for second order nonlinear differential equations

We prove nonoscillation theorems for the second order Emden-Fowler equation (E): , , where and . It is shown that when is nondecreasing for any and is bounded above, then (E) is nonoscillatory. This improves a well-known result of Belohorec in the sublinear case, i.e. when and .

     

A new HPM for ordinary differential equations

In this paper, we introduce a new version of the homotopy perturbation method (NHPM) that efficiently solves linear and non‐linear ordinary differential equations. Several examples, including Euler‐Lagrange, Bernoulli and Ricatti differential equations, are given to demonstrate the efficiency of the new method. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

On the asymptotic behavior of nonoscillatory solutions of second order quasilinear ordinary differential equations

In this paper second order quasilinear ordinary differential equations are considered, and a necessary and sufficient condition for the existence of a slowly growing positive solution is established. Moreover, the precise asymptotic forms as t→∞ of slowly growing positive solutions and slowly decaying positive solutions are obtained.     

Twist periodic solutions of second order singular differential equations

We study the twist periodic solutions of second order singular differential equations. Such twist periodic solutions are stable in the sense of Lyapunov and present much interesting dynamical features around them. The proof is based on the third-order approximation method. The estimates of periodic solutions of Ermakov-Pinney equations and the estimates on rotation numbers of Hill equations play an important role in the analysis.     

Conditional linearizability criteria for a system of third-order ordinary differential equations

We provide linearizability criteria for a class of systems of two third-order ordinary differential equations that is cubically nonlinear in the first derivative, by differentiating a system of second-order quadratically nonlinear ordinary differential equations and using the original system to replace the second derivatives. The procedure developed splits into two cases: those for which the coefficients are constant and those for which they are variables. Both cases are discussed and examples given.     

Positive solutions of three-point boundary value problems for systems of nonlinear second order ordinary differential equations

In this paper, we study the three-point boundary value problems for systems of nonlinear second order ordinary differential equations of the form

Under some conditions, we show the existence and multiplicity of positive solutions of the above problem by applying the fixed point index theory in cones.