An approximate method for integration of ordinary differential equations

An approximate analytic method of solving a Cauchy problem for normal systems of ordinary differential equations is considered. The method is based on the approximation of the solution by partial sums of shifted Chebyshev series. The coefficients of the series are determined by an iterative process using Markov quadrature formulas.     

Numerical integration of ordinary differential equations on manifolds

Summary This paper is concerned with the problem of developing numerical integration algorithms for differential equations that, when viewed as equations in some Euclidean space, naturally evolve on some embedded submanifold. It is desired to construct algorithms whose iterates also evolve on the same manifold. These algorithms can therefore be viewed as integrating ordinary differential equations on manifolds. The basic method “decouples” the computation of flows on the submanifold from the numerical integration process. It is shown that two classes of single-step and multistep algorithms can be posed and analyzed theoretically, using the concept of “freezing” the coefficients of differential operators obtained from the defining vector field. Explicit third-order algorithms are derived, with additional equations augmenting those of their classical counterparts, obtained from “obstructions” defined by nonvanishing Lie brackets.     

Liapunov functions and error bounds for approximate solutions of ordinary differential equations

Summary It is shown that Liapunov functions may be used to obtain error bounds for approximate solutions of systems of ordinary differential equations. These error bounds may reflect the behaviour of the error more accurately than other bounds.     

High order methods for the numerical integration of ordinary differential equations

Summary High order implicit integration formulae with a large region of absolute stability are developed for the approximate numerical integration of both stiff and non-stiff systems of ordinary differential equations. The algorithms derived behave essentially like one step methods and are demonstrated by direct application to certain particular examples.     

A multi-step method for the numerical integration of ordinary differential equations

Summary In this paper we develop a multi-step method of order nine for obtaining an approximate solution of the initial value problemy'=f(x,y),y((x₀)=y ₀. The present method makes use of the second derivatives, namely, at the grid points. A sufficient criterion for the convergence of the iteration procedure is established. Analysis of the discretization error is performed. Various numerical examples are presented to demonstrate the practical usefulness of our integration method.

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Zusammenfassung In dieser Arbeit entwickeln wir eine mehrschrittige Methode der neunten Ordnung, um eine angenäherte Lösung des Anfangswertproblemsy'=f(x, y), y(x ₀)=y ₀. zu erhalten. Diese Methode bedient sich der Ableitungen zweiter Ordnung an den Schnittpunkten, d.h. . Ein hinreichendes Kriterium für die Konvergenz des Iterationsprozesses wird aufgestellt. Eine Analyse des Diskretionsfehlers ist durchgeführt. Verschiedene numerische Beispiele sollen den praktischen Nutzen unserer Integrationsmethode beweisen.

     

The use of Chebyshev series for approximate analytic solution of ordinary differential equations

Application of Chebyshev series to solve ordinary differential equations is described. This approach is based on the approximation of the solution to a given Cauchy problem and its derivatives by partial sums of shifted Chebyshev series. The coefficients of the series are determined by an iterative process using Markov quadrature formulas. It is shown that the proposed approach can be applied to formulate an approximate analytical method for solving Cauchy problems. A number of examples are considered to illustrate the obtaining of approximate analytical solutions in the form of partial sums of shifted Chebyshev series.     

Hypercomplex analysis and integration of systems of ordinary differential equations

We review the theory of hypercomplex numbers and hypercomplex analysis with the ultimate goal of applying them to issues related to the integration of systems of ordinary differential equations (ODEs). We introduce the notion of hypercomplexification, which allows the lifting of some results known for scalar ODEs to systems of ODEs. In particular, we provide another approach to the construction of superposition laws for some Riccati‐type systems, we obtain invariants of Abel‐type systems, we derive integrable Ermakov systems through hypercomplexification, we address the problem of linearization by hypercomplexification, and we provide a solution to the inverse problem of the calculus of variations for some systems of ODEs. Copyright © 2016 John Wiley & Sons, Ltd.     

On the approximate solution of the singular problem of Cauchy for ordinary differential equations

In the present paper an approximate solution of the singular problem of Cauchy for the ordinary differential equation of mth order is constructed and, by the method of finite differences, sufficient conditions are found for the convergence to the exact solution when the mesh width tends to zero.     

Asymptotic expansions of periodic solutions of ordinary differential equations with large high-frequency terms

We consider the problem on the periodic solutions of a system of ordinary differential equations of arbitrary order n containing terms oscillating at a frequency ω ? 1 with coefficients of the order of ω ^n/2. For this problem, we construct the averaged (limit) problem and justify the averaging method as well as another efficient algorithm for constructing the complete asymptotics of the solution.     

Asymptotic expansions for solutions of n-th order linear ordinary differential equations with two turning points

Asymptotic expansions for solutions of n-th order linear differential equation with two turning points are constructed in Olver's form. Analytic properties of the coefficients of the series obtained are investigated. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 186, pp. 172–179, 1990.     

An algorithm for analytical integration of singularly perturbed systems of ordinary differential equations

Translated from Vychislitel'nye Kompleksy i Modelirovanie Slozhnykh Sistem, pp. 194–198, Moscow State University, 1989.