Solutions Of Autonomous Differential Equations Research Articles

Didactic demostration of the tubular flow theorem in the solution of autonomous differential equations to model environmental pollutants

El proposito del estudio fue la demostracion didactica del teorema del flujo tubular en solucion de ecuaciones diferenciales autonomas para modelar contaminantes ambientales. Se aplico metodos de investigacion con enfoque cuantitativo, teoricos como el analitico-sintetico, asi como el inductivo-deductivo. La relacion establecida fue entre el campo vectorial, teoria cualitativa de las ecuaciones diferenciales autonomas, conjugacion topologica y foliaciones. Se demostro la conjugacion topologica entre un campo vectorial diferenciable y un campo constante no nulo. El teorema del flujo tubular demostro el comportamiento de las soluciones del sistema de trayectorias que conjeturan sobre la vecindad de un punto regular, es decir, un punto que no anula al campo vectorial el cual define la ecuacion diferencial autonoma. La demostracion didactica sobre el teorema del flujo tubular proporciono el comportamiento geometrico local de las orbitas de un campo vectorial en la vecindad de un punto regular.

Asymptotic solutions of a fourth—order analogue for the Painlevé equations

Asymptotic solutions of a fourth-order analogue for the Painlevé equations that is self-similar reduction of the modified Sawada-Kotera and Kaup-Kupershmidt equation is considered. The Boutroux variables of two types have been found which allows us to find asymptotic solutions of the equation in the neighbourhood of the infinity. It was shown that asymptotic of self-similar solution for the modified Sawada-Kotera and Kaup-Kupershmidt equations can be determined as solutions of autonomous differential equations. Asymptotic solutions expressed by elementary functions have been found too. Besides asymptotic solutions expressed by logarithmic derivative of two elliptic Weierstrass functions have been found. Connection between obtained asymptotic solutions and asymptotic solutions of the Sawada-Kotera and Kaup-Kupershmidt equations has been discussed.

An error analysis for a certain class of iterative methods

We provide local convergence results in affine form for in-exact Newton-like as well as quasi-Newton iterative methods in a Banach space setting. We use hypotheses on the second or on the first andmth Frechet-derivative (m ≥ 2 an integer) of the operator involved. Our results allow a wider choice of starting points since our radius of convergence can be larger than the corresponding one given in earlier results using hypotheses on the first-Frechet-derivative only. A numerical example is provided to illustrate this fact. Our results apply when the method is, for example, a difference Newton-like or update-Newton method. Furthermore, our results have direct applications to the solution of autonomous differential equations.

Conditions for the Technical Stability of Solutions of Autonomous Differential Equations with a Discontinuous Control

Conditions for the Technical Stability of Solutions of Autonomous Differential Equations with a Discontinuous Control

Local convergence of inexact Newton-like-iterative methods and applications

We provide local convergence results in affine form for inexact Newton-like as well as quasi-Newton iterative methods in a Banach space setting. We use hypotheses on the second or on the first and second Fréchet-derivative of the operator involved. Our results allow a wider choice of starting points since our radius of convergence can be larger than the corresponding one given in earlier results using hypotheses on the first-Fréchet-derivative only. A numerical example is provided to illustrate this fact. Our results apply when the method is, for example, a difference Newton-like or update-Newton method. Furthermore, our results have direct applications to the solution of autonomous differential equations.

A fast algorithm for computing lie series solutions of autonomous differential equations

The solution of any real variable autonomous differential equation with the initial condition can be expressed as a Lie series under the assumption of convergence of the series, where and . Lie series solutions of multivariate systems of autonomous differential equations are defined analogously through partial, instead of ordinary derivatives. The main advantage of using truncated Lie series as approximate solutions of systems of autonomous differential equations is that it provides a systematic way of obtaining solutions that are in an explicit functional form. However, the actual computation of coefficients of Lie series solutions by directly using the mathematical definition of the operator is a computationally intractable process for most practically useful autonomous differential equations. In this paper we first present a fast algorithm that computes the coefficients of the first N terms of the Lie series solution of an autonomous differential equation. The algorithm is then extended, with an analysis of computational complexity and storage requirements, to Lie series solutions of multi-variable autonomous systems. An example illustrates the actual application of the algorithm to a simple yet useful type of autonomous nonlinear differential equation.

A Note on the Asymptotic Stability of Periodic Solutions of Autonomous Differential Equations

In this note we provide two examples to illustrate that the asymptotic stability of a periodic solution of an autonomous differential equation is not necessarily determined by the sign of the real part of the eigenvalues of the linear variational equation.

Periodic solutions of autonomous differential equations in higher dimensional spaces

Periodic solutions of autonomous differential equations in higher dimensional spaces