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.     

Spectral Asymptotics for Higher-Order Ordinary Differential Equations

This paper gives one-term componentwise asymptotics for theM and spectral matrices of a self-adjoint realisation of aneven-order ordinary differential expression. The underlyinginterval is assumed to have at least one regular endpoint, andthe boundary conditions are supposed to be separated. Furthermore,the weight function and the reciprocal of the highest-ordercoefficient are supposed to be of regular variation at the regularendpoint, in the sense of Bingham, Goldie and Teugels. 1991Mathematics Subject Classification: 34B24, 34E05.     

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.     

On Implicit Ordinary Differential Equations

In a recent paper Fox & Mayers discuss the numerical solutionof implicit ordinary differential equations of the form f(x,y(x), y'(x)) = 0. They find that numerical methods can be veryunreliable near the point where f_y' = 0. In this paper we givea theoretical analysis of the problem which enables us to explainwhen to expect numerical difficulties. We suggest a possibleline of approach for the solution of such problems, and discusssome numerical examples. Research supported by the National Science Foundation, the Officeof Naval Research, the Army Research, and the Air Force Officeof Scientific Research. Travel funding provided by the Universityof Toronto and the British Council.     

Mobius Transforms for Linear Ordinary Differential Equations

The Mobius transforms for linear ordinary differential equations of the second order are examined. It is shown that this transform has some quasi-isospectral properties. Solutions of the Heun equation with one apparent singularity are constructed. Bibliography: 15 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 308, 2004, pp. 67–88.     

Attractors of Ordinary Differential Equations

We obtain sufficient conditions for the existence of polynomial attractors and polynomial equilibrium.     

Orthogonal Polynomials and Second-order Differential Equations

We show that a sequence of polynomials can be eigenfunctionsof a second-order differential operator only under severe restrictions.     

Numerical Solution of Arbitrary-Order Ordinary Differential and Integro-Differential Equations with Separated Boundary Conditions Using Optimal Control Technique

In this paper, a new method for solving arbitrary order ordinary differential equations and integro-differential equations of Fredholm and Volterra kind is presented. In the proposed method, these equations with separated boundary conditions are converted to a parametric optimization problem subject to algebraic constraints. Finally, control and state variables will be approximated by a Chebychev series. In this method, a new idea has been used, which offers us the ability of applying the mentioned method for almost all kinds of ordinary differential and integro-differential equations with different types of boundary conditions. The accuracy and efficiency of the proposed numerical technique have been illustrated by solving some test problems.     

Stability of Numerical Methods for Ordinary Differential Equations

Variable stepsize stability results are found for three representative multivalue methods. For the second order BDF method, a best possible result is found for a maximum stepsize ratio that will still guarantee A(0)-stability behaviour. It is found that under this same restriction, A()-stability holds for 70°. For a new two stage two value first order method, which is L-stable for constant stepsize, A(0)-stability is maintained for stepsize ratios as high as aproximately 2.94. For the third order BDF method, a best possible result of (1/2)(1+ ) is found for a ratio bound that will still guarantee zero-stability.     

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.     

Problems for Ordinary Differential Equations with Transmission Conditions

We investigate a boundary-functional problem with transmission conditions for ordinary differential-operator equation in Sobolev spaces with a weight. We prove an isomorphism, coerciveness with respect to the spectral parameter, completeness and Abel basis of a system of root functions of the problem. Obtained results in the article are new even in case of Sobolev spaces without the weight.     

A Generalized Smoother for Linear Ordinary Differential Equations

Ordinary differential equations (ODEs) are equalities involving a function and its derivatives that define the evolution of the function over a prespecified domain. The applications of ODEs range from simulation and prediction to control and diagnosis in diverse fields such as engineering, physics, medicine, and finance. Parameter estimation is often required to calibrate these theoretical models to data. While there are many methods for estimating ODE parameters from partially observed data, they are invariably subject to several problems including high computational cost, complex estimation procedures, biased estimates, and large sampling variance. We propose a method that overcomes these issues and produces estimates of the ODE parameters that have less bias, a smaller sampling variance, and a 10-fold improvement in computational efficiency. The package GenPen containing the Matlab code to perform the methods described in this article is available online.     

Stability of Adams-Type Formulae for Second-Order Ordinary Differential Equations

The stability of methods for systems of second-order equationsis discussed. Stability regions are obtained for a single equationand the existence of stable step-sizes is shown for systems.An example is used as an illustration of the effect of the usualorder selection strategies on stability and accuracy.