Search results for "ordinary differential equation"
showing 10 items of 98 documents
Figures of equilibrium in close binary systems
1992
The equilibrium configurations of close binary systems are analyzed. The autogravitational, centrifugal and tidal potentials are expanded in Clairaut's coordinates. From the set of the total potential angular terms an integral equations system is derived. The reduction of them to ordinary differential equations and the determination of the boundary conditions allow a formulation of the problem in terms of a single variable.
Minimally implicit Runge-Kutta methods for Resistive Relativistic MHD
2016
The Relativistic Resistive Magnetohydrodynamic (RRMHD) equations are a hyperbolic system of partial differential equations used to describe the dynamics of relativistic magnetized fluids with a finite conductivity. Close to the ideal magnetohydrodynamic regime, the source term proportional to the conductivity becomes potentially stiff and cannot be handled with standard explicit time integration methods. We propose a new class of methods to deal with the stiffness fo the system, which we name Minimally Implicit Runge-Kutta methods. These methods avoid the development of numerical instabilities without increasing the computational costs in comparison with explicit methods, need no iterative …
Integration of a Dirac comb and the Bernoulli polynomials
2016
Abstract For any positive integer n , we consider the ordinary differential equations of the form y ( n ) = 1 − Ш + F where Ш denotes the Dirac comb distribution and F is a piecewise- C ∞ periodic function with null average integral. We prove the existence and uniqueness of periodic solutions of maximal regularity. Above all, these solutions are given by means of finite explicit formulae involving a minimal number of Bernoulli polynomials. We generalize this approach to a larger class of differential equations for which the computation of periodic solutions is also sharp, finite and effective.
Oscillatory Behavior of Second-Order Nonlinear Neutral Differential Equations
2014
Published version of an article in the journal: Abstract and Applied Analysis. Also available from the publisher at: http://dx.doi.org/10.1155/2014/143614 Open Access We study oscillatory behavior of solutions to a class of second-order nonlinear neutral differential equations under the assumptions that allow applications to differential equations with delayed and advanced arguments. New theorems do not need several restrictive assumptions required in related results reported in the literature. Several examples are provided to show that the results obtained are sharp even for second-order ordinary differential equations and improve related contributions to the subject.
Dynamical Models of Interrelation in a Class of Artificial Networks
2020
The system of ordinary differential equations that models a type of artificial networks is considered. The system consists of a sigmoidal function that depends on linear combinations of the arguments minus the linear part. The linear combinations of the arguments are described by the regulatory matrix W. For the three-dimensional cases, several types of matrices W are considered and the behavior of solutions of the system is analyzed. The attractive sets are constructed for most cases. The illustrative examples are provided. The list of references consists of 12 items.
Splineapproximationen von beliebigem Defekt zur numerischen L�sung gew�hnlicher Differentialgleichungen. Teil III
1980
In the first part [5] a general procedure is presented to obtain polynomial spline approximations of arbitrary defect for the solution of the initial value problem of ordinary differential equations. The essential result is a divergence theorem in dependence of the polynomial degree and the defect of the spline functions. In this second part the convergent procedures are investigated and two convergence theorems are proved. Furthermore the question is treated, whether the convergent procedures are appropriate for the numerical solution of stiff equations. The paper is finished by a convergence theorem for a procedure producing spline approximations in a natural way by the discrete approxima…
Maximale Konvergenzordnung bei der numerischen Lösung von Anfangswertproblemen mit Splines
1982
In [10] a general procedureV is presented to obtain spline approximations by collocation for the solutions of initial value problems for first order ordinary differential equations. In this paper the attainable order of convergence with respect to the maximum norm is characterized in dependence of the parameters involved inV; in particular the appropriate choice of the collocation points is considered.
On the stability of spline-collocation methods of multivalue type
1987
In this paper the general classV of spline-collocation methods for first order systems of ordinary differential equations is investigated. The methods can in part be regarded as so-called multivalue methods. This type contains the generalized singly-implicit methods treated by Butcher.
Constrained control of a nonlinear two point boundary value problem, I
1994
In this paper we consider an optimal control problem for a nonlinear second order ordinary differential equation with integral constraints. A necessary optimality condition in form of the Pontryagin minimum principle is derived. The proof is based on McShane-variations of the optimal control, a thorough study of their behaviour in dependence of some denning parameters, a generalized Green formula for second order ordinary differential equations with measurable coefficients and certain tools of convex analysis.
A third integrating factor for indefinite integrals of special functions
2020
An integrating factor f ~ x is presented involving the terms in y ′ ′ x and q x y x of the general homogenous second-order linear ordinary differential equation. The new integrating factors obey se...