Laguerre Tau Methods for Solving Higher-Order Ordinary Differential Equations

In this paper, we present two numerical methods for solving higher-order differential equations using the Laguerre Tau method. These methods generate linear systems, which can be solved by Gauss elimination with maximal partial pivoting strategy. Results of some numerical experiments and theoretical analysis are presented.     

Spectral Deferred Correction Methods for Ordinary Differential Equations

We introduce a new class of methods for the Cauchy problem for ordinary differential equations (ODEs). We begin by converting the original ODE into the corresponding Picard equation and apply a deferred correction procedure in the integral formulation, driven by either the explicit or the implicit Euler marching scheme. The approach results in algorithms of essentially arbitrary order accuracy for both non-stiff and stiff problems; their performance is illustrated with several numerical examples. For non-stiff problems, the stability behavior of the obtained explicit schemes is very satisfactory and algorithms with orders between 8 and 20 should be competitive with the best existing ones. In our preliminary experiments with stiff problems, a simple adaptive implementation of the method demonstrates performance comparable to that of a state-of-the-art extrapolation code (at least, at moderate to high precision).Deferred correction methods based on the Picard equation appear to be promising candidates for further investigation.     

A Chebyshev-Gauss Spectral Collocation Method for Ordinary Differential Equations

In this paper, we introduce an efficient Chebyshev-Gauss spectral collocation method for initial value problems of ordinary differential equations. We first propose a single interval method and analyze its convergence. We then develop a multi-interval method. The suggested algorithms enjoy spectral accuracy and can be implemented in stable and efficient manners. Some numerical comparisons with some popular methods are given to demonstrate the effectiveness of this approach.     

Spectral Asymptotics of Solutions of a $$2times 2$$ System of First-Order Ordinary Differential Equations

Mathematical Notes -     

Spectral Analysis of Higher-Order Differential Operators II: Fourth-Order Equations

We investigate the location and nature of the spectrum of thefourth-order self-adjoint equation (p₀ y')'+(p₁ y')'+qy=zwy subject to certain asymptotic assumptions on the coefficients.The main tools are the theory of asymptotic integration andthe Titchmarsh–Weyl M-matrix. Asymptotic integration yieldsasymptotic formulae for the solutions of the differential equationwhich are then used to derive properties of the M-matrix. Thecharacterisation of spectral properties in terms of the boundarybehaviour of M leads to the desired results.     

Hamiltonization of Higher-Order Nonlinear Ordinary Differential Equations and the Jacobi Last Multiplier

It is known that Jacobi’s last multiplier is directly connected to the deduction of a Lagrangian via Rao’s formula (Madhava Rao in Proc. Benaras Math. Soc. (N.S.) 2:53–59, 1940). In this paper we explicitly demonstrate that it also plays an important role in Hamiltonian theory. In particular, we apply the recent results obtained by Torres del Castillo (J. Phys. A Math. Theor. 43:265202, 2009) and deduce the Hamiltonian of a second-order ODE of the Lienard type, namely, [(x)\ddot]+f(x)[(x)\dot]²+g(x)=0\ddot{x}+f(x){\dot{x}}^{2}+g(x)=0. In addition, we consider cases where the coefficient functions may also depend on the independent variable t. We illustrate our construction with various examples taken from astrophysics, cosmology and the Painlevé-Gambier class of differential equations. Finally we discuss the Hamiltonization of third-order equations using Nambu-Hamiltonian mechanics.     

On the Number of Zeros of Nonoscillatory Solutions to Higher-Order Linear Ordinary Differential Equations

 This paper is concerned with the problem of counting the number of zeros of bounded nonoscillatory solutions of higher-order linear ordinary differential equations involving a parameter. It is shown that a recent result for the second-order case is still valid for the higher-order case.     

Almost-Normed Spaces of Functions with Given Asymptotics, Lagrangian Asymptotics, and Their Application to Ordinary Differential Equations

The concept of almost-normed spaces is introduced. It is proved that the space of sufficiently smooth functions asymptotically approximating to polynomials (of degrees no higher than a given one) as their argument tends to infinity is an almost-normed space. It is demonstrated that this space is a complete metric space with respect to the metrics generated by the almost-norm introduced. The space of functions strongly asymptotically approximating to polynomials is defined, and its embedding into the space of functions asymptotically approximating to polynomials is proved. The results obtained give a new approach to studying boundary-value problems with asymptotic initial value data at singular points of ordinary differential equations.     

On the Number of Zeros of Nonoscillatory Solutions to Higher-Order Linear Ordinary Differential Equations

 This paper is concerned with the problem of counting the number of zeros of bounded nonoscillatory solutions of higher-order linear ordinary differential equations involving a parameter. It is shown that a recent result for the second-order case is still valid for the higher-order case. Received September 20, 2001; in revised form February 8, 2002 Published online July 12, 2002     

Reducible Ordinary Differential Equations

The class of reducible differential equations under consideration here includes the class of symmetric systems, and examples show that the inclusion is proper. We first discuss reducibility, as well as the stronger concept of complete reducibility, from the viewpoint of Lie algebras of vector fields and their invariants, and find Lie algebra conditions for reducibility which generalize the conditions in the symmetric case. Completely reducible equations are shown to correspond to a special class of abelian Lie algebras. Then we consider the inverse problem of determining all vector fields which are reducible by some given map. We find conditions imposed on the vector fields by the map, and present an algorithmic access for a given polynomial or local analytic map to Next, reducibility of polynomial systems is discussed, with applications to local reducibility near a stationary point. We find necessary conditions for reducibility, including restrictions for possible reduction maps to a one-dimensional equation.     

一类高阶线性变系数常微分方程的通解

利用升阶法研究了一类高阶线性变系数常微分方程,给出了齐次方程的通解公式,并讨论了非齐次方程待定的特解.     

复高阶微分方程的解

利用亚纯函数的Nevanlinna值分布理论以及最大模原理,讨论了一类复高阶微分方程的代数体解以及一类复高阶微分方程组的超越亚纯解的存在性问题,得到了两个结论.还推广了一些文献的结论,例子表明该文的结论是精确的.     

Precise Spectral Asymptotics for Logistic Equations of Population Dynamics

The semilinear elliptic eigenvalue problem with superlinearpure power nonlinearity is considered. This problem is treatedfrom the standpoint of L²-theory and the precise asymptoticformula for the eigenvalue parameter = () as is established,where is the L²-norm of the solution u associated with . 2000Mathematics Subject Classification 35P30 (primary), 35J60 (secondary).     

Regular Spectral Problems of Hyperbolic Type for a System of First-Order Ordinary Differential Equations

Mathematical Notes -     

Envelope Solutions for Implicit Ordinary Differential Equations

In recent papers the numerical solution of implicit ordinarydifferential equations of the form f(x, y(x), y'(x))=0 has beendiscussed. In this paper we address the problem of computingnumerically the so-called envelope solutions to these equations.In particular we suggest a numerical method for the solutionof this problem-one which is in spirit a predictor-correctormethod. We discuss the numerical difficulties encountered andgive some numerical examples.